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

Optimal. Leaf size=341 \[ -\frac{2 c x^2 \sqrt{a+b x} \left (-5 a^2 d^2+12 a b c d+b^2 c^2\right )}{3 b^2 d (c+d x)^{3/2} (b c-a d)^3}+\frac{\sqrt{a+b x} \left (d x (b c-a d) \left (-15 a^3 d^3+35 a^2 b c d^2-9 a b^2 c^2 d+5 b^3 c^3\right )+c \left (15 a^4 d^4-40 a^3 b c d^3+18 a^2 b^2 c^2 d^2-40 a b^3 c^3 d+15 b^4 c^4\right )\right )}{3 b^3 d^3 \sqrt{c+d x} (b c-a d)^4}-\frac{5 (a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{7/2} d^{7/2}}+\frac{2 a x^3 (11 b c-5 a d)}{3 b^2 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^2}+\frac{2 a x^4}{3 b (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)} \]

[Out]

(2*a*x^4)/(3*b*(b*c - a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2)) + (2*a*(11*b*c - 5*a
*d)*x^3)/(3*b^2*(b*c - a*d)^2*Sqrt[a + b*x]*(c + d*x)^(3/2)) - (2*c*(b^2*c^2 + 1
2*a*b*c*d - 5*a^2*d^2)*x^2*Sqrt[a + b*x])/(3*b^2*d*(b*c - a*d)^3*(c + d*x)^(3/2)
) + (Sqrt[a + b*x]*(c*(15*b^4*c^4 - 40*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 - 40*a^3
*b*c*d^3 + 15*a^4*d^4) + d*(b*c - a*d)*(5*b^3*c^3 - 9*a*b^2*c^2*d + 35*a^2*b*c*d
^2 - 15*a^3*d^3)*x))/(3*b^3*d^3*(b*c - a*d)^4*Sqrt[c + d*x]) - (5*(b*c + a*d)*Ar
cTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(b^(7/2)*d^(7/2))

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Rubi [A]  time = 0.976025, antiderivative size = 341, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{2 c x^2 \sqrt{a+b x} \left (-5 a^2 d^2+12 a b c d+b^2 c^2\right )}{3 b^2 d (c+d x)^{3/2} (b c-a d)^3}+\frac{\sqrt{a+b x} \left (d x (b c-a d) \left (-15 a^3 d^3+35 a^2 b c d^2-9 a b^2 c^2 d+5 b^3 c^3\right )+c \left (15 a^4 d^4-40 a^3 b c d^3+18 a^2 b^2 c^2 d^2-40 a b^3 c^3 d+15 b^4 c^4\right )\right )}{3 b^3 d^3 \sqrt{c+d x} (b c-a d)^4}-\frac{5 (a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{7/2} d^{7/2}}+\frac{2 a x^3 (11 b c-5 a d)}{3 b^2 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^2}+\frac{2 a x^4}{3 b (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

(2*a*x^4)/(3*b*(b*c - a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2)) + (2*a*(11*b*c - 5*a
*d)*x^3)/(3*b^2*(b*c - a*d)^2*Sqrt[a + b*x]*(c + d*x)^(3/2)) - (2*c*(b^2*c^2 + 1
2*a*b*c*d - 5*a^2*d^2)*x^2*Sqrt[a + b*x])/(3*b^2*d*(b*c - a*d)^3*(c + d*x)^(3/2)
) + (Sqrt[a + b*x]*(c*(15*b^4*c^4 - 40*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 - 40*a^3
*b*c*d^3 + 15*a^4*d^4) + d*(b*c - a*d)*(5*b^3*c^3 - 9*a*b^2*c^2*d + 35*a^2*b*c*d
^2 - 15*a^3*d^3)*x))/(3*b^3*d^3*(b*c - a*d)^4*Sqrt[c + d*x]) - (5*(b*c + a*d)*Ar
cTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(b^(7/2)*d^(7/2))

