\(\int \frac {x^3}{(c+d x)^{5/2} \sqrt {a-b x^2}} \, dx\) [1521]

Optimal result

Integrand size = 25, antiderivative size = 447 \[ \int \frac {x^3}{(c+d x)^{5/2} \sqrt {a-b x^2}} \, dx=-\frac {2 c^3 \sqrt {a-b x^2}}{3 d^2 \left (b c^2-a d^2\right ) (c+d x)^{3/2}}+\frac {2 c^2 \left (5 b c^2-9 a d^2\right ) \sqrt {a-b x^2}}{3 d^2 \left (b c^2-a d^2\right )^2 \sqrt {c+d x}}-\frac {2 \sqrt {a} \left (8 b^2 c^4-15 a b c^2 d^2+3 a^2 d^4\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 \sqrt {b} d^3 \left (b c^2-a d^2\right )^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {2 \sqrt {a} c \left (8 b c^2-9 a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 \sqrt {b} d^3 \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

-2/3*c^3*(-b*x^2+a)^(1/2)/d^2/(-a*d^2+b*c^2)/(d*x+c)^(3/2)+2/3*c^2*(-9*a*d 
^2+5*b*c^2)*(-b*x^2+a)^(1/2)/d^2/(-a*d^2+b*c^2)^2/(d*x+c)^(1/2)-2/3*a^(1/2 
)*(3*a^2*d^4-15*a*b*c^2*d^2+8*b^2*c^4)*(d*x+c)^(1/2)*(1-b*x^2/a)^(1/2)*Ell 
ipticE(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))/b^(1/2)/d^3/(-a*d^2+b*c^2)^2/(b^(1/2)*(d*x+c)/(b^(1/ 
2)*c+a^(1/2)*d))^(1/2)/(-b*x^2+a)^(1/2)+2/3*a^(1/2)*c*(-9*a*d^2+8*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))/b^(1/2)/d^3/(-a*d^2+b*c^2)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 24.43 (sec) , antiderivative size = 531, normalized size of antiderivative = 1.19 \[ \int \frac {x^3}{(c+d x)^{5/2} \sqrt {a-b x^2}} \, dx=\frac {2 \sqrt {a-b x^2} \left (-3 b c^4+6 a c^2 d^2-\frac {3 a^2 d^4}{b}+\frac {-b c^5+a c^3 d^2}{c+d x}+\frac {i \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (8 b^2 c^4-15 a b c^2 d^2+3 a^2 d^4\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{d^2 \left (a-b x^2\right )}+\frac {i \sqrt {a} \left (8 b^2 c^4-2 \sqrt {a} b^{3/2} c^3 d-15 a b c^2 d^2+6 a^{3/2} \sqrt {b} c d^3+3 a^2 d^4\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{\sqrt {b} d \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-a+b x^2\right )}\right )}{3 \left (b c^2 d-a d^3\right )^2 \sqrt {c+d x}} \] Input:

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

Output:

(2*Sqrt[a - b*x^2]*(-3*b*c^4 + 6*a*c^2*d^2 - (3*a^2*d^4)/b + (-(b*c^5) + a 
*c^3*d^2)/(c + d*x) + (I*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(8*b^2*c^4 - 15*a* 
b*c^2*d^2 + 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)*EllipticE[I*ArcSinh[ 
Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sq 
rt[b]*c - Sqrt[a]*d)])/(d^2*(a - b*x^2)) + (I*Sqrt[a]*(8*b^2*c^4 - 2*Sqrt[ 
a]*b^(3/2)*c^3*d - 15*a*b*c^2*d^2 + 6*a^(3/2)*Sqrt[b]*c*d^3 + 3*a^2*d^4)*S 
qrt[(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)/Sq 
rt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)])/( 
Sqrt[b]*d*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(-a + b*x^2))))/(3*(b*c^2*d - a*d 
^3)^2*Sqrt[c + d*x])
 

Rubi [A] (verified)

Time = 0.86 (sec) , antiderivative size = 446, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {603, 27, 2182, 27, 600, 509, 508, 327, 512, 511, 321}

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.

