\(\int \frac {(c+d x)^{3/2} \sqrt {a-b x^2}}{x^5} \, dx\) [1450]

Optimal result

Integrand size = 25, antiderivative size = 601 \[ \int \frac {(c+d x)^{3/2} \sqrt {a-b x^2}}{x^5} \, dx=-\frac {d \sqrt {c+d x} \sqrt {a-b x^2}}{8 x^3}+\frac {\left (\frac {4 b c}{a}-\frac {d^2}{c}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{32 x^2}+\frac {d \left (28 b+\frac {3 a d^2}{c^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{64 a x}-\frac {(c+d x)^{3/2} \sqrt {a-b x^2}}{4 x^4}-\frac {\sqrt {b} d \left (28 b c^2+3 a d^2\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{64 \sqrt {a} c^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {\sqrt {b} d \left (36 b c^2+a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{64 \sqrt {a} c \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {\left (\frac {16 b^2 c^2}{a}+24 b d^2-\frac {3 a d^4}{c^2}\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{64 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

-1/8*d*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/x^3+1/32*(4*b*c/a-d^2/c)*(d*x+c)^(1/ 
2)*(-b*x^2+a)^(1/2)/x^2+1/64*d*(28*b+3*a*d^2/c^2)*(d*x+c)^(1/2)*(-b*x^2+a) 
^(1/2)/a/x-1/4*(d*x+c)^(3/2)*(-b*x^2+a)^(1/2)/x^4-1/64*b^(1/2)*d*(3*a*d^2+ 
28*b*c^2)*(d*x+c)^(1/2)*(1-b*x^2/a)^(1/2)*EllipticE(1/2*(1-b^(1/2)*x/a^(1/ 
2))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)*d/(b^(1/2)*c+a^(1/2)*d))^(1/2))/a^(1/2) 
/c^2/(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d))^(1/2)/(-b*x^2+a)^(1/2)+1/64*b 
^(1/2)*d*(a*d^2+36*b*c^2)*(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d))^(1/2)*(1 
-b*x^2/a)^(1/2)*EllipticF(1/2*(1-b^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2^(1/2)* 
(a^(1/2)*d/(b^(1/2)*c+a^(1/2)*d))^(1/2))/a^(1/2)/c/(d*x+c)^(1/2)/(-b*x^2+a 
)^(1/2)+1/64*(16*b^2*c^2/a+24*b*d^2-3*a*d^4/c^2)*(b^(1/2)*(d*x+c)/(b^(1/2) 
*c+a^(1/2)*d))^(1/2)*((-b*x^2+a)/a)^(1/2)*EllipticPi(1/2*(1-b^(1/2)*x/a^(1 
/2))^(1/2)*2^(1/2),2,2^(1/2)*(a^(1/2)*d/(b^(1/2)*c+a^(1/2)*d))^(1/2))/(d*x 
+c)^(1/2)/(-b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 24.88 (sec) , antiderivative size = 1256, normalized size of antiderivative = 2.09 \[ \int \frac {(c+d x)^{3/2} \sqrt {a-b x^2}}{x^5} \, dx =\text {Too large to display} \] Input:

Integrate[((c + d*x)^(3/2)*Sqrt[a - b*x^2])/x^5,x]
 

Output:

(Sqrt[a - b*x^2]*(((c + d*x)*(4*b*c^2*x^2*(2*c + 7*d*x) - a*(16*c^3 + 24*c 
^2*d*x + 2*c*d^2*x^2 - 3*d^3*x^3)))/(a*c^2*x^4) + (-28*b^2*c^5*Sqrt[-c + ( 
Sqrt[a]*d)/Sqrt[b]] + 25*a*b*c^3*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] + 3*a^ 
2*c*d^4*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] + 56*b^2*c^4*Sqrt[-c + (Sqrt[a]*d)/ 
Sqrt[b]]*(c + d*x) + 6*a*b*c^2*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x 
) - 28*b^2*c^3*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x)^2 - 3*a*b*c*d^2*Sq 
rt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x)^2 + I*Sqrt[b]*c*(28*b^(3/2)*c^3 - 2 
8*Sqrt[a]*b*c^2*d + 3*a*Sqrt[b]*c*d^2 - 3*a^(3/2)*d^3)*Sqrt[(d*(Sqrt[a]/Sq 
rt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + 
 d*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d* 
x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)] - I*(8*b^2*c^4 - 28* 
Sqrt[a]*b^(3/2)*c^3*d + 26*a*b*c^2*d^2 - 3*a^(3/2)*Sqrt[b]*c*d^3 - 3*a^2*d 
^4)*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[b] 
- d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-c + (Sqrt[a]* 
d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d 
)] + (16*I)*b^2*c^4*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqr 
t[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticPi[(Sqrt[b]*c)/ 
(Sqrt[b]*c - Sqrt[a]*d), I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + 
 d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)] + (24*I)*a*b*c^2* 
d^2*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[...
 

