\(\int \frac {A+B x+C x^2}{x^3 \sqrt {c+d x} \sqrt {a-b x^2}} \, dx\) [191]

Optimal result

Integrand size = 35, antiderivative size = 515 \[ \int \frac {A+B x+C x^2}{x^3 \sqrt {c+d x} \sqrt {a-b x^2}} \, dx=-\frac {A \sqrt {c+d x} \sqrt {a-b x^2}}{2 a c x^2}-\frac {(4 B c-3 A d) \sqrt {c+d x} \sqrt {a-b x^2}}{4 a c^2 x}+\frac {\sqrt {b} (4 B c-3 A d) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{4 \sqrt {a} c^2 \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}-\frac {\sqrt {b} (4 B c-A d) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{4 \sqrt {a} c \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\left (4 A b c^2+8 a c^2 C-4 a B c d+3 a A d^2\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{4 a c^2 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

-1/2*A*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/a/c/x^2-1/4*(-3*A*d+4*B*c)*(d*x+c)^( 
1/2)*(-b*x^2+a)^(1/2)/a/c^2/x+1/4*b^(1/2)*(-3*A*d+4*B*c)*(d*x+c)^(1/2)*((- 
b*x^2+a)/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/((d*x+c)/(c+a^(1/2) 
*d/b^(1/2)))^(1/2)/(-b*x^2+a)^(1/2)-1/4*b^(1/2)*(-A*d+4*B*c)*((d*x+c)/(c+a 
^(1/2)*d/b^(1/2)))^(1/2)*((-b*x^2+a)/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)-1/4*(3*A*a*d^2+4*A*b*c^2-4*B*a*c*d+8 
*C*a*c^2)*((d*x+c)/(c+a^(1/2)*d/b^(1/2)))^(1/2)*((-b*x^2+a)/a)^(1/2)*Ellip 
ticPi(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))/a/c^2/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 27.69 (sec) , antiderivative size = 1392, normalized size of antiderivative = 2.70 \[ \int \frac {A+B x+C x^2}{x^3 \sqrt {c+d x} \sqrt {a-b x^2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(Sqrt[a - b*x^2]*(-(((c + d*x)*(2*A*c + 4*B*c*x - 3*A*d*x))/(a*c^2*x^2)) + 
 (4*b*B*c^4*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] - 3*A*b*c^3*d*Sqrt[-c + (Sqrt[a 
]*d)/Sqrt[b]] - 4*a*B*c^2*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] + 3*a*A*c*d^3 
*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] - 8*b*B*c^3*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] 
*(c + d*x) + 6*A*b*c^2*d*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x) + 4*b*B* 
c^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x)^2 - 3*A*b*c*d*Sqrt[-c + (Sqrt 
[a]*d)/Sqrt[b]]*(c + d*x)^2 - I*Sqrt[b]*c*(Sqrt[b]*c - Sqrt[a]*d)*(4*B*c - 
 3*A*d)*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)*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*d*(A*(2*b*c^2 + 3*Sqrt[a]*Sqrt[b]*c*d + 3*a*d^2) - 4*(Sqrt[a]*S 
qrt[b]*B*c^2 + a*c*(-2*c*C + B*d)))*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)*Ellipti 
cF[I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + S 
qrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)] - (4*I)*A*b*c^2*d*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)*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)] - (8*I)*a*c^2*C*d*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)*Ellipt...
 

Rubi [A] (verified)

Time = 2.83 (sec) , antiderivative size = 570, normalized size of antiderivative = 1.11, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.486, Rules used = {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.

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

\(\Big \downarrow \) 2352

\(\displaystyle -\frac {\int -\frac {A b d x^2+2 c (A b+2 a C) x+a (4 B c-3 A d)}{x^2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{4 a c}-\frac {A \sqrt {a-b x^2} \sqrt {c+d x}}{2 a c x^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {A b d x^2+2 c (A b+2 a C) x+a (4 B c-3 A d)}{x^2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{4 a c}-\frac {A \sqrt {a-b x^2} \sqrt {c+d x}}{2 a c x^2}\)

