3.1519 \(\int \frac{1}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx\)

Optimal. Leaf size=135 \[ \frac{32 b d^2 \sqrt{a+b x}}{3 \sqrt{c+d x} (b c-a d)^4}+\frac{16 d^2 \sqrt{a+b x}}{3 (c+d x)^{3/2} (b c-a d)^3}+\frac{4 d}{\sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^2}-\frac{2}{3 (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)} \]

[Out]

-2/(3*(b*c - a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2)) + (4*d)/((b*c - a*d)^2*Sqrt[a
+ b*x]*(c + d*x)^(3/2)) + (16*d^2*Sqrt[a + b*x])/(3*(b*c - a*d)^3*(c + d*x)^(3/
2)) + (32*b*d^2*Sqrt[a + b*x])/(3*(b*c - a*d)^4*Sqrt[c + d*x])

_______________________________________________________________________________________

Rubi [A]  time = 0.118135, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{32 b d^2 \sqrt{a+b x}}{3 \sqrt{c+d x} (b c-a d)^4}+\frac{16 d^2 \sqrt{a+b x}}{3 (c+d x)^{3/2} (b c-a d)^3}+\frac{4 d}{\sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^2}-\frac{2}{3 (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)^(5/2)*(c + d*x)^(5/2)),x]

[Out]

-2/(3*(b*c - a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2)) + (4*d)/((b*c - a*d)^2*Sqrt[a
+ b*x]*(c + d*x)^(3/2)) + (16*d^2*Sqrt[a + b*x])/(3*(b*c - a*d)^3*(c + d*x)^(3/
2)) + (32*b*d^2*Sqrt[a + b*x])/(3*(b*c - a*d)^4*Sqrt[c + d*x])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 21.1596, size = 121, normalized size = 0.9 \[ \frac{32 b d^{2} \sqrt{a + b x}}{3 \sqrt{c + d x} \left (a d - b c\right )^{4}} - \frac{16 d^{2} \sqrt{a + b x}}{3 \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )^{3}} + \frac{4 d}{\sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )^{2}} + \frac{2}{3 \left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**(5/2)/(d*x+c)**(5/2),x)

[Out]

32*b*d**2*sqrt(a + b*x)/(3*sqrt(c + d*x)*(a*d - b*c)**4) - 16*d**2*sqrt(a + b*x)
/(3*(c + d*x)**(3/2)*(a*d - b*c)**3) + 4*d/(sqrt(a + b*x)*(c + d*x)**(3/2)*(a*d
- b*c)**2) + 2/(3*(a + b*x)**(3/2)*(c + d*x)**(3/2)*(a*d - b*c))

_______________________________________________________________________________________

Mathematica [A]  time = 0.162794, size = 118, normalized size = 0.87 \[ \sqrt{a+b x} \sqrt{c+d x} \left (\frac{16 b^2 d}{3 (a+b x) (b c-a d)^4}-\frac{2 b^2}{3 (a+b x)^2 (b c-a d)^3}+\frac{16 b d^2}{3 (c+d x) (b c-a d)^4}+\frac{2 d^2}{3 (c+d x)^2 (b c-a d)^3}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)^(5/2)*(c + d*x)^(5/2)),x]

[Out]

Sqrt[a + b*x]*Sqrt[c + d*x]*((-2*b^2)/(3*(b*c - a*d)^3*(a + b*x)^2) + (16*b^2*d)
/(3*(b*c - a*d)^4*(a + b*x)) + (2*d^2)/(3*(b*c - a*d)^3*(c + d*x)^2) + (16*b*d^2
)/(3*(b*c - a*d)^4*(c + d*x)))