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

\(\Big \downarrow \) 603

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2182

\(\displaystyle -\frac {\frac {2 \int -\frac {2 a c \left (\frac {b c^2}{d}-3 a d\right )-\left (-\frac {8 b^2 c^4}{d^2}+15 a b c^2-3 a^2 d^2\right ) x}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{b c^2-a d^2}+\frac {2 c^2 \sqrt {a-b x^2} \left (9 a-\frac {5 b c^2}{d^2}\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}-\frac {2 c^3 \sqrt {a-b x^2}}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {2 c^2 \sqrt {a-b x^2} \left (9 a-\frac {5 b c^2}{d^2}\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}-\frac {\int \frac {2 a c \left (\frac {b c^2}{d}-3 a d\right )-\left (-\frac {8 b^2 c^4}{d^2}+15 a b c^2-3 a^2 d^2\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{b c^2-a d^2}}{3 \left (b c^2-a d^2\right )}-\frac {2 c^3 \sqrt {a-b x^2}}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 600

\(\displaystyle -\frac {\frac {2 c^2 \sqrt {a-b x^2} \left (9 a-\frac {5 b c^2}{d^2}\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}-\frac {\frac {\left (3 a^2 d^4-15 a b c^2 d^2+8 b^2 c^4\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d^3}-\frac {c \left (8 b c^2-9 a d^2\right ) \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d^3}}{b c^2-a d^2}}{3 \left (b c^2-a d^2\right )}-\frac {2 c^3 \sqrt {a-b x^2}}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 509

\(\displaystyle -\frac {\frac {2 c^2 \sqrt {a-b x^2} \left (9 a-\frac {5 b c^2}{d^2}\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}-\frac {\frac {\sqrt {1-\frac {b x^2}{a}} \left (3 a^2 d^4-15 a b c^2 d^2+8 b^2 c^4\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d^3 \sqrt {a-b x^2}}-\frac {c \left (8 b c^2-9 a d^2\right ) \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d^3}}{b c^2-a d^2}}{3 \left (b c^2-a d^2\right )}-\frac {2 c^3 \sqrt {a-b x^2}}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 508

\(\displaystyle -\frac {\frac {2 c^2 \sqrt {a-b x^2} \left (9 a-\frac {5 b c^2}{d^2}\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}-\frac {-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a^2 d^4-15 a b c^2 d^2+8 b^2 c^4\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d^3 \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {c \left (8 b c^2-9 a d^2\right ) \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d^3}}{b c^2-a d^2}}{3 \left (b c^2-a d^2\right )}-\frac {2 c^3 \sqrt {a-b x^2}}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 327

\(\displaystyle -\frac {\frac {2 c^2 \sqrt {a-b x^2} \left (9 a-\frac {5 b c^2}{d^2}\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}-\frac {-\frac {c \left (8 b c^2-9 a d^2\right ) \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d^3}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a^2 d^4-15 a b c^2 d^2+8 b^2 c^4\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d^3 \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{b c^2-a d^2}}{3 \left (b c^2-a d^2\right )}-\frac {2 c^3 \sqrt {a-b x^2}}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 512

\(\displaystyle -\frac {\frac {2 c^2 \sqrt {a-b x^2} \left (9 a-\frac {5 b c^2}{d^2}\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}-\frac {-\frac {c \sqrt {1-\frac {b x^2}{a}} \left (8 b c^2-9 a d^2\right ) \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d^3 \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a^2 d^4-15 a b c^2 d^2+8 b^2 c^4\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d^3 \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{b c^2-a d^2}}{3 \left (b c^2-a d^2\right )}-\frac {2 c^3 \sqrt {a-b x^2}}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 511

\(\displaystyle -\frac {\frac {2 c^2 \sqrt {a-b x^2} \left (9 a-\frac {5 b c^2}{d^2}\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}-\frac {\frac {2 \sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \left (8 b c^2-9 a d^2\right ) \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d^3 \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a^2 d^4-15 a b c^2 d^2+8 b^2 c^4\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d^3 \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{b c^2-a d^2}}{3 \left (b c^2-a d^2\right )}-\frac {2 c^3 \sqrt {a-b x^2}}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 321

