\(\int \frac {x}{\sqrt {c+d x} (a-b x^2)^{5/2}} \, dx\) [1584]

Optimal result

Integrand size = 23, antiderivative size = 421 \[ \int \frac {x}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\frac {(c-d x) \sqrt {c+d x}}{3 \left (b c^2-a d^2\right ) \left (a-b x^2\right )^{3/2}}-\frac {d \sqrt {c+d x} \left (4 a c d-\left (b c^2+3 a d^2\right ) x\right )}{6 a \left (b c^2-a d^2\right )^2 \sqrt {a-b x^2}}+\frac {d \left (b c^2+3 a d^2\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{6 \sqrt {a} \sqrt {b} \left (b c^2-a d^2\right )^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {c d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{6 \sqrt {a} \sqrt {b} \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

1/3*(-d*x+c)*(d*x+c)^(1/2)/(-a*d^2+b*c^2)/(-b*x^2+a)^(3/2)-1/6*d*(d*x+c)^( 
1/2)*(4*a*c*d-(3*a*d^2+b*c^2)*x)/a/(-a*d^2+b*c^2)^2/(-b*x^2+a)^(1/2)+1/6*d 
*(3*a*d^2+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)/b^(1/2)/(-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)-1/6*c*d*(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)/b^(1/2)/(-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 = 12.23 (sec) , antiderivative size = 552, normalized size of antiderivative = 1.31 \[ \int \frac {x}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\frac {\sqrt {a-b x^2} \left (\frac {(c+d x) \left (2 a \left (b c^2-a d^2\right ) (c-d x)+d \left (a-b x^2\right ) \left (b c^2 x+a d (-4 c+3 d x)\right )\right )}{\left (a-b x^2\right )^2}+\frac {d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (b c^2+3 a d^2\right ) \left (a-b x^2\right )+i \sqrt {b} \left (b^{3/2} c^3-\sqrt {a} b c^2 d+3 a \sqrt {b} c d^2-3 a^{3/2} d^3\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )+i \sqrt {a} \sqrt {b} d \left (b c^2-4 \sqrt {a} \sqrt {b} c d+3 a d^2\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{b \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (a-b x^2\right )}\right )}{6 a \left (b c^2-a d^2\right )^2 \sqrt {c+d x}} \] Input:

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

Output:

(Sqrt[a - b*x^2]*(((c + d*x)*(2*a*(b*c^2 - a*d^2)*(c - d*x) + d*(a - b*x^2 
)*(b*c^2*x + a*d*(-4*c + 3*d*x))))/(a - b*x^2)^2 + (d^2*Sqrt[-c + (Sqrt[a] 
*d)/Sqrt[b]]*(b*c^2 + 3*a*d^2)*(a - b*x^2) + I*Sqrt[b]*(b^(3/2)*c^3 - Sqrt 
[a]*b*c^2*d + 3*a*Sqrt[b]*c*d^2 - 3*a^(3/2)*d^3)*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*Sqrt[a]*Sqrt[b]*d*(b*c 
^2 - 4*Sqrt[a]*Sqrt[b]*c*d + 3*a*d^2)*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)*Ellip 
ticF[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)])/(b*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(a 
 - b*x^2))))/(6*a*(b*c^2 - a*d^2)^2*Sqrt[c + d*x])
 

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 433, normalized size of antiderivative = 1.03, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {593, 27, 686, 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.

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

\(\Big \downarrow \) 593

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

\(\Big \downarrow \) 27

\(\displaystyle \frac {d \int \frac {c-3 d x}{\sqrt {c+d x} \left (a-b x^2\right )^{3/2}}dx}{6 \left (b c^2-a d^2\right )}+\frac {\sqrt {c+d x} (c-d x)}{3 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 686

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 600

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

\(\Big \downarrow \) 509

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

\(\Big \downarrow \) 508

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

\(\Big \downarrow \) 327

\(\displaystyle \frac {d \left (-\frac {d \left (-\frac {c \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a d^2+b c^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{2 a \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (4 a c d-x \left (3 a d^2+b c^2\right )\right )}{a \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\right )}{6 \left (b c^2-a d^2\right )}+\frac {\sqrt {c+d x} (c-d x)}{3 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 512

