3.794 \(\int \frac{x^4}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx\)

Optimal. Leaf size=241 \[ -\frac{2 c \sqrt{a+b x} \left (c (a d+b c) \left (3 a^2 d^2-14 a b c d+3 b^2 c^2\right )+2 d x \left (3 a^3 d^3-12 a^2 b c d^2-a b^2 c^2 d+2 b^3 c^3\right )\right )}{3 b^2 d^2 (c+d x)^{3/2} (b c-a d)^4}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{5/2} d^{5/2}}+\frac{2 a x^2 (3 b c-a d)}{b^2 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^2}+\frac{2 a x^3}{3 b (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)} \]

[Out]

(2*a*x^3)/(3*b*(b*c - a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2)) + (2*a*(3*b*c - a*d)
*x^2)/(b^2*(b*c - a*d)^2*Sqrt[a + b*x]*(c + d*x)^(3/2)) - (2*c*Sqrt[a + b*x]*(c*
(b*c + a*d)*(3*b^2*c^2 - 14*a*b*c*d + 3*a^2*d^2) + 2*d*(2*b^3*c^3 - a*b^2*c^2*d
- 12*a^2*b*c*d^2 + 3*a^3*d^3)*x))/(3*b^2*d^2*(b*c - a*d)^4*(c + d*x)^(3/2)) + (2
*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(b^(5/2)*d^(5/2))

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Rubi [A]  time = 0.572732, antiderivative size = 241, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{2 c \sqrt{a+b x} \left (c (a d+b c) \left (3 a^2 d^2-14 a b c d+3 b^2 c^2\right )+2 d x \left (3 a^3 d^3-12 a^2 b c d^2-a b^2 c^2 d+2 b^3 c^3\right )\right )}{3 b^2 d^2 (c+d x)^{3/2} (b c-a d)^4}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{5/2} d^{5/2}}+\frac{2 a x^2 (3 b c-a d)}{b^2 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^2}+\frac{2 a x^3}{3 b (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[x^4/((a + b*x)^(5/2)*(c + d*x)^(5/2)),x]

[Out]

(2*a*x^3)/(3*b*(b*c - a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2)) + (2*a*(3*b*c - a*d)
*x^2)/(b^2*(b*c - a*d)^2*Sqrt[a + b*x]*(c + d*x)^(3/2)) - (2*c*Sqrt[a + b*x]*(c*
(b*c + a*d)*(3*b^2*c^2 - 14*a*b*c*d + 3*a^2*d^2) + 2*d*(2*b^3*c^3 - a*b^2*c^2*d
- 12*a^2*b*c*d^2 + 3*a^3*d^3)*x))/(3*b^2*d^2*(b*c - a*d)^4*(c + d*x)^(3/2)) + (2
*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(b^(5/2)*d^(5/2))

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Rubi in Sympy [A]  time = 51.4445, size = 238, normalized size = 0.99 \[ - \frac{2 a x^{3}}{3 b \left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )} - \frac{2 a x^{2} \left (a d - 3 b c\right )}{b^{2} \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )^{2}} - \frac{16 c \sqrt{a + b x} \left (\frac{3 c \left (a d + b c\right ) \left (3 a^{2} d^{2} - 14 a b c d + 3 b^{2} c^{2}\right )}{8} + \frac{3 d x \left (3 a^{3} d^{3} - 12 a^{2} b c d^{2} - a b^{2} c^{2} d + 2 b^{3} c^{3}\right )}{4}\right )}{9 b^{2} d^{2} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )^{4}} + \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{b^{\frac{5}{2}} d^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(b*x+a)**(5/2)/(d*x+c)**(5/2),x)

[Out]

-2*a*x**3/(3*b*(a + b*x)**(3/2)*(c + d*x)**(3/2)*(a*d - b*c)) - 2*a*x**2*(a*d -
3*b*c)/(b**2*sqrt(a + b*x)*(c + d*x)**(3/2)*(a*d - b*c)**2) - 16*c*sqrt(a + b*x)
*(3*c*(a*d + b*c)*(3*a**2*d**2 - 14*a*b*c*d + 3*b**2*c**2)/8 + 3*d*x*(3*a**3*d**
3 - 12*a**2*b*c*d**2 - a*b**2*c**2*d + 2*b**3*c**3)/4)/(9*b**2*d**2*(c + d*x)**(
3/2)*(a*d - b*c)**4) + 2*atanh(sqrt(b)*sqrt(c + d*x)/(sqrt(d)*sqrt(a + b*x)))/(b
**(5/2)*d**(5/2))