\(\Big \downarrow \) 2352

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2351

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

\(\Big \downarrow \) 600

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

\(\Big \downarrow \) 509

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

\(\Big \downarrow \) 508

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

\(\Big \downarrow \) 327

\(\displaystyle \frac {\frac {a \left (3 a A d^2-4 a B c d+8 a c^2 C+4 A b c^2\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx+a b c (4 B c-A d) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx+\frac {2 a^{3/2} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} (4 B c-3 A d) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} (4 B c-3 A d)}{c x}}{4 a c}-\frac {A \sqrt {a-b x^2} \sqrt {c+d x}}{2 a c x^2}\)

\(\Big \downarrow \) 512

\(\displaystyle \frac {\frac {a \left (3 a A d^2-4 a B c d+8 a c^2 C+4 A b c^2\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx+\frac {a b c \sqrt {1-\frac {b x^2}{a}} (4 B c-A d) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{\sqrt {a-b x^2}}+\frac {2 a^{3/2} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} (4 B c-3 A d) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} (4 B c-3 A d)}{c x}}{4 a c}-\frac {A \sqrt {a-b x^2} \sqrt {c+d x}}{2 a c x^2}\)

\(\Big \downarrow \) 511

\(\displaystyle \frac {\frac {-\frac {2 a^{3/2} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} (4 B c-A d) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {a-b x^2} \sqrt {c+d x}}+a \left (3 a A d^2-4 a B c d+8 a c^2 C+4 A b c^2\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx+\frac {2 a^{3/2} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} (4 B c-3 A d) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} (4 B c-3 A d)}{c x}}{4 a c}-\frac {A \sqrt {a-b x^2} \sqrt {c+d x}}{2 a c x^2}\)

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 633

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

\(\Big \downarrow \) 632

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

\(\Big \downarrow \) 186

\(\displaystyle \frac {\frac {-\frac {2 a \sqrt {1-\frac {b x^2}{a}} \left (3 a A d^2-4 a B c d+8 a c^2 C+4 A b c^2\right ) \int \frac {\sqrt {a}}{\sqrt {b} x \sqrt {\frac {\sqrt {b} x}{\sqrt {a}}+1} \sqrt {c+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}}}d\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {a-b x^2}}-\frac {2 a^{3/2} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} (4 B c-A d) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {c+d x}}+\frac {2 a^{3/2} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} (4 B c-3 A d) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} (4 B c-3 A d)}{c x}}{4 a c}-\frac {A \sqrt {a-b x^2} \sqrt {c+d x}}{2 a c x^2}\)

\(\Big \downarrow \) 413

\(\displaystyle \frac {\frac {-\frac {2 a \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \left (3 a A d^2-4 a B c d+8 a c^2 C+4 A b c^2\right ) \int \frac {\sqrt {a}}{\sqrt {b} x \sqrt {\frac {\sqrt {b} x}{\sqrt {a}}+1} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} c+\sqrt {a} d}}}d\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}-\frac {2 a^{3/2} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} (4 B c-A d) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {c+d x}}+\frac {2 a^{3/2} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} (4 B c-3 A d) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} (4 B c-3 A d)}{c x}}{4 a c}-\frac {A \sqrt {a-b x^2} \sqrt {c+d x}}{2 a c x^2}\)

\(\Big \downarrow \) 412

\(\displaystyle \frac {\frac {-\frac {2 a^{3/2} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} (4 B c-A d) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {c+d x}}+\frac {2 a^{3/2} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} (4 B c-3 A d) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {2 a \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \left (3 a A d^2-4 a B c d+8 a c^2 C+4 A b c^2\right ) \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}}{2 a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x} (4 B c-3 A d)}{c x}}{4 a c}-\frac {A \sqrt {a-b x^2} \sqrt {c+d x}}{2 a c x^2}\)

Input:

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

Output:

-1/2*(A*Sqrt[c + d*x]*Sqrt[a - b*x^2])/(a*c*x^2) + (-(((4*B*c - 3*A*d)*Sqr 
t[c + d*x]*Sqrt[a - b*x^2])/(c*x)) + ((2*a^(3/2)*Sqrt[b]*(4*B*c - 3*A*d)*S 
qrt[c + d*x]*Sqrt[1 - (b*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqr 
t[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^(3/2)*Sqrt[b]*c*(4*B*c 
- A*d)*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*(4*A*b*c^2 + 8* 
a*c^2*C - 4*a*B*c*d + 3*a*A*d^2)*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[Sq 
rt[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)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 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. \(902\) vs. \(2(426)=852\).

