\(\int \frac {x}{\sqrt {c+d x} (a-b x^2)^{3/2}} \, dx\) [1557]

Optimal result

Integrand size = 23, antiderivative size = 190 \[ \int \frac {x}{\sqrt {c+d x} \left (a-b x^2\right )^{3/2}} \, dx=\frac {(c-d x) \sqrt {c+d x}}{\left (b c^2-a d^2\right ) \sqrt {a-b x^2}}-\frac {\sqrt {a} d \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {b} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 11.28 (sec) , antiderivative size = 451, normalized size of antiderivative = 2.37 \[ \int \frac {x}{\sqrt {c+d x} \left (a-b x^2\right )^{3/2}} \, dx=\frac {\sqrt {-\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \left (\sqrt {b} \sqrt {\frac {d \left (\sqrt {a}+\sqrt {b} x\right )}{-\sqrt {b} c+\sqrt {a} d}} (-c+d x) \sqrt {-\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}}+i d \sqrt {\frac {d \left (\sqrt {a}-\sqrt {b} x\right )}{\sqrt {b} c+\sqrt {a} d}} \left (\sqrt {a}+\sqrt {b} x\right ) E\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )-i d \sqrt {\frac {d \left (\sqrt {a}-\sqrt {b} x\right )}{\sqrt {b} c+\sqrt {a} d}} \left (\sqrt {a}+\sqrt {b} x\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )\right )}{\left (b^{3/2} c-\sqrt {a} b d\right ) \sqrt {\frac {d \left (\sqrt {a}+\sqrt {b} x\right )}{-\sqrt {b} c+\sqrt {a} d}} \sqrt {c+d x} \sqrt {a-b x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {593, 27, 509, 508, 327}

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.

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

\(\Big \downarrow \) 593

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 509

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

\(\Big \downarrow \) 508

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

\(\Big \downarrow \) 327

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

Input:

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

Output:

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(624\) vs. \(2(159)=318\).

Time = 3.69 (sec) , antiderivative size = 625, normalized size of antiderivative = 3.29

method	result	size
default	\(\frac {\left (a \sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}\, \sqrt {\frac {\left (-b x +\sqrt {a b}\right ) d}{d \sqrt {a b}+b c}}\, \sqrt {\frac {\left (b x +\sqrt {a b}\right ) d}{d \sqrt {a b}-b c}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}, \sqrt {-\frac {d \sqrt {a b}-b c}{d \sqrt {a b}+b c}}\right ) d^{2}-\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}\, \sqrt {\frac {\left (-b x +\sqrt {a b}\right ) d}{d \sqrt {a b}+b c}}\, \sqrt {\frac {\left (b x +\sqrt {a b}\right ) d}{d \sqrt {a b}-b c}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}, \sqrt {-\frac {d \sqrt {a b}-b c}{d \sqrt {a b}+b c}}\right ) b \,c^{2}-\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}\, \sqrt {\frac {\left (-b x +\sqrt {a b}\right ) d}{d \sqrt {a b}+b c}}\, \sqrt {\frac {\left (b x +\sqrt {a b}\right ) d}{d \sqrt {a b}-b c}}\, \operatorname {EllipticE}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}, \sqrt {-\frac {d \sqrt {a b}-b c}{d \sqrt {a b}+b c}}\right ) a \,d^{2}+\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}\, \sqrt {\frac {\left (-b x +\sqrt {a b}\right ) d}{d \sqrt {a b}+b c}}\, \sqrt {\frac {\left (b x +\sqrt {a b}\right ) d}{d \sqrt {a b}-b c}}\, \operatorname {EllipticE}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {a b}-b c}}, \sqrt {-\frac {d \sqrt {a b}-b c}{d \sqrt {a b}+b c}}\right ) b \,c^{2}+b \,x^{2} d^{2}-b \,c^{2}\right ) \sqrt {-b \,x^{2}+a}\, \sqrt {d x +c}}{b \left (a \,d^{2}-b \,c^{2}\right ) \left (-b d \,x^{3}-b c \,x^{2}+a d x +a c \right )}\)	\(625\)
elliptic	\(\frac {\sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}\, \left (-\frac {2 \left (-b d x -b c \right ) \left (\frac {d x}{2 \left (a \,d^{2}-b \,c^{2}\right ) b}-\frac {c}{2 \left (a \,d^{2}-b \,c^{2}\right ) b}\right )}{\sqrt {\left (x^{2}-\frac {a}{b}\right ) \left (-b d x -b c \right )}}-\frac {d c \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{\left (a \,d^{2}-b \,c^{2}\right ) \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}-\frac {d^{2} \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \left (\left (-\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{b}\right )}{\left (a \,d^{2}-b \,c^{2}\right ) \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )}{\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\)	\(647\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.36 \[ \int \frac {x}{\sqrt {c+d x} \left (a-b x^2\right )^{3/2}} \, dx=\frac {2 \, {\left (b c x^{2} - a c\right )} \sqrt {-b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right ) - 3 \, {\left (b d x^{2} - a d\right )} \sqrt {-b d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )\right ) - 3 \, {\left (b d x - b c\right )} \sqrt {-b x^{2} + a} \sqrt {d x + c}}{3 \, {\left (a b^{2} c^{2} - a^{2} b d^{2} - {\left (b^{3} c^{2} - a b^{2} d^{2}\right )} x^{2}\right )}} \] Input:

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

Output:

1/3*(2*(b*c*x^2 - a*c)*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*(b*d* 
x^2 - a*d)*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*(b*d*x - 
 b*c)*sqrt(-b*x^2 + a)*sqrt(d*x + c))/(a*b^2*c^2 - a^2*b*d^2 - (b^3*c^2 - 
a*b^2*d^2)*x^2)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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