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Rubi in Sympy [A]  time = 92.2933, size = 342, normalized size = 1. \[ - \frac{2 a x^{4}}{3 b \left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )} - \frac{2 a x^{3} \left (5 a d - 11 b c\right )}{3 b^{2} \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )^{2}} - \frac{2 c x^{2} \sqrt{a + b x} \left (5 a^{2} d^{2} - 12 a b c d - b^{2} c^{2}\right )}{3 b^{2} d \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )^{3}} + \frac{16 \sqrt{a + b x} \left (\frac{3 c \left (15 a^{4} d^{4} - 40 a^{3} b c d^{3} + 18 a^{2} b^{2} c^{2} d^{2} - 40 a b^{3} c^{3} d + 15 b^{4} c^{4}\right )}{16} + \frac{3 d x \left (a d - b c\right ) \left (15 a^{3} d^{3} - 35 a^{2} b c d^{2} + 9 a b^{2} c^{2} d - 5 b^{3} c^{3}\right )}{16}\right )}{9 b^{3} d^{3} \sqrt{c + d x} \left (a d - b c\right )^{4}} - \frac{5 \left (a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{b^{\frac{7}{2}} d^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a*x**4/(3*b*(a + b*x)**(3/2)*(c + d*x)**(3/2)*(a*d - b*c)) - 2*a*x**3*(5*a*d
- 11*b*c)/(3*b**2*sqrt(a + b*x)*(c + d*x)**(3/2)*(a*d - b*c)**2) - 2*c*x**2*sqrt
(a + b*x)*(5*a**2*d**2 - 12*a*b*c*d - b**2*c**2)/(3*b**2*d*(c + d*x)**(3/2)*(a*d
- b*c)**3) + 16*sqrt(a + b*x)*(3*c*(15*a**4*d**4 - 40*a**3*b*c*d**3 + 18*a**2*b
**2*c**2*d**2 - 40*a*b**3*c**3*d + 15*b**4*c**4)/16 + 3*d*x*(a*d - b*c)*(15*a**3
*d**3 - 35*a**2*b*c*d**2 + 9*a*b**2*c**2*d - 5*b**3*c**3)/16)/(9*b**3*d**3*sqrt(
c + d*x)*(a*d - b*c)**4) - 5*(a*d + b*c)*atanh(sqrt(b)*sqrt(c + d*x)/(sqrt(d)*sq
rt(a + b*x)))/(b**(7/2)*d**(7/2))

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Mathematica [A]  time = 1.00763, size = 214, normalized size = 0.63 \[ \frac{1}{3} \sqrt{a+b x} \sqrt{c+d x} \left (\frac{2 a^5}{b^3 (a+b x)^2 (b c-a d)^3}+\frac{2 a^4 (7 a d-15 b c)}{b^3 (a+b x) (b c-a d)^4}+\frac{2 c^5}{d^3 (c+d x)^2 (a d-b c)^3}+\frac{2 c^4 (7 b c-15 a d)}{d^3 (c+d x) (b c-a d)^4}+\frac{3}{b^3 d^3}\right )-\frac{5 (a d+b c) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{2 b^{7/2} d^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(3/(b^3*d^3) + (2*a^5)/(b^3*(b*c - a*d)^3*(a + b*x)
^2) + (2*a^4*(-15*b*c + 7*a*d))/(b^3*(b*c - a*d)^4*(a + b*x)) + (2*c^5)/(d^3*(-(
b*c) + a*d)^3*(c + d*x)^2) + (2*c^4*(7*b*c - 15*a*d))/(d^3*(b*c - a*d)^4*(c + d*
x))))/3 - (5*(b*c + a*d)*Log[b*c + a*d + 2*b*d*x + 2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*
x]*Sqrt[c + d*x]])/(2*b^(7/2)*d^(7/2))

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Maple [B]  time = 0.052, size = 2748, normalized size = 8.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(80*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^5*b*c^3*d^3+15*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^5*c^7+15*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^7*d^7
+15*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^7*c^7+15*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^7*c^2*d^5-30*a^6*c^2*d^4*((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)*a^4*b^2*c^4*d^2+80*((b*x+a)*(d*x+c))^(1/2)*(b*d
)^(1/2)*a^3*b^3*c^5*d-6*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*x^4*a^4*b^2*d^6-6*((
b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*x^4*b^6*c^4*d^2-40*((b*x+a)*(d*x+c))^(1/2)*(b*
d)^(1/2)*x^3*a^5*b*d^6-40*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*x^3*b^6*c^5*d+15*l
n(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*b*c*d^6-135*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^5*b^2*c^2*d^5+105*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^3*c^3*d^4+105*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^4*c^4*d^3-135*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
^5*c^5*d^2+15*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*b^6*c^6*d-60*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*b*c^2*d^5-30*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^2*c^3*d^4+120*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^3*c^4*d^3-30*l
n(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^4*c^5*d^2-60*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^5*c^6*d-30*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^5*c^4*d^3-60*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^6*c^5*d^2-60*x*a^6*c*d^5*(
(b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-60*x*a*b^5*c^6*((b*x+a)*(d*x+c))^(1/2)*(b*d)^
(1/2)-45*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^3*c*d^6+30*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^4*c^2*d^5+30*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^5*c^3*d^4-45*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^6*c^4*d^3-60
*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^2*c*d^6-30*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^3*c^2*d^5+120*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^4*c^3*d^4+120*((b*x+a)*(d*x+c))^(1/2
)*(b*d)^(1/2)*x*a^2*b^4*c^5*d+24*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*x^4*a^3*b^3
*c*d^5-36*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*x^4*a^2*b^4*c^2*d^4+24*((b*x+a)*(d
*x+c))^(1/2)*(b*d)^(1/2)*x^4*a*b^5*c^3*d^3+96*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2
)*x^3*a^4*b^2*c*d^5-24*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*x^3*a^3*b^3*c^2*d^4-2
4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*x^3*a^2*b^4*c^3*d^3+96*((b*x+a)*(d*x+c))^(
1/2)*(b*d)^(1/2)*x^3*a*b^5*c^4*d^2+174*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*x^2*a
^4*b^2*c^2*d^4-96*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*x^2*a^3*b^3*c^3*d^3+174*((
b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*x^2*a^2*b^4*c^4*d^2+120*((b*x+a)*(d*x+c))^(1/2
)*(b*d)^(1/2)*x*a^5*b*c^2*d^4+36*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*x*a^4*b^2*c
^3*d^3+36*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*x*a^3*b^3*c^4*d^2-30*a^2*b^4*c^6*(
(b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+15*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^5*b^2*d^7+15*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^7*c^5*d^2+30*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^6*b*d^7+30
*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^7*c^6*d+30*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^7*c*d^6+30*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^6*c^7-45*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*b*c^3*d^4+30*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^2*c^4*d^3+30*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^3*c^5*d^2-45*l
n(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
^4*c^6*d-30*x^2*a^6*d^6*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-30*x^2*b^6*c^6*((b*x
+a)*(d*x+c))^(1/2)*(b*d)^(1/2))/((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)/b^3/d^3