Rubi [A] (verified)

Time = 2.44 (sec) , antiderivative size = 665, normalized size of antiderivative = 1.11, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {628, 2352, 27, 2352, 25, 2352, 25, 2351, 600, 509, 508, 327, 512, 511, 321, 633, 632, 186, 413, 412}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\sqrt {a-b x^2} (c+d x)^{3/2}}{x^5} \, dx\)

\(\Big \downarrow \) 628

\(\displaystyle \frac {1}{8} \int \frac {-8 b d^2 x^3-11 b c d x^2-2 \left (b c^2-4 a d^2\right ) x+9 a c d}{x^4 \sqrt {c+d x} \sqrt {a-b x^2}}dx-\frac {c \sqrt {a-b x^2} \sqrt {c+d x}}{4 x^4}\)

\(\Big \downarrow \) 2352

\(\displaystyle \frac {1}{8} \left (-\frac {\int \frac {3 \left (10 a b d x c^2+7 a b d^2 x^2 c+a \left (4 b c^2-a d^2\right ) c\right )}{x^3 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{6 a c}-\frac {3 d \sqrt {a-b x^2} \sqrt {c+d x}}{x^3}\right )-\frac {c \sqrt {a-b x^2} \sqrt {c+d x}}{4 x^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{8} \left (-\frac {\int \frac {10 a b d x c^2+7 a b d^2 x^2 c+a \left (4 b c^2-a d^2\right ) c}{x^3 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{2 a c}-\frac {3 d \sqrt {a-b x^2} \sqrt {c+d x}}{x^3}\right )-\frac {c \sqrt {a-b x^2} \sqrt {c+d x}}{4 x^4}\)

\(\Big \downarrow \) 2352

\(\displaystyle \frac {1}{8} \left (-\frac {-\frac {\int -\frac {c d \left (28 b c^2+3 a d^2\right ) a^2+b c d \left (4 b c^2-a d^2\right ) x^2 a+2 b c^2 \left (4 b c^2+13 a d^2\right ) x a}{x^2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{4 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} \left (4 b c^2-a d^2\right )}{2 x^2}}{2 a c}-\frac {3 d \sqrt {a-b x^2} \sqrt {c+d x}}{x^3}\right )-\frac {c \sqrt {a-b x^2} \sqrt {c+d x}}{4 x^4}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{8} \left (-\frac {\frac {\int \frac {c d \left (28 b c^2+3 a d^2\right ) a^2+b c d \left (4 b c^2-a d^2\right ) x^2 a+2 b c^2 \left (4 b c^2+13 a d^2\right ) x a}{x^2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{4 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} \left (4 b c^2-a d^2\right )}{2 x^2}}{2 a c}-\frac {3 d \sqrt {a-b x^2} \sqrt {c+d x}}{x^3}\right )-\frac {c \sqrt {a-b x^2} \sqrt {c+d x}}{4 x^4}\)

\(\Big \downarrow \) 2352

\(\displaystyle \frac {1}{8} \left (-\frac {\frac {-\frac {\int -\frac {-b c d^2 \left (28 b c^2+3 a d^2\right ) x^2 a^2+c \left (16 b^2 c^4+24 a b d^2 c^2-3 a^2 d^4\right ) a^2+2 b c^2 d \left (4 b c^2-a d^2\right ) x a^2}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx}{2 a c}-\frac {a d \sqrt {a-b x^2} \sqrt {c+d x} \left (3 a d^2+28 b c^2\right )}{x}}{4 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} \left (4 b c^2-a d^2\right )}{2 x^2}}{2 a c}-\frac {3 d \sqrt {a-b x^2} \sqrt {c+d x}}{x^3}\right )-\frac {c \sqrt {a-b x^2} \sqrt {c+d x}}{4 x^4}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{8} \left (-\frac {\frac {\frac {\int \frac {-b c d^2 \left (28 b c^2+3 a d^2\right ) x^2 a^2+c \left (16 b^2 c^4+24 a b d^2 c^2-3 a^2 d^4\right ) a^2+2 b c^2 d \left (4 b c^2-a d^2\right ) x a^2}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx}{2 a c}-\frac {a d \sqrt {a-b x^2} \sqrt {c+d x} \left (3 a d^2+28 b c^2\right )}{x}}{4 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} \left (4 b c^2-a d^2\right )}{2 x^2}}{2 a c}-\frac {3 d \sqrt {a-b x^2} \sqrt {c+d x}}{x^3}\right )-\frac {c \sqrt {a-b x^2} \sqrt {c+d x}}{4 x^4}\)