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Mathematica [A]  time = 0.928883, size = 194, normalized size = 0.8 \[ \frac{2}{3} \sqrt{a+b x} \sqrt{c+d x} \left (-\frac{a^4}{b^2 (a+b x)^2 (b c-a d)^3}-\frac{4 a^3 (a d-3 b c)}{b^2 (a+b x) (b c-a d)^4}-\frac{c^4}{d^2 (c+d x)^2 (a d-b c)^3}-\frac{4 c^3 (b c-3 a d)}{d^2 (c+d x) (b c-a d)^4}\right )+\frac{\log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{b^{5/2} d^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/((a + b*x)^(5/2)*(c + d*x)^(5/2)),x]

[Out]

(2*Sqrt[a + b*x]*Sqrt[c + d*x]*(-(a^4/(b^2*(b*c - a*d)^3*(a + b*x)^2)) - (4*a^3*
(-3*b*c + a*d))/(b^2*(b*c - a*d)^4*(a + b*x)) - c^4/(d^2*(-(b*c) + a*d)^3*(c + d
*x)^2) - (4*c^3*(b*c - 3*a*d))/(d^2*(b*c - a*d)^4*(c + d*x))))/3 + Log[b*c + a*d
+ 2*b*d*x + 2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*Sqrt[c + d*x]]/(b^(5/2)*d^(5/2))

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Maple [B]  time = 0.049, size = 2089, normalized size = 8.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(b*x+a)^(5/2)/(d*x+c)^(5/2),x)

[Out]

1/3*(-12*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1
/2))*x^4*a^3*b^3*c*d^5+3*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a
*d+b*c)/(b*d)^(1/2))*x^2*a^6*d^6+3*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*
d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x^2*b^6*c^6+3*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))
^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^6*c^2*d^4+3*ln(1/2*(2*b*d*x+2*((b*x+a
)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^2*b^4*c^6+24*x^3*a^3*b^2*c*
d^4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+24*x^3*a*b^4*c^3*d^2*((b*x+a)*(d*x+c))^(
1/2)*(b*d)^(1/2)+36*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*x*a^2*b^3*c^4*d+6*((b*x+
a)*(d*x+c))^(1/2)*(b*d)^(1/2)*x^2*a^4*b*c*d^4+48*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(
1/2)*x^2*a^3*b^2*c^2*d^3+48*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*x^2*a^2*b^3*c^3*
d^2+6*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*x^2*a*b^4*c^4*d+36*((b*x+a)*(d*x+c))^(
1/2)*(b*d)^(1/2)*x*a^4*b*c^2*d^3+48*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*x*a^3*b^
2*c^3*d^2+18*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d
)^(1/2))*a^4*b^2*c^4*d^2-18*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2
)+a*d+b*c)/(b*d)^(1/2))*x*a^2*b^4*c^5*d-12*x*a^5*c*d^4*((b*x+a)*(d*x+c))^(1/2)*(
b*d)^(1/2)+48*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*
d)^(1/2))*x^2*a^3*b^3*c^3*d^3-27*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)
^(1/2)+a*d+b*c)/(b*d)^(1/2))*x^2*a^2*b^4*c^4*d^2-18*ln(1/2*(2*b*d*x+2*((b*x+a)*(
d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x*a^5*b*c^2*d^4+12*ln(1/2*(2*b*d
*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x*a^4*b^2*c^3*d^3
-8*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*x^3*a^4*b*d^5-8*((b*x+a)*(d*x+c))^(1/2)*(
b*d)^(1/2)*x^3*b^5*c^4*d+22*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^4*b*c^3*d^2+22
*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^3*b^2*c^4*d-12*x*a*b^4*c^5*((b*x+a)*(d*x+
c))^(1/2)*(b*d)^(1/2)+12*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a
*d+b*c)/(b*d)^(1/2))*x*a^3*b^3*c^4*d^2+18*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1
/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x^4*a^2*b^4*c^2*d^4-12*ln(1/2*(2*b*d*x+2*(
(b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x^4*a*b^5*c^3*d^3-18*ln
(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x^3*a^
4*b^2*c*d^5+12*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b
*d)^(1/2))*x^3*a^3*b^3*c^2*d^4+12*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d
)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x^3*a^2*b^4*c^3*d^3-18*ln(1/2*(2*b*d*x+2*((b*x+a)*
(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x^3*a*b^5*c^4*d^2-27*ln(1/2*(2*
b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x^2*a^4*b^2*c^
2*d^4-12*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1
/2))*a^3*b^3*c^5*d-6*x^2*a^5*d^5*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-6*x^2*b^5*c
^5*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-6*a^5*c^2*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*
d)^(1/2)-6*a^2*b^3*c^5*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+3*ln(1/2*(2*b*d*x+2*(
(b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x^4*a^4*b^2*d^6+3*ln(1/
2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x^4*b^6*c
^4*d^2+6*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1
/2))*x^3*a^5*b*d^6+6*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b
*c)/(b*d)^(1/2))*x^3*b^6*c^5*d+6*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)
^(1/2)+a*d+b*c)/(b*d)^(1/2))*x*a^6*c*d^5+6*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(
1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x*a*b^5*c^6-12*ln(1/2*(2*b*d*x+2*((b*x+a)
*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^5*b*c^3*d^3)/((b*x+a)*(d*x+c
))^(1/2)/(a*d-b*c)^4/(b*d)^(1/2)/(b*x+a)^(3/2)/(d*x+c)^(3/2)/d^2/b^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/((b*x + a)^(5/2)*(d*x + c)^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.99927, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/((b*x + a)^(5/2)*(d*x + c)^(5/2)),x, algorithm="fricas")