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.98589, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/12*(4*(15*a^2*b^4*c^6 - 40*a^3*b^3*c^5*d + 18*a^4*b^2*c^4*d^2 - 40*a^5*b*c^3*
d^3 + 15*a^6*c^2*d^4 + 3*(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 + 4*(5*b^6*c^5*d - 12*a*b^5*c^4*d^2 + 3*a^2*b^4
*c^3*d^3 + 3*a^3*b^3*c^2*d^4 - 12*a^4*b^2*c*d^5 + 5*a^5*b*d^6)*x^3 + 3*(5*b^6*c^
6 - 29*a^2*b^4*c^4*d^2 + 16*a^3*b^3*c^3*d^3 - 29*a^4*b^2*c^2*d^4 + 5*a^6*d^6)*x^
2 + 6*(5*a*b^5*c^6 - 10*a^2*b^4*c^5*d - 3*a^3*b^3*c^4*d^2 - 3*a^4*b^2*c^3*d^3 -
10*a^5*b*c^2*d^4 + 5*a^6*c*d^5)*x)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 15*(a
^2*b^5*c^7 - 3*a^3*b^4*c^6*d + 2*a^4*b^3*c^5*d^2 + 2*a^5*b^2*c^4*d^3 - 3*a^6*b*c
^3*d^4 + a^7*c^2*d^5 + (b^7*c^5*d^2 - 3*a*b^6*c^4*d^3 + 2*a^2*b^5*c^3*d^4 + 2*a^
3*b^4*c^2*d^5 - 3*a^4*b^3*c*d^6 + a^5*b^2*d^7)*x^4 + 2*(b^7*c^6*d - 2*a*b^6*c^5*
d^2 - a^2*b^5*c^4*d^3 + 4*a^3*b^4*c^3*d^4 - a^4*b^3*c^2*d^5 - 2*a^5*b^2*c*d^6 +
a^6*b*d^7)*x^3 + (b^7*c^7 + a*b^6*c^6*d - 9*a^2*b^5*c^5*d^2 + 7*a^3*b^4*c^4*d^3
+ 7*a^4*b^3*c^3*d^4 - 9*a^5*b^2*c^2*d^5 + a^6*b*c*d^6 + a^7*d^7)*x^2 + 2*(a*b^6*
c^7 - 2*a^2*b^5*c^6*d - a^3*b^4*c^5*d^2 + 4*a^4*b^3*c^4*d^3 - a^5*b^2*c^3*d^4 -
2*a^6*b*c^2*d^5 + a^7*c*d^6)*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^7*c^6*d^3 - 4*a^3*b^6*c^5*d^4 + 6*a^4*b^5*c
^4*d^5 - 4*a^5*b^4*c^3*d^6 + a^6*b^3*c^2*d^7 + (b^9*c^4*d^5 - 4*a*b^8*c^3*d^6 +
6*a^2*b^7*c^2*d^7 - 4*a^3*b^6*c*d^8 + a^4*b^5*d^9)*x^4 + 2*(b^9*c^5*d^4 - 3*a*b^
8*c^4*d^5 + 2*a^2*b^7*c^3*d^6 + 2*a^3*b^6*c^2*d^7 - 3*a^4*b^5*c*d^8 + a^5*b^4*d^
9)*x^3 + (b^9*c^6*d^3 - 9*a^2*b^7*c^4*d^5 + 16*a^3*b^6*c^3*d^6 - 9*a^4*b^5*c^2*d
^7 + a^6*b^3*d^9)*x^2 + 2*(a*b^8*c^6*d^3 - 3*a^2*b^7*c^5*d^4 + 2*a^3*b^6*c^4*d^5
+ 2*a^4*b^5*c^3*d^6 - 3*a^5*b^4*c^2*d^7 + a^6*b^3*c*d^8)*x)*sqrt(b*d)), 1/6*(2*