\(\Big \downarrow \) 2351

\(\displaystyle \frac {1}{8} \left (-\frac {\frac {\frac {a^2 c \left (-3 a^2 d^4+24 a b c^2 d^2+16 b^2 c^4\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx+\int \frac {8 a^2 b^2 d c^4-2 a^3 b d^3 c^2-a^2 b d^2 \left (28 b c^2+3 a d^2\right ) x c}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{2 a c}-\frac {a d \sqrt {a-b x^2} \sqrt {c+d x} \left (3 a d^2+28 b c^2\right )}{x}}{4 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} \left (4 b c^2-a d^2\right )}{2 x^2}}{2 a c}-\frac {3 d \sqrt {a-b x^2} \sqrt {c+d x}}{x^3}\right )-\frac {c \sqrt {a-b x^2} \sqrt {c+d x}}{4 x^4}\)

\(\Big \downarrow \) 600

\(\displaystyle \frac {1}{8} \left (-\frac {\frac {\frac {a^2 c \left (-3 a^2 d^4+24 a b c^2 d^2+16 b^2 c^4\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx+a^2 b c^2 d \left (a d^2+36 b c^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx-a^2 b c d \left (3 a d^2+28 b c^2\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{2 a c}-\frac {a d \sqrt {a-b x^2} \sqrt {c+d x} \left (3 a d^2+28 b c^2\right )}{x}}{4 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} \left (4 b c^2-a d^2\right )}{2 x^2}}{2 a c}-\frac {3 d \sqrt {a-b x^2} \sqrt {c+d x}}{x^3}\right )-\frac {c \sqrt {a-b x^2} \sqrt {c+d x}}{4 x^4}\)

\(\Big \downarrow \) 509

\(\displaystyle \frac {1}{8} \left (-\frac {\frac {\frac {a^2 c \left (-3 a^2 d^4+24 a b c^2 d^2+16 b^2 c^4\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx+a^2 b c^2 d \left (a d^2+36 b c^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx-\frac {a^2 b c d \sqrt {1-\frac {b x^2}{a}} \left (3 a d^2+28 b c^2\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{\sqrt {a-b x^2}}}{2 a c}-\frac {a d \sqrt {a-b x^2} \sqrt {c+d x} \left (3 a d^2+28 b c^2\right )}{x}}{4 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} \left (4 b c^2-a d^2\right )}{2 x^2}}{2 a c}-\frac {3 d \sqrt {a-b x^2} \sqrt {c+d x}}{x^3}\right )-\frac {c \sqrt {a-b x^2} \sqrt {c+d x}}{4 x^4}\)

\(\Big \downarrow \) 508

\(\displaystyle \frac {1}{8} \left (-\frac {\frac {\frac {\frac {2 a^{5/2} \sqrt {b} c d \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a d^2+28 b c^2\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}+a^2 c \left (-3 a^2 d^4+24 a b c^2 d^2+16 b^2 c^4\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx+a^2 b c^2 d \left (a d^2+36 b c^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{2 a c}-\frac {a d \sqrt {a-b x^2} \sqrt {c+d x} \left (3 a d^2+28 b c^2\right )}{x}}{4 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} \left (4 b c^2-a d^2\right )}{2 x^2}}{2 a c}-\frac {3 d \sqrt {a-b x^2} \sqrt {c+d x}}{x^3}\right )-\frac {c \sqrt {a-b x^2} \sqrt {c+d x}}{4 x^4}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {1}{8} \left (-\frac {\frac {\frac {a^2 c \left (-3 a^2 d^4+24 a b c^2 d^2+16 b^2 c^4\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx+a^2 b c^2 d \left (a d^2+36 b c^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx+\frac {2 a^{5/2} \sqrt {b} c d \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a d^2+28 b c^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 a c}-\frac {a d \sqrt {a-b x^2} \sqrt {c+d x} \left (3 a d^2+28 b c^2\right )}{x}}{4 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} \left (4 b c^2-a d^2\right )}{2 x^2}}{2 a c}-\frac {3 d \sqrt {a-b x^2} \sqrt {c+d x}}{x^3}\right )-\frac {c \sqrt {a-b x^2} \sqrt {c+d x}}{4 x^4}\)