Time = 5.22 (sec) , antiderivative size = 903, normalized size of antiderivative = 1.75

method	result	size
elliptic	\(\frac {\sqrt {\left (-b \,x^{2}+a \right ) \left (d x +c \right )}\, \left (-\frac {A \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{2 c a \,x^{2}}+\frac {\left (3 A d -4 B c \right ) \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{4 a \,c^{2} x}+\frac {b d A \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 )}{2 a c \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {b d \left (3 A d -4 B c \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 )}{4 a \,c^{2} \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}-\frac {\left (3 A a \,d^{2}+4 b A \,c^{2}-4 B a c d +8 C a \,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}}}\, 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 )}{4 c^{3} a \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )}{\sqrt {-b \,x^{2}+a}\, \sqrt {d x +c}}\)	\(903\)
risch	\(-\frac {\sqrt {-b \,x^{2}+a}\, \sqrt {d x +c}\, \left (-3 A d x +4 B c x +2 A c \right )}{4 a \,c^{2} x^{2}}+\frac {\left (-\frac {\left (3 A a \,d^{2}+4 b A \,c^{2}-4 B a c d +8 C a \,c^{2}\right ) \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \operatorname {EllipticPi}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, 2, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {3 A \,d^{2} \sqrt {a b}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \left (\left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )-\frac {c \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )}{d}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {2 A c d \sqrt {a b}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}-\frac {4 B c d \sqrt {a b}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \left (\left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )-\frac {c \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )}{d}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right ) \sqrt {\left (-b \,x^{2}+a \right ) \left (d x +c \right )}}{8 c^{2} a \sqrt {-b \,x^{2}+a}\, \sqrt {d x +c}}\)	\(921\)
default	\(\text {Expression too large to display}\)	\(2823\)

Input:

int((C*x^2+B*x+A)/x^3/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2),x,method=_RETURNVERBO 
SE)
 

Output:

((-b*x^2+a)*(d*x+c))^(1/2)/(-b*x^2+a)^(1/2)/(d*x+c)^(1/2)*(-1/2*A/c/a/x^2* 
(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)+1/4*(3*A*d-4*B*c)/a/c^2*(-b*d*x^3-b*c*x 
^2+a*d*x+a*c)^(1/2)/x+1/2*b*d/a/c*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))+1/4*b*d*(3*A*d-4*B*c)/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))*Ellipti 
cE(((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)))-1/4/c^ 
3/a*(3*A*a*d^2+4*A*b*c^2-4*B*a*c*d+8*C*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)*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]

\[ \int \frac {A+B x+C x^2}{x^3 \sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\int { \frac {C x^{2} + B x + A}{\sqrt {-b x^{2} + a} \sqrt {d x + c} x^{3}} \,d x } \] Input:

integrate((C*x^2+B*x+A)/x^3/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2),x, algorithm="f 
ricas")
 

Output:

integral(-(C*x^2 + B*x + A)*sqrt(-b*x^2 + a)*sqrt(d*x + c)/(b*d*x^6 + b*c* 
x^5 - a*d*x^4 - a*c*x^3), x)
 

Sympy [F]

\[ \int \frac {A+B x+C x^2}{x^3 \sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\int \frac {A + B x + C x^{2}}{x^{3} \sqrt {a - b x^{2}} \sqrt {c + d x}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {A+B x+C x^2}{x^3 \sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\int { \frac {C x^{2} + B x + A}{\sqrt {-b x^{2} + a} \sqrt {d x + c} x^{3}} \,d x } \] Input:

integrate((C*x^2+B*x+A)/x^3/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2),x, algorithm="m 
axima")
 

Output:

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

Giac [F]

\[ \int \frac {A+B x+C x^2}{x^3 \sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\int { \frac {C x^{2} + B x + A}{\sqrt {-b x^{2} + a} \sqrt {d x + c} x^{3}} \,d x } \] Input:

integrate((C*x^2+B*x+A)/x^3/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2),x, algorithm="g 
iac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2}{x^3 \sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\int \frac {C\,x^2+B\,x+A}{x^3\,\sqrt {a-b\,x^2}\,\sqrt {c+d\,x}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {A+B x+C x^2}{x^3 \sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\int \frac {C \,x^{2}+B x +A}{x^{3} \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}d x \] Input:

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

Output:

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