\(\displaystyle \frac {d \left (-\frac {d \left (-\frac {c \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a d^2+b c^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{2 a \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (4 a c d-x \left (3 a d^2+b c^2\right )\right )}{a \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\right )}{6 \left (b c^2-a d^2\right )}+\frac {\sqrt {c+d x} (c-d x)}{3 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 511

\(\displaystyle \frac {d \left (-\frac {d \left (\frac {2 \sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a d^2+b c^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{2 a \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (4 a c d-x \left (3 a d^2+b c^2\right )\right )}{a \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\right )}{6 \left (b c^2-a d^2\right )}+\frac {\sqrt {c+d x} (c-d x)}{3 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {d \left (-\frac {d \left (\frac {2 \sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (3 a d^2+b c^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{2 a \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (4 a c d-x \left (3 a d^2+b c^2\right )\right )}{a \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\right )}{6 \left (b c^2-a d^2\right )}+\frac {\sqrt {c+d x} (c-d x)}{3 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}\)

Input:

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

Output:

((c - d*x)*Sqrt[c + d*x])/(3*(b*c^2 - a*d^2)*(a - b*x^2)^(3/2)) + (d*(-((S 
qrt[c + d*x]*(4*a*c*d - (b*c^2 + 3*a*d^2)*x))/(a*(b*c^2 - a*d^2)*Sqrt[a - 
b*x^2])) - (d*((-2*Sqrt[a]*(b*c^2 + 3*a*d^2)*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*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqr 
t[a]*d)]*Sqrt[a - b*x^2]) + (2*Sqrt[a]*c*(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)])/(Sqr 
t[b]*d*Sqrt[c + d*x]*Sqrt[a - b*x^2])))/(2*a*(b*c^2 - a*d^2))))/(6*(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 593

Int[(x_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
 Simp[(c + d*x)^(n + 1)*(c - d*x)*((a + b*x^2)^(p + 1)/(2*(p + 1)*(b*c^2 + 
a*d^2))), x] - Simp[d/(2*(p + 1)*(b*c^2 + a*d^2))   Int[(c + d*x)^n*(a + b* 
x^2)^(p + 1)*(c*n - d*(n + 2*p + 4)*x), x], x] /; FreeQ[{a, b, c, d, n}, x] 
 && LtQ[p, -1] && NeQ[b*c^2 + a*d^2, 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 686

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + 
a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[ 
1/(2*a*c*(p + 1)*(c*d^2 + a*e^2))   Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Sim 
p[f*(c^2*d^2*(2*p + 3) + a*c*e^2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f 
+ a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ 
[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(817\) vs. \(2(351)=702\).

Time = 7.18 (sec) , antiderivative size = 818, normalized size of antiderivative = 1.94

method	result	size
elliptic	\(\frac {\sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}\, \left (\frac {\left (\frac {d x}{3 \left (a \,d^{2}-b \,c^{2}\right ) b^{2}}-\frac {c}{3 \left (a \,d^{2}-b \,c^{2}\right ) b^{2}}\right ) \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{\left (x^{2}-\frac {a}{b}\right )^{2}}-\frac {2 \left (-b d x -b c \right ) \left (\frac {d \left (3 a \,d^{2}+b \,c^{2}\right ) x}{12 a b \left (a \,d^{2}-b \,c^{2}\right )^{2}}-\frac {c \,d^{2}}{3 \left (a \,d^{2}-b \,c^{2}\right )^{2} b}\right )}{\sqrt {\left (x^{2}-\frac {a}{b}\right ) \left (-b d x -b c \right )}}+\frac {2 \left (-\frac {c d}{6 a \left (a \,d^{2}-b \,c^{2}\right )}+\frac {d^{3} c}{3 \left (a \,d^{2}-b \,c^{2}\right )^{2}}-\frac {c d \left (3 a \,d^{2}+b \,c^{2}\right )}{6 a \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 {d^{2} \left (3 a \,d^{2}+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 )}{6 a \left (a \,d^{2}-b \,c^{2}\right )^{2} \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )}{\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\)	\(818\)
default	\(\text {Expression too large to display}\)	\(2179\)

Input:

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

Output:

((d*x+c)*(-b*x^2+a))^(1/2)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)*((1/3*d/(a*d^2-b 
*c^2)/b^2*x-1/3*c/(a*d^2-b*c^2)/b^2)*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)/(x 
^2-a/b)^2-2*(-b*d*x-b*c)*(1/12*d*(3*a*d^2+b*c^2)/a/b/(a*d^2-b*c^2)^2*x-1/3 
*c*d^2/(a*d^2-b*c^2)^2/b)/((x^2-a/b)*(-b*d*x-b*c))^(1/2)+2*(-1/6*c*d/a/(a* 
d^2-b*c^2)+1/3*d^3*c/(a*d^2-b*c^2)^2-1/6*c*d*(3*a*d^2+b*c^2)/a/(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))-1/6*d^2*(3*a*d^2+b*c^2)/a/(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 = 0.10 (sec) , antiderivative size = 503, normalized size of antiderivative = 1.19 \[ \int \frac {x}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=-\frac {{\left (a^{2} b c^{3} - 9 \, a^{3} c d^{2} + {\left (b^{3} c^{3} - 9 \, a b^{2} c d^{2}\right )} x^{4} - 2 \, {\left (a b^{2} c^{3} - 9 \, a^{2} b c d^{2}\right )} x^{2}\right )} \sqrt {-b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right ) + 3 \, {\left (a^{2} b c^{2} d + 3 \, a^{3} d^{3} + {\left (b^{3} c^{2} d + 3 \, a b^{2} d^{3}\right )} x^{4} - 2 \, {\left (a b^{2} c^{2} d + 3 \, a^{2} b d^{3}\right )} x^{2}\right )} \sqrt {-b d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )\right ) - 3 \, {\left (4 \, a b^{2} c d^{2} x^{2} + 2 \, a b^{2} c^{3} - 6 \, a^{2} b c d^{2} - {\left (b^{3} c^{2} d + 3 \, a b^{2} d^{3}\right )} x^{3} - {\left (a b^{2} c^{2} d - 5 \, a^{2} b d^{3}\right )} x\right )} \sqrt {-b x^{2} + a} \sqrt {d x + c}}{18 \, {\left (a^{3} b^{3} c^{4} - 2 \, a^{4} b^{2} c^{2} d^{2} + a^{5} b d^{4} + {\left (a b^{5} c^{4} - 2 \, a^{2} b^{4} c^{2} d^{2} + a^{3} b^{3} d^{4}\right )} x^{4} - 2 \, {\left (a^{2} b^{4} c^{4} - 2 \, a^{3} b^{3} c^{2} d^{2} + a^{4} b^{2} d^{4}\right )} x^{2}\right )}} \] Input:

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

Output:

-1/18*((a^2*b*c^3 - 9*a^3*c*d^2 + (b^3*c^3 - 9*a*b^2*c*d^2)*x^4 - 2*(a*b^2 
*c^3 - 9*a^2*b*c*d^2)*x^2)*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*( 
a^2*b*c^2*d + 3*a^3*d^3 + (b^3*c^2*d + 3*a*b^2*d^3)*x^4 - 2*(a*b^2*c^2*d + 
 3*a^2*b*d^3)*x^2)*sqrt(-b*d)*weierstrassZeta(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*a*b^2*c*d^2*x^2 + 2*a*b^2*c^3 - 6*a^2*b*c*d^2 - (b^3*c^2*d + 3*a*b^2*d^ 
3)*x^3 - (a*b^2*c^2*d - 5*a^2*b*d^3)*x)*sqrt(-b*x^2 + a)*sqrt(d*x + c))/(a 
^3*b^3*c^4 - 2*a^4*b^2*c^2*d^2 + a^5*b*d^4 + (a*b^5*c^4 - 2*a^2*b^4*c^2*d^ 
2 + a^3*b^3*d^4)*x^4 - 2*(a^2*b^4*c^4 - 2*a^3*b^3*c^2*d^2 + a^4*b^2*d^4)*x 
^2)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x}{-b^{3} d \,x^{7}-b^{3} c \,x^{6}+3 a \,b^{2} d \,x^{5}+3 a \,b^{2} c \,x^{4}-3 a^{2} b d \,x^{3}-3 a^{2} b c \,x^{2}+a^{3} d x +a^{3} c}d x \] Input:

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

Output:

int((sqrt(c + d*x)*sqrt(a - b*x**2)*x)/(a**3*c + a**3*d*x - 3*a**2*b*c*x** 
2 - 3*a**2*b*d*x**3 + 3*a*b**2*c*x**4 + 3*a*b**2*d*x**5 - b**3*c*x**6 - b* 
*3*d*x**7),x)