\(\Big \downarrow \) 512

\(\displaystyle \frac {1}{8} \left (-\frac {\frac {\frac {a^2 c \left (-3 a^2 d^4+24 a b c^2 d^2+16 b^2 c^4\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx+\frac {a^2 b c^2 d \sqrt {1-\frac {b x^2}{a}} \left (a d^2+36 b c^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{\sqrt {a-b x^2}}+\frac {2 a^{5/2} \sqrt {b} c d \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a d^2+28 b c^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 a c}-\frac {a d \sqrt {a-b x^2} \sqrt {c+d x} \left (3 a d^2+28 b c^2\right )}{x}}{4 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} \left (4 b c^2-a d^2\right )}{2 x^2}}{2 a c}-\frac {3 d \sqrt {a-b x^2} \sqrt {c+d x}}{x^3}\right )-\frac {c \sqrt {a-b x^2} \sqrt {c+d x}}{4 x^4}\)

\(\Big \downarrow \) 511

\(\displaystyle \frac {1}{8} \left (-\frac {\frac {\frac {-\frac {2 a^{5/2} \sqrt {b} c^2 d \sqrt {1-\frac {b x^2}{a}} \left (a d^2+36 b c^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {a-b x^2} \sqrt {c+d x}}+a^2 c \left (-3 a^2 d^4+24 a b c^2 d^2+16 b^2 c^4\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx+\frac {2 a^{5/2} \sqrt {b} c d \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a d^2+28 b c^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 a c}-\frac {a d \sqrt {a-b x^2} \sqrt {c+d x} \left (3 a d^2+28 b c^2\right )}{x}}{4 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} \left (4 b c^2-a d^2\right )}{2 x^2}}{2 a c}-\frac {3 d \sqrt {a-b x^2} \sqrt {c+d x}}{x^3}\right )-\frac {c \sqrt {a-b x^2} \sqrt {c+d x}}{4 x^4}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {1}{8} \left (-\frac {\frac {\frac {a^2 c \left (-3 a^2 d^4+24 a b c^2 d^2+16 b^2 c^4\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx-\frac {2 a^{5/2} \sqrt {b} c^2 d \sqrt {1-\frac {b x^2}{a}} \left (a d^2+36 b c^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {c+d x}}+\frac {2 a^{5/2} \sqrt {b} c d \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a d^2+28 b c^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 a c}-\frac {a d \sqrt {a-b x^2} \sqrt {c+d x} \left (3 a d^2+28 b c^2\right )}{x}}{4 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} \left (4 b c^2-a d^2\right )}{2 x^2}}{2 a c}-\frac {3 d \sqrt {a-b x^2} \sqrt {c+d x}}{x^3}\right )-\frac {c \sqrt {a-b x^2} \sqrt {c+d x}}{4 x^4}\)

\(\Big \downarrow \) 633

\(\displaystyle \frac {1}{8} \left (-\frac {\frac {\frac {\frac {a^2 c \sqrt {1-\frac {b x^2}{a}} \left (-3 a^2 d^4+24 a b c^2 d^2+16 b^2 c^4\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{\sqrt {a-b x^2}}-\frac {2 a^{5/2} \sqrt {b} c^2 d \sqrt {1-\frac {b x^2}{a}} \left (a d^2+36 b c^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {c+d x}}+\frac {2 a^{5/2} \sqrt {b} c d \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a d^2+28 b c^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 a c}-\frac {a d \sqrt {a-b x^2} \sqrt {c+d x} \left (3 a d^2+28 b c^2\right )}{x}}{4 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} \left (4 b c^2-a d^2\right )}{2 x^2}}{2 a c}-\frac {3 d \sqrt {a-b x^2} \sqrt {c+d x}}{x^3}\right )-\frac {c \sqrt {a-b x^2} \sqrt {c+d x}}{4 x^4}\)