(15*a^2*b^4*c^6 - 40*a^3*b^3*c^5*d + 18*a^4*b^2*c^4*d^2 - 40*a^5*b*c^3*d^3 + 15*
a^6*c^2*d^4 + 3*(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 + 4*(5*b^6*c^5*d - 12*a*b^5*c^4*d^2 + 3*a^2*b^4*c^3*d^3
+ 3*a^3*b^3*c^2*d^4 - 12*a^4*b^2*c*d^5 + 5*a^5*b*d^6)*x^3 + 3*(5*b^6*c^6 - 29*a^
2*b^4*c^4*d^2 + 16*a^3*b^3*c^3*d^3 - 29*a^4*b^2*c^2*d^4 + 5*a^6*d^6)*x^2 + 6*(5*
a*b^5*c^6 - 10*a^2*b^4*c^5*d - 3*a^3*b^3*c^4*d^2 - 3*a^4*b^2*c^3*d^3 - 10*a^5*b*
c^2*d^4 + 5*a^6*c*d^5)*x)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c) - 15*(a^2*b^5*c
^7 - 3*a^3*b^4*c^6*d + 2*a^4*b^3*c^5*d^2 + 2*a^5*b^2*c^4*d^3 - 3*a^6*b*c^3*d^4 +
a^7*c^2*d^5 + (b^7*c^5*d^2 - 3*a*b^6*c^4*d^3 + 2*a^2*b^5*c^3*d^4 + 2*a^3*b^4*c^
2*d^5 - 3*a^4*b^3*c*d^6 + a^5*b^2*d^7)*x^4 + 2*(b^7*c^6*d - 2*a*b^6*c^5*d^2 - a^
2*b^5*c^4*d^3 + 4*a^3*b^4*c^3*d^4 - a^4*b^3*c^2*d^5 - 2*a^5*b^2*c*d^6 + a^6*b*d^
7)*x^3 + (b^7*c^7 + a*b^6*c^6*d - 9*a^2*b^5*c^5*d^2 + 7*a^3*b^4*c^4*d^3 + 7*a^4*
b^3*c^3*d^4 - 9*a^5*b^2*c^2*d^5 + a^6*b*c*d^6 + a^7*d^7)*x^2 + 2*(a*b^6*c^7 - 2*
a^2*b^5*c^6*d - a^3*b^4*c^5*d^2 + 4*a^4*b^3*c^4*d^3 - a^5*b^2*c^3*d^4 - 2*a^6*b*
c^2*d^5 + a^7*c*d^6)*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^7*c^6*d^3 - 4*a^3*b^6*c^5*d^4 + 6*a^4*b^5*c^4*d^
5 - 4*a^5*b^4*c^3*d^6 + a^6*b^3*c^2*d^7 + (b^9*c^4*d^5 - 4*a*b^8*c^3*d^6 + 6*a^2
*b^7*c^2*d^7 - 4*a^3*b^6*c*d^8 + a^4*b^5*d^9)*x^4 + 2*(b^9*c^5*d^4 - 3*a*b^8*c^4
*d^5 + 2*a^2*b^7*c^3*d^6 + 2*a^3*b^6*c^2*d^7 - 3*a^4*b^5*c*d^8 + a^5*b^4*d^9)*x^
3 + (b^9*c^6*d^3 - 9*a^2*b^7*c^4*d^5 + 16*a^3*b^6*c^3*d^6 - 9*a^4*b^5*c^2*d^7 +
a^6*b^3*d^9)*x^2 + 2*(a*b^8*c^6*d^3 - 3*a^2*b^7*c^5*d^4 + 2*a^3*b^6*c^4*d^5 + 2*
a^4*b^5*c^3*d^6 - 3*a^5*b^4*c^2*d^7 + a^6*b^3*c*d^8)*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**5/(b*x+a)**(5/2)/(d*x+c)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.684266, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x