\(\displaystyle -\frac {\frac {2 c^2 \sqrt {a-b x^2} \left (9 a-\frac {5 b c^2}{d^2}\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}-\frac {\frac {2 \sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \left (8 b c^2-9 a d^2\right ) \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d^3 \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a^2 d^4-15 a b c^2 d^2+8 b^2 c^4\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d^3 \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{b c^2-a d^2}}{3 \left (b c^2-a d^2\right )}-\frac {2 c^3 \sqrt {a-b x^2}}{3 d^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

Input:

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

Output:

(-2*c^3*Sqrt[a - b*x^2])/(3*d^2*(b*c^2 - a*d^2)*(c + d*x)^(3/2)) - ((2*c^2 
*(9*a - (5*b*c^2)/d^2)*Sqrt[a - b*x^2])/((b*c^2 - a*d^2)*Sqrt[c + d*x]) - 
((-2*Sqrt[a]*(8*b^2*c^4 - 15*a*b*c^2*d^2 + 3*a^2*d^4)*Sqrt[c + d*x]*Sqrt[1 
 - (b*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2* 
d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(Sqrt[b]*d^3*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt 
[b]*c + Sqrt[a]*d)]*Sqrt[a - b*x^2]) + (2*Sqrt[a]*c*(8*b*c^2 - 9*a*d^2)*(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[b]*d^3*Sqrt[c + d*x]*Sqrt[a - b*x^2]))/(b 
*c^2 - a*d^2))/(3*(b*c^2 - a*d^2))
 

Defintions of rubi rules used

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 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 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 603

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol 
] :> With[{Qx = PolynomialQuotient[x^m, c + d*x, x], R = PolynomialRemainde 
r[x^m, c + d*x, x]}, Simp[d*R*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/((n + 
1)*(b*c^2 + a*d^2))), x] + Simp[1/((n + 1)*(b*c^2 + a*d^2))   Int[(c + d*x) 
^(n + 1)*(a + b*x^2)^p*ExpandToSum[(n + 1)*(b*c^2 + a*d^2)*Qx + b*c*R*(n + 
1) - b*d*R*(n + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, p}, x] && IGt 
Q[m, 1] && LtQ[n, -1] && NeQ[b*c^2 + a*d^2, 0]
 

rule 2182

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
 With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, 
 d + e*x, x]}, Simp[e*R*(d + e*x)^(m + 1)*((a + b*x^2)^(p + 1)/((m + 1)*(b* 
d^2 + a*e^2))), x] + Simp[1/((m + 1)*(b*d^2 + a*e^2))   Int[(d + e*x)^(m + 
1)*(a + b*x^2)^p*ExpandToSum[(m + 1)*(b*d^2 + a*e^2)*Qx + b*d*R*(m + 1) - b 
*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, d, e, p}, x] && PolyQ[Pq, 
 x] && NeQ[b*d^2 + a*e^2, 0] && LtQ[m, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(764\) vs. \(2(377)=754\).

Time = 5.92 (sec) , antiderivative size = 765, normalized size of antiderivative = 1.71

method	result	size
elliptic	\(\frac {\sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}\, \left (\frac {2 c^{3} \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{3 d^{4} \left (a \,d^{2}-b \,c^{2}\right ) \left (x +\frac {c}{d}\right )^{2}}-\frac {2 \left (-b d \,x^{2}+a d \right ) c^{2} \left (9 a \,d^{2}-5 b \,c^{2}\right )}{3 d^{3} \left (a \,d^{2}-b \,c^{2}\right )^{2} \sqrt {\left (x +\frac {c}{d}\right ) \left (-b d \,x^{2}+a d \right )}}+\frac {2 \left (-\frac {2 c}{d^{3}}-\frac {b \,c^{3}}{3 d^{3} \left (a \,d^{2}-b \,c^{2}\right )}-\frac {b \,c^{3} \left (9 a \,d^{2}-5 b \,c^{2}\right )}{3 d^{3} \left (a \,d^{2}-b \,c^{2}\right )^{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 )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {2 \left (\frac {1}{d^{2}}-\frac {b \,c^{2} \left (9 a \,d^{2}-5 b \,c^{2}\right )}{3 d^{2} \left (a \,d^{2}-b \,c^{2}\right )^{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 )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )}{\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\)	\(765\)
default	\(\text {Expression too large to display}\)	\(3036\)