\(\Big \downarrow \) 632

\(\displaystyle \frac {1}{8} \left (-\frac {\frac {\frac {\frac {a^2 c \sqrt {1-\frac {b x^2}{a}} \left (-3 a^2 d^4+24 a b c^2 d^2+16 b^2 c^4\right ) \int \frac {1}{x \sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}} \sqrt {\frac {\sqrt {b} x}{\sqrt {a}}+1} \sqrt {c+d x}}dx}{\sqrt {a-b x^2}}-\frac {2 a^{5/2} \sqrt {b} c^2 d \sqrt {1-\frac {b x^2}{a}} \left (a d^2+36 b c^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {c+d x}}+\frac {2 a^{5/2} \sqrt {b} c d \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a d^2+28 b c^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 a c}-\frac {a d \sqrt {a-b x^2} \sqrt {c+d x} \left (3 a d^2+28 b c^2\right )}{x}}{4 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} \left (4 b c^2-a d^2\right )}{2 x^2}}{2 a c}-\frac {3 d \sqrt {a-b x^2} \sqrt {c+d x}}{x^3}\right )-\frac {c \sqrt {a-b x^2} \sqrt {c+d x}}{4 x^4}\)

\(\Big \downarrow \) 186

\(\displaystyle \frac {1}{8} \left (-\frac {\frac {\frac {-\frac {2 a^2 c \sqrt {1-\frac {b x^2}{a}} \left (-3 a^2 d^4+24 a b c^2 d^2+16 b^2 c^4\right ) \int \frac {\sqrt {a}}{\sqrt {b} x \sqrt {\frac {\sqrt {b} x}{\sqrt {a}}+1} \sqrt {c+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}}}d\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {a-b x^2}}-\frac {2 a^{5/2} \sqrt {b} c^2 d \sqrt {1-\frac {b x^2}{a}} \left (a d^2+36 b c^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {c+d x}}+\frac {2 a^{5/2} \sqrt {b} c d \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a d^2+28 b c^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 a c}-\frac {a d \sqrt {a-b x^2} \sqrt {c+d x} \left (3 a d^2+28 b c^2\right )}{x}}{4 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} \left (4 b c^2-a d^2\right )}{2 x^2}}{2 a c}-\frac {3 d \sqrt {a-b x^2} \sqrt {c+d x}}{x^3}\right )-\frac {c \sqrt {a-b x^2} \sqrt {c+d x}}{4 x^4}\)

\(\Big \downarrow \) 413

\(\displaystyle \frac {1}{8} \left (-\frac {\frac {\frac {-\frac {2 a^2 c \sqrt {1-\frac {b x^2}{a}} \left (-3 a^2 d^4+24 a b c^2 d^2+16 b^2 c^4\right ) \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \int \frac {\sqrt {a}}{\sqrt {b} x \sqrt {\frac {\sqrt {b} x}{\sqrt {a}}+1} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} c+\sqrt {a} d}}}d\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}-\frac {2 a^{5/2} \sqrt {b} c^2 d \sqrt {1-\frac {b x^2}{a}} \left (a d^2+36 b c^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {c+d x}}+\frac {2 a^{5/2} \sqrt {b} c d \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a d^2+28 b c^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 a c}-\frac {a d \sqrt {a-b x^2} \sqrt {c+d x} \left (3 a d^2+28 b c^2\right )}{x}}{4 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} \left (4 b c^2-a d^2\right )}{2 x^2}}{2 a c}-\frac {3 d \sqrt {a-b x^2} \sqrt {c+d x}}{x^3}\right )-\frac {c \sqrt {a-b x^2} \sqrt {c+d x}}{4 x^4}\)

\(\Big \downarrow \) 412

\(\displaystyle \frac {1}{8} \left (-\frac {\frac {\frac {-\frac {2 a^{5/2} \sqrt {b} c^2 d \sqrt {1-\frac {b x^2}{a}} \left (a d^2+36 b c^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {c+d x}}+\frac {2 a^{5/2} \sqrt {b} c d \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a d^2+28 b c^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {2 a^2 c \sqrt {1-\frac {b x^2}{a}} \left (-3 a^2 d^4+24 a b c^2 d^2+16 b^2 c^4\right ) \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}}{2 a c}-\frac {a d \sqrt {a-b x^2} \sqrt {c+d x} \left (3 a d^2+28 b c^2\right )}{x}}{4 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} \left (4 b c^2-a d^2\right )}{2 x^2}}{2 a c}-\frac {3 d \sqrt {a-b x^2} \sqrt {c+d x}}{x^3}\right )-\frac {c \sqrt {a-b x^2} \sqrt {c+d x}}{4 x^4}\)

Input:

Int[((c + d*x)^(3/2)*Sqrt[a - b*x^2])/x^5,x]
 

Output:

-1/4*(c*Sqrt[c + d*x]*Sqrt[a - b*x^2])/x^4 + ((-3*d*Sqrt[c + d*x]*Sqrt[a - 
 b*x^2])/x^3 - (-1/2*((4*b*c^2 - a*d^2)*Sqrt[c + d*x]*Sqrt[a - b*x^2])/x^2 
 + (-((a*d*(28*b*c^2 + 3*a*d^2)*Sqrt[c + d*x]*Sqrt[a - b*x^2])/x) + ((2*a^ 
(5/2)*Sqrt[b]*c*d*(28*b*c^2 + 3*a*d^2)*Sqrt[c + d*x]*Sqrt[1 - (b*x^2)/a]*E 
llipticE[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c) 
/Sqrt[a] + d)])/(Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[a 
- b*x^2]) - (2*a^(5/2)*Sqrt[b]*c^2*d*(36*b*c^2 + a*d^2)*Sqrt[(Sqrt[b]*(c + 
 d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[1 - (b*x^2)/a]*EllipticF[ArcSin[Sqrt[ 
1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(Sqrt 
[c + d*x]*Sqrt[a - b*x^2]) - (2*a^2*c*(16*b^2*c^4 + 24*a*b*c^2*d^2 - 3*a^2 
*d^4)*Sqrt[1 - (b*x^2)/a]*Sqrt[1 - (Sqrt[a]*d*(1 - (Sqrt[b]*x)/Sqrt[a]))/( 
Sqrt[b]*c + Sqrt[a]*d)]*EllipticPi[2, ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]] 
/Sqrt[2]], (2*Sqrt[a]*d)/(Sqrt[b]*c + Sqrt[a]*d)])/(Sqrt[a - b*x^2]*Sqrt[c 
 + (Sqrt[a]*d)/Sqrt[b] - (Sqrt[a]*d*(1 - (Sqrt[b]*x)/Sqrt[a]))/Sqrt[b]]))/ 
(2*a*c))/(4*a*c))/(2*a*c))/8
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 186

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_ 
)]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[-2   Subst[Int[1/(Simp[b*c - a*d 
- b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + f*(x^2/d), x]]*Sqrt[Simp[(d*g - c*h)/ 
d + h*(x^2/d), x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, 
g, h}, x] && GtQ[(d*e - c*f)/d, 0]
 

rule 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 412

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x 
_)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* 
(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, 
 f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && S 
implerSqrtQ[-f/e, -d/c])
 

rule 413

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x 
_)^2]), x_Symbol] :> Simp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2]   Int[1/((a + 
 b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, 
e, f}, x] &&  !GtQ[c, 0]
 

rule 508

Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> With[{q 
 = Rt[-b/a, 2]}, Simp[-2*(Sqrt[c + d*x]/(Sqrt[a]*q*Sqrt[q*((c + d*x)/(d + c 
*q))]))   Subst[Int[Sqrt[1 - 2*d*(x^2/(d + c*q))]/Sqrt[1 - x^2], x], x, Sqr 
t[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[a, 0]
 

rule 509

Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[Sq 
rt[1 + b*(x^2/a)]/Sqrt[a + b*x^2]   Int[Sqrt[c + d*x]/Sqrt[1 + b*(x^2/a)], 
x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] &&  !GtQ[a, 0]
 

rule 511

Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Wit 
h[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[q*((c + d*x)/(d + c*q))]/(Sqrt[a]*q*Sqrt 
[c + d*x]))   Subst[Int[1/(Sqrt[1 - 2*d*(x^2/(d + c*q))]*Sqrt[1 - x^2]), x] 
, x, Sqrt[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ 
a, 0]
 

rule 512

Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim 
p[Sqrt[1 + b*(x^2/a)]/Sqrt[a + b*x^2]   Int[1/(Sqrt[c + d*x]*Sqrt[1 + b*(x^ 
2/a)]), x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] &&  !GtQ[a, 0]
 

rule 600

Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] 
), x_Symbol] :> Simp[B/d   Int[Sqrt[c + d*x]/Sqrt[a + b*x^2], x], x] - Simp 
[(B*c - A*d)/d   Int[1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] /; FreeQ[{a, 
 b, c, d, A, B}, x] && NegQ[b/a]
 