[Out]

[-1/6*(4*(3*a^2*b^3*c^5 - 11*a^3*b^2*c^4*d - 11*a^4*b*c^3*d^2 + 3*a^5*c^2*d^3 +
4*(b^5*c^4*d - 3*a*b^4*c^3*d^2 - 3*a^3*b^2*c*d^4 + a^4*b*d^5)*x^3 + 3*(b^5*c^5 -
a*b^4*c^4*d - 8*a^2*b^3*c^3*d^2 - 8*a^3*b^2*c^2*d^3 - a^4*b*c*d^4 + a^5*d^5)*x^
2 + 6*(a*b^4*c^5 - 3*a^2*b^3*c^4*d - 4*a^3*b^2*c^3*d^2 - 3*a^4*b*c^2*d^3 + a^5*c
*d^4)*x)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) - 3*(a^2*b^4*c^6 - 4*a^3*b^3*c^5*
d + 6*a^4*b^2*c^4*d^2 - 4*a^5*b*c^3*d^3 + a^6*c^2*d^4 + (b^6*c^4*d^2 - 4*a*b^5*c
^3*d^3 + 6*a^2*b^4*c^2*d^4 - 4*a^3*b^3*c*d^5 + a^4*b^2*d^6)*x^4 + 2*(b^6*c^5*d -
3*a*b^5*c^4*d^2 + 2*a^2*b^4*c^3*d^3 + 2*a^3*b^3*c^2*d^4 - 3*a^4*b^2*c*d^5 + a^5
*b*d^6)*x^3 + (b^6*c^6 - 9*a^2*b^4*c^4*d^2 + 16*a^3*b^3*c^3*d^3 - 9*a^4*b^2*c^2*
d^4 + a^6*d^6)*x^2 + 2*(a*b^5*c^6 - 3*a^2*b^4*c^5*d + 2*a^3*b^3*c^4*d^2 + 2*a^4*
b^2*c^3*d^3 - 3*a^5*b*c^2*d^4 + a^6*c*d^5)*x)*log(4*(2*b^2*d^2*x + b^2*c*d + a*b
*d^2)*sqrt(b*x + a)*sqrt(d*x + c) + (8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d
^2 + 8*(b^2*c*d + a*b*d^2)*x)*sqrt(b*d)))/((a^2*b^6*c^6*d^2 - 4*a^3*b^5*c^5*d^3
+ 6*a^4*b^4*c^4*d^4 - 4*a^5*b^3*c^3*d^5 + a^6*b^2*c^2*d^6 + (b^8*c^4*d^4 - 4*a*b
^7*c^3*d^5 + 6*a^2*b^6*c^2*d^6 - 4*a^3*b^5*c*d^7 + a^4*b^4*d^8)*x^4 + 2*(b^8*c^5
*d^3 - 3*a*b^7*c^4*d^4 + 2*a^2*b^6*c^3*d^5 + 2*a^3*b^5*c^2*d^6 - 3*a^4*b^4*c*d^7
+ a^5*b^3*d^8)*x^3 + (b^8*c^6*d^2 - 9*a^2*b^6*c^4*d^4 + 16*a^3*b^5*c^3*d^5 - 9*
a^4*b^4*c^2*d^6 + a^6*b^2*d^8)*x^2 + 2*(a*b^7*c^6*d^2 - 3*a^2*b^6*c^5*d^3 + 2*a^
3*b^5*c^4*d^4 + 2*a^4*b^4*c^3*d^5 - 3*a^5*b^3*c^2*d^6 + a^6*b^2*c*d^7)*x)*sqrt(b
*d)), -1/3*(2*(3*a^2*b^3*c^5 - 11*a^3*b^2*c^4*d - 11*a^4*b*c^3*d^2 + 3*a^5*c^2*d