_______________________________________________________________________________________

Maple [A]  time = 0.012, size = 169, normalized size = 1.3 \[ -{\frac{-32\,{b}^{3}{d}^{3}{x}^{3}-48\,a{b}^{2}{d}^{3}{x}^{2}-48\,{b}^{3}c{d}^{2}{x}^{2}-12\,{a}^{2}b{d}^{3}x-72\,a{b}^{2}c{d}^{2}x-12\,{b}^{3}{c}^{2}dx+2\,{a}^{3}{d}^{3}-18\,{a}^{2}bc{d}^{2}-18\,a{b}^{2}{c}^{2}d+2\,{b}^{3}{c}^{3}}{3\,{d}^{4}{a}^{4}-12\,b{d}^{3}c{a}^{3}+18\,{b}^{2}{d}^{2}{c}^{2}{a}^{2}-12\,{b}^{3}d{c}^{3}a+3\,{b}^{4}{c}^{4}} \left ( bx+a \right ) ^{-{\frac{3}{2}}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*(-16*b^3*d^3*x^3-24*a*b^2*d^3*x^2-24*b^3*c*d^2*x^2-6*a^2*b*d^3*x-36*a*b^2*c
*d^2*x-6*b^3*c^2*d*x+a^3*d^3-9*a^2*b*c*d^2-9*a*b^2*c^2*d+b^3*c^3)/(b*x+a)^(3/2)/
(d*x+c)^(3/2)/(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.731149, size = 603, normalized size = 4.47 \[ \frac{2 \,{\left (16 \, b^{3} d^{3} x^{3} - b^{3} c^{3} + 9 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - a^{3} d^{3} + 24 \,{\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 6 \,{\left (b^{3} c^{2} d + 6 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{3 \,{\left (a^{2} b^{4} c^{6} - 4 \, a^{3} b^{3} c^{5} d + 6 \, a^{4} b^{2} c^{4} d^{2} - 4 \, a^{5} b c^{3} d^{3} + a^{6} c^{2} d^{4} +{\left (b^{6} c^{4} d^{2} - 4 \, a b^{5} c^{3} d^{3} + 6 \, a^{2} b^{4} c^{2} d^{4} - 4 \, a^{3} b^{3} c d^{5} + a^{4} b^{2} d^{6}\right )} x^{4} + 2 \,{\left (b^{6} c^{5} d - 3 \, a b^{5} c^{4} d^{2} + 2 \, a^{2} b^{4} c^{3} d^{3} + 2 \, a^{3} b^{3} c^{2} d^{4} - 3 \, a^{4} b^{2} c d^{5} + a^{5} b d^{6}\right )} x^{3} +{\left (b^{6} c^{6} - 9 \, a^{2} b^{4} c^{4} d^{2} + 16 \, a^{3} b^{3} c^{3} d^{3} - 9 \, a^{4} b^{2} c^{2} d^{4} + a^{6} d^{6}\right )} x^{2} + 2 \,{\left (a b^{5} c^{6} - 3 \, a^{2} b^{4} c^{5} d + 2 \, a^{3} b^{3} c^{4} d^{2} + 2 \, a^{4} b^{2} c^{3} d^{3} - 3 \, a^{5} b c^{2} d^{4} + a^{6} c d^{5}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(16*b^3*d^3*x^3 - b^3*c^3 + 9*a*b^2*c^2*d + 9*a^2*b*c*d^2 - a^3*d^3 + 24*(b^
3*c*d^2 + a*b^2*d^3)*x^2 + 6*(b^3*c^2*d + 6*a*b^2*c*d^2 + a^2*b*d^3)*x)*sqrt(b*x
+ a)*sqrt(d*x + c)/(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)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x\right )^{\frac{5}{2}} \left (c + d x\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x+a)**(5/2)/(d*x+c)**(5/2),x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.406717, size = 718, normalized size = 5.32 \[ -\frac{\sqrt{b x + a}{\left (\frac{8 \,{\left (b^{7} c^{3} d^{4}{\left | b \right |} - 3 \, a b^{6} c^{2} d^{5}{\left | b \right |} + 3 \, a^{2} b^{5} c d^{6}{\left | b \right |} - a^{3} b^{4} d^{7}{\left | b \right |}\right )}{\left (b x + a\right )}}{b^{8} c^{2} d^{4} - 2 \, a b^{7} c d^{5} + a^{2} b^{6} d^{6}} + \frac{9 \,{\left (b^{8} c^{4} d^{3}{\left | b \right |} - 4 \, a b^{7} c^{3} d^{4}{\left | b \right |} + 6 \, a^{2} b^{6} c^{2} d^{5}{\left | b \right |} - 4 \, a^{3} b^{5} c d^{6}{\left | b \right |} + a^{4} b^{4} d^{7}{\left | b \right |}\right )}}{b^{8} c^{2} d^{4} - 2 \, a b^{7} c d^{5} + a^{2} b^{6} d^{6}}\right )}}{24 \,{\left (b^{2} c +{\left (b x + a\right )} b d - a b d\right )}^{\frac{3}{2}}} + \frac{8 \,{\left (4 \, \sqrt{b d} b^{7} c^{2} d - 8 \, \sqrt{b d} a b^{6} c d^{2} + 4 \, \sqrt{b d} a^{2} b^{5} d^{3} - 9 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} b^{5} c d + 9 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{4} d^{2} + 3 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{4} b^{3} d\right )}}{3 \,{\left (b^{3} c^{3}{\left | b \right |} - 3 \, a b^{2} c^{2} d{\left | b \right |} + 3 \, a^{2} b c d^{2}{\left | b \right |} - a^{3} d^{3}{\left | b \right |}\right )}{\left (b^{2} c - a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/24*sqrt(b*x + a)*(8*(b^7*c^3*d^4*abs(b) - 3*a*b^6*c^2*d^5*abs(b) + 3*a^2*b^5*
c*d^6*abs(b) - a^3*b^4*d^7*abs(b))*(b*x + a)/(b^8*c^2*d^4 - 2*a*b^7*c*d^5 + a^2*
b^6*d^6) + 9*(b^8*c^4*d^3*abs(b) - 4*a*b^7*c^3*d^4*abs(b) + 6*a^2*b^6*c^2*d^5*ab
s(b) - 4*a^3*b^5*c*d^6*abs(b) + a^4*b^4*d^7*abs(b))/(b^8*c^2*d^4 - 2*a*b^7*c*d^5
+ a^2*b^6*d^6))/(b^2*c + (b*x + a)*b*d - a*b*d)^(3/2) + 8/3*(4*sqrt(b*d)*b^7*c^
2*d - 8*sqrt(b*d)*a*b^6*c*d^2 + 4*sqrt(b*d)*a^2*b^5*d^3 - 9*sqrt(b*d)*(sqrt(b*d)
*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^5*c*d + 9*sqrt(b*d)*(s
qrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^4*d^2 + 3*sq
rt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^3*d)
/((b^3*c^3*abs(b) - 3*a*b^2*c^2*d*abs(b) + 3*a^2*b*c*d^2*abs(b) - a^3*d^3*abs(b)
)*(b^2*c - a*b*d - (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d
))^2)^3)