rule 628

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*Sqrt[(a_) + (b_.)*(x_)^2], x 
_Symbol] :> Simp[c^(n - 1/2)*(e*x)^(m + 1)*Sqrt[c + d*x]*(Sqrt[a + b*x^2]/( 
e*(m + 1))), x] - Simp[1/(2*e*(m + 1))   Int[((e*x)^(m + 1)/(Sqrt[c + d*x]* 
Sqrt[a + b*x^2]))*ExpandToSum[(2*a*c^(n + 1/2)*(m + 1) + a*c^(n - 1/2)*d*(2 
*m + 3)*x + 2*b*c^(n + 1/2)*(m + 2)*x^2 + b*c^(n - 1/2)*d*(2*m + 5)*x^3 - 2 
*a*(m + 1)*(c + d*x)^(n + 1/2) - 2*b*(m + 1)*x^2*(c + d*x)^(n + 1/2))/x, x] 
, x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[n + 3/2, 0] && LtQ[m, -1] && 
IntegerQ[2*m]
 

rule 632

Int[1/((x_)*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] : 
> With[{q = Rt[-b/a, 2]}, Simp[1/Sqrt[a]   Int[1/(x*Sqrt[c + d*x]*Sqrt[1 - 
q*x]*Sqrt[1 + q*x]), x], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ 
a, 0]
 

rule 633

Int[1/((x_)*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] : 
> Simp[Sqrt[1 + b*(x^2/a)]/Sqrt[a + b*x^2]   Int[1/(x*Sqrt[c + d*x]*Sqrt[1 
+ b*(x^2/a)]), x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] &&  !GtQ[a, 0]
 

rule 2351

Int[((Px_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_.))/(x_), x_S 
ymbol] :> Int[PolynomialQuotient[Px, x, x]*(c + d*x)^n*(a + b*x^2)^p, x] + 
Simp[PolynomialRemainder[Px, x, x]   Int[(c + d*x)^n*((a + b*x^2)^p/x), x], 
 x] /; FreeQ[{a, b, c, d, n, p}, x] && PolynomialQ[Px, x]
 

rule 2352

Int[((Px_)*((e_.)*(x_))^(m_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x 
_)^2]), x_Symbol] :> With[{Px0 = Coefficient[Px, x, 0]}, Simp[Px0*(e*x)^(m 
+ 1)*Sqrt[c + d*x]*(Sqrt[a + b*x^2]/(a*c*e*(m + 1))), x] + Simp[1/(2*a*c*e* 
(m + 1))   Int[((e*x)^(m + 1)/(Sqrt[c + d*x]*Sqrt[a + b*x^2]))*ExpandToSum[ 
2*a*c*(m + 1)*((Px - Px0)/x) - Px0*(a*d*(2*m + 3) + 2*b*c*(m + 2)*x + b*d*( 
2*m + 5)*x^2), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolynomialQ[Px, 
x] && LtQ[m, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(994\) vs. \(2(494)=988\).

Time = 3.99 (sec) , antiderivative size = 995, normalized size of antiderivative = 1.66

method	result	size
elliptic	\(\frac {\sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}\, \left (-\frac {c \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{4 x^{4}}-\frac {3 d \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{8 x^{3}}-\frac {\left (a \,d^{2}-4 b \,c^{2}\right ) \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{32 a c \,x^{2}}+\frac {d \left (3 a \,d^{2}+28 b \,c^{2}\right ) \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{64 a \,c^{2} x}+\frac {d b \left (a \,d^{2}-4 b \,c^{2}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{32 c a \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {b \,d^{2} \left (3 a \,d^{2}+28 b \,c^{2}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \left (\left (-\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{b}\right )}{64 a \,c^{2} \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}-\frac {\left (3 a^{2} d^{4}-24 b \,c^{2} d^{2} a -16 b^{2} c^{4}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, d \operatorname {EllipticPi}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, -\frac {\left (-\frac {c}{d}+\frac {\sqrt {a b}}{b}\right ) d}{c}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{64 c^{3} a \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )}{\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\)	\(995\)
risch	\(\text {Expression too large to display}\)	\(1129\)
default	\(\text {Expression too large to display}\)	\(2389\)

Input:

int((d*x+c)^(3/2)*(-b*x^2+a)^(1/2)/x^5,x,method=_RETURNVERBOSE)
 

Output:

((d*x+c)*(-b*x^2+a))^(1/2)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)*(-1/4*c/x^4*(-b* 
d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)-3/8*d/x^3*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2 
)-1/32*(a*d^2-4*b*c^2)/a/c*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)/x^2+1/64/a*d 
*(3*a*d^2+28*b*c^2)/c^2*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)/x+1/32*d*b/c*(a 
*d^2-4*b*c^2)/a*(c/d-1/b*(a*b)^(1/2))*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2 
)*((x-1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2)*((x+1/b*(a*b)^(1/2))/ 
(-c/d+1/b*(a*b)^(1/2)))^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)*EllipticF 
(((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2),((-c/d+1/b*(a*b)^(1/2))/(-c/d-1/b*( 
a*b)^(1/2)))^(1/2))+1/64*b*d^2*(3*a*d^2+28*b*c^2)/a/c^2*(c/d-1/b*(a*b)^(1/ 
2))*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*((x-1/b*(a*b)^(1/2))/(-c/d-1/b*( 
a*b)^(1/2)))^(1/2)*((x+1/b*(a*b)^(1/2))/(-c/d+1/b*(a*b)^(1/2)))^(1/2)/(-b* 
d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)*((-c/d-1/b*(a*b)^(1/2))*EllipticE(((x+c/d)/ 
(c/d-1/b*(a*b)^(1/2)))^(1/2),((-c/d+1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2) 
))^(1/2))+1/b*(a*b)^(1/2)*EllipticF(((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2), 
((-c/d+1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2)))-1/64/c^3*(3*a^2*d^ 
4-24*a*b*c^2*d^2-16*b^2*c^4)/a*(c/d-1/b*(a*b)^(1/2))*((x+c/d)/(c/d-1/b*(a* 
b)^(1/2)))^(1/2)*((x-1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2)*((x+1/ 
b*(a*b)^(1/2))/(-c/d+1/b*(a*b)^(1/2)))^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^ 
(1/2)*d*EllipticPi(((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2),-(-c/d+1/b*(a*b)^ 
(1/2))/c*d,((-c/d+1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2)))
 

Fricas [F(-1)]

Timed out. \[ \int \frac {(c+d x)^{3/2} \sqrt {a-b x^2}}{x^5} \, dx=\text {Timed out} \] Input:

integrate((d*x+c)^(3/2)*(-b*x^2+a)^(1/2)/x^5,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {(c+d x)^{3/2} \sqrt {a-b x^2}}{x^5} \, dx=\int \frac {\sqrt {a - b x^{2}} \left (c + d x\right )^{\frac {3}{2}}}{x^{5}}\, dx \] Input:

integrate((d*x+c)**(3/2)*(-b*x**2+a)**(1/2)/x**5,x)
 

Output:

Integral(sqrt(a - b*x**2)*(c + d*x)**(3/2)/x**5, x)
 

Maxima [F]

\[ \int \frac {(c+d x)^{3/2} \sqrt {a-b x^2}}{x^5} \, dx=\int { \frac {\sqrt {-b x^{2} + a} {\left (d x + c\right )}^{\frac {3}{2}}}{x^{5}} \,d x } \] Input:

integrate((d*x+c)^(3/2)*(-b*x^2+a)^(1/2)/x^5,x, algorithm="maxima")
 

Output:

integrate(sqrt(-b*x^2 + a)*(d*x + c)^(3/2)/x^5, x)
 

Giac [F]

\[ \int \frac {(c+d x)^{3/2} \sqrt {a-b x^2}}{x^5} \, dx=\int { \frac {\sqrt {-b x^{2} + a} {\left (d x + c\right )}^{\frac {3}{2}}}{x^{5}} \,d x } \] Input:

integrate((d*x+c)^(3/2)*(-b*x^2+a)^(1/2)/x^5,x, algorithm="giac")
 

Output:

integrate(sqrt(-b*x^2 + a)*(d*x + c)^(3/2)/x^5, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^{3/2} \sqrt {a-b x^2}}{x^5} \, dx=\int \frac {\sqrt {a-b\,x^2}\,{\left (c+d\,x\right )}^{3/2}}{x^5} \,d x \] Input:

int(((a - b*x^2)^(1/2)*(c + d*x)^(3/2))/x^5,x)
 

Output:

int(((a - b*x^2)^(1/2)*(c + d*x)^(3/2))/x^5, x)
 

Reduce [F]

\[ \int \frac {(c+d x)^{3/2} \sqrt {a-b x^2}}{x^5} \, dx=\int \frac {\left (d x +c \right )^{\frac {3}{2}} \sqrt {-b \,x^{2}+a}}{x^{5}}d x \] Input:

int((d*x+c)^(3/2)*(-b*x^2+a)^(1/2)/x^5,x)
 

Output:

int((d*x+c)^(3/2)*(-b*x^2+a)^(1/2)/x^5,x)