^3 + 4*(b^5*c^4*d - 3*a*b^4*c^3*d^2 - 3*a^3*b^2*c*d^4 + a^4*b*d^5)*x^3 + 3*(b^5*
c^5 - a*b^4*c^4*d - 8*a^2*b^3*c^3*d^2 - 8*a^3*b^2*c^2*d^3 - a^4*b*c*d^4 + a^5*d^
5)*x^2 + 6*(a*b^4*c^5 - 3*a^2*b^3*c^4*d - 4*a^3*b^2*c^3*d^2 - 3*a^4*b*c^2*d^3 +
a^5*c*d^4)*x)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c) - 3*(a^2*b^4*c^6 - 4*a^3*b^
3*c^5*d + 6*a^4*b^2*c^4*d^2 - 4*a^5*b*c^3*d^3 + a^6*c^2*d^4 + (b^6*c^4*d^2 - 4*a
*b^5*c^3*d^3 + 6*a^2*b^4*c^2*d^4 - 4*a^3*b^3*c*d^5 + a^4*b^2*d^6)*x^4 + 2*(b^6*c
^5*d - 3*a*b^5*c^4*d^2 + 2*a^2*b^4*c^3*d^3 + 2*a^3*b^3*c^2*d^4 - 3*a^4*b^2*c*d^5
+ a^5*b*d^6)*x^3 + (b^6*c^6 - 9*a^2*b^4*c^4*d^2 + 16*a^3*b^3*c^3*d^3 - 9*a^4*b^
2*c^2*d^4 + a^6*d^6)*x^2 + 2*(a*b^5*c^6 - 3*a^2*b^4*c^5*d + 2*a^3*b^3*c^4*d^2 +
2*a^4*b^2*c^3*d^3 - 3*a^5*b*c^2*d^4 + a^6*c*d^5)*x)*arctan(1/2*(2*b*d*x + b*c +
a*d)*sqrt(-b*d)/(sqrt(b*x + a)*sqrt(d*x + c)*b*d)))/((a^2*b^6*c^6*d^2 - 4*a^3*b^
5*c^5*d^3 + 6*a^4*b^4*c^4*d^4 - 4*a^5*b^3*c^3*d^5 + a^6*b^2*c^2*d^6 + (b^8*c^4*d
^4 - 4*a*b^7*c^3*d^5 + 6*a^2*b^6*c^2*d^6 - 4*a^3*b^5*c*d^7 + a^4*b^4*d^8)*x^4 +
2*(b^8*c^5*d^3 - 3*a*b^7*c^4*d^4 + 2*a^2*b^6*c^3*d^5 + 2*a^3*b^5*c^2*d^6 - 3*a^4
*b^4*c*d^7 + a^5*b^3*d^8)*x^3 + (b^8*c^6*d^2 - 9*a^2*b^6*c^4*d^4 + 16*a^3*b^5*c^
3*d^5 - 9*a^4*b^4*c^2*d^6 + a^6*b^2*d^8)*x^2 + 2*(a*b^7*c^6*d^2 - 3*a^2*b^6*c^5*
d^3 + 2*a^3*b^5*c^4*d^4 + 2*a^4*b^4*c^3*d^5 - 3*a^5*b^3*c^2*d^6 + a^6*b^2*c*d^7)
*x)*sqrt(-b*d))]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(b*x+a)**(5/2)/(d*x+c)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.672138, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/((b*x + a)^(5/2)*(d*x + c)^(5/2)),x, algorithm="giac")

[Out]

sage0*x