Input:

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

Output:

((d*x+c)*(-b*x^2+a))^(1/2)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)*(2/3/d^4/(a*d^2- 
b*c^2)*c^3*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)/(x+c/d)^2-2/3*(-b*d*x^2+a*d) 
/d^3/(a*d^2-b*c^2)^2*c^2*(9*a*d^2-5*b*c^2)/((x+c/d)*(-b*d*x^2+a*d))^(1/2)+ 
2*(-2*c/d^3-1/3*b/d^3*c^3/(a*d^2-b*c^2)-1/3*b/d^3*c^3*(9*a*d^2-5*b*c^2)/(a 
*d^2-b*c^2)^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)*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))+2*(1/d^2-1/3*b*c^2/d^2*(9*a*d^2-5*b*c^2)/(a*d^2-b*c^2)^ 
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) 
)))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 535, normalized size of antiderivative = 1.20 \[ \int \frac {x^3}{(c+d x)^{5/2} \sqrt {a-b x^2}} \, dx=\frac {2 \, {\left ({\left (8 \, b^{2} c^{7} - 21 \, a b c^{5} d^{2} + 21 \, a^{2} c^{3} d^{4} + {\left (8 \, b^{2} c^{5} d^{2} - 21 \, a b c^{3} d^{4} + 21 \, a^{2} c d^{6}\right )} x^{2} + 2 \, {\left (8 \, b^{2} c^{6} d - 21 \, a b c^{4} d^{3} + 21 \, a^{2} c^{2} d^{5}\right )} x\right )} \sqrt {-b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right ) + 3 \, {\left (8 \, b^{2} c^{6} d - 15 \, a b c^{4} d^{3} + 3 \, a^{2} c^{2} d^{5} + {\left (8 \, b^{2} c^{4} d^{3} - 15 \, a b c^{2} d^{5} + 3 \, a^{2} d^{7}\right )} x^{2} + 2 \, {\left (8 \, b^{2} c^{5} d^{2} - 15 \, a b c^{3} d^{4} + 3 \, a^{2} c d^{6}\right )} x\right )} \sqrt {-b d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )\right ) + 3 \, {\left (4 \, b^{2} c^{5} d^{2} - 8 \, a b c^{3} d^{4} + {\left (5 \, b^{2} c^{4} d^{3} - 9 \, a b c^{2} d^{5}\right )} x\right )} \sqrt {-b x^{2} + a} \sqrt {d x + c}\right )}}{9 \, {\left (b^{3} c^{6} d^{4} - 2 \, a b^{2} c^{4} d^{6} + a^{2} b c^{2} d^{8} + {\left (b^{3} c^{4} d^{6} - 2 \, a b^{2} c^{2} d^{8} + a^{2} b d^{10}\right )} x^{2} + 2 \, {\left (b^{3} c^{5} d^{5} - 2 \, a b^{2} c^{3} d^{7} + a^{2} b c d^{9}\right )} x\right )}} \] Input:

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

Output:

2/9*((8*b^2*c^7 - 21*a*b*c^5*d^2 + 21*a^2*c^3*d^4 + (8*b^2*c^5*d^2 - 21*a* 
b*c^3*d^4 + 21*a^2*c*d^6)*x^2 + 2*(8*b^2*c^6*d - 21*a*b*c^4*d^3 + 21*a^2*c 
^2*d^5)*x)*sqrt(-b*d)*weierstrassPInverse(4/3*(b*c^2 + 3*a*d^2)/(b*d^2), - 
8/27*(b*c^3 - 9*a*c*d^2)/(b*d^3), 1/3*(3*d*x + c)/d) + 3*(8*b^2*c^6*d - 15 
*a*b*c^4*d^3 + 3*a^2*c^2*d^5 + (8*b^2*c^4*d^3 - 15*a*b*c^2*d^5 + 3*a^2*d^7 
)*x^2 + 2*(8*b^2*c^5*d^2 - 15*a*b*c^3*d^4 + 3*a^2*c*d^6)*x)*sqrt(-b*d)*wei 
erstrassZeta(4/3*(b*c^2 + 3*a*d^2)/(b*d^2), -8/27*(b*c^3 - 9*a*c*d^2)/(b*d 
^3), weierstrassPInverse(4/3*(b*c^2 + 3*a*d^2)/(b*d^2), -8/27*(b*c^3 - 9*a 
*c*d^2)/(b*d^3), 1/3*(3*d*x + c)/d)) + 3*(4*b^2*c^5*d^2 - 8*a*b*c^3*d^4 + 
(5*b^2*c^4*d^3 - 9*a*b*c^2*d^5)*x)*sqrt(-b*x^2 + a)*sqrt(d*x + c))/(b^3*c^ 
6*d^4 - 2*a*b^2*c^4*d^6 + a^2*b*c^2*d^8 + (b^3*c^4*d^6 - 2*a*b^2*c^2*d^8 + 
 a^2*b*d^10)*x^2 + 2*(b^3*c^5*d^5 - 2*a*b^2*c^3*d^7 + a^2*b*c*d^9)*x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(2*sqrt(c + d*x)*sqrt(a - b*x**2)*a*d - 4*sqrt(c + d*x)*sqrt(a - b*x**2)*b 
*c*x - int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**2)/(a*c**3 + 3*a*c**2*d*x + 
3*a*c*d**2*x**2 + a*d**3*x**3 - b*c**3*x**2 - 3*b*c**2*d*x**3 - 3*b*c*d**2 
*x**4 - b*d**3*x**5),x)*a*b*c**2*d**2 - 2*int((sqrt(c + d*x)*sqrt(a - b*x* 
*2)*x**2)/(a*c**3 + 3*a*c**2*d*x + 3*a*c*d**2*x**2 + a*d**3*x**3 - b*c**3* 
x**2 - 3*b*c**2*d*x**3 - 3*b*c*d**2*x**4 - b*d**3*x**5),x)*a*b*c*d**3*x - 
int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**2)/(a*c**3 + 3*a*c**2*d*x + 3*a*c*d 
**2*x**2 + a*d**3*x**3 - b*c**3*x**2 - 3*b*c**2*d*x**3 - 3*b*c*d**2*x**4 - 
 b*d**3*x**5),x)*a*b*d**4*x**2 - 8*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x** 
2)/(a*c**3 + 3*a*c**2*d*x + 3*a*c*d**2*x**2 + a*d**3*x**3 - b*c**3*x**2 - 
3*b*c**2*d*x**3 - 3*b*c*d**2*x**4 - b*d**3*x**5),x)*b**2*c**4 - 16*int((sq 
rt(c + d*x)*sqrt(a - b*x**2)*x**2)/(a*c**3 + 3*a*c**2*d*x + 3*a*c*d**2*x** 
2 + a*d**3*x**3 - b*c**3*x**2 - 3*b*c**2*d*x**3 - 3*b*c*d**2*x**4 - b*d**3 
*x**5),x)*b**2*c**3*d*x - 8*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**2)/(a*c 
**3 + 3*a*c**2*d*x + 3*a*c*d**2*x**2 + a*d**3*x**3 - b*c**3*x**2 - 3*b*c** 
2*d*x**3 - 3*b*c*d**2*x**4 - b*d**3*x**5),x)*b**2*c**2*d**2*x**2 + 3*int(( 
sqrt(c + d*x)*sqrt(a - b*x**2))/(a*c**3 + 3*a*c**2*d*x + 3*a*c*d**2*x**2 + 
 a*d**3*x**3 - b*c**3*x**2 - 3*b*c**2*d*x**3 - 3*b*c*d**2*x**4 - b*d**3*x* 
*5),x)*a**2*c**2*d**2 + 6*int((sqrt(c + d*x)*sqrt(a - b*x**2))/(a*c**3 + 3 
*a*c**2*d*x + 3*a*c*d**2*x**2 + a*d**3*x**3 - b*c**3*x**2 - 3*b*c**2*d*...