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

Optimal. Leaf size=158 \[ -\frac{2 c}{3 d (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}-\frac{16 d \sqrt{a+b x} (a d+b c)}{3 \sqrt{c+d x} (b c-a d)^4}-\frac{8 (a d+b c)}{3 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^3}+\frac{2 (a d+b c)}{3 d (a+b x)^{3/2} \sqrt{c+d x} (b c-a d)^2} \]

[Out]

(-2*c)/(3*d*(b*c - a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2)) + (2*(b*c + a*d))/(3*d*
(b*c - a*d)^2*(a + b*x)^(3/2)*Sqrt[c + d*x]) - (8*(b*c + a*d))/(3*(b*c - a*d)^3*
Sqrt[a + b*x]*Sqrt[c + d*x]) - (16*d*(b*c + a*d)*Sqrt[a + b*x])/(3*(b*c - a*d)^4
*Sqrt[c + d*x])

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Rubi [A]  time = 0.202901, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ -\frac{2 c}{3 d (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}-\frac{16 d \sqrt{a+b x} (a d+b c)}{3 \sqrt{c+d x} (b c-a d)^4}-\frac{8 (a d+b c)}{3 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^3}+\frac{2 (a d+b c)}{3 d (a+b x)^{3/2} \sqrt{c+d x} (b c-a d)^2} \]

Antiderivative was successfully verified.

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

[Out]

(-2*c)/(3*d*(b*c - a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2)) + (2*(b*c + a*d))/(3*d*
(b*c - a*d)^2*(a + b*x)^(3/2)*Sqrt[c + d*x]) - (8*(b*c + a*d))/(3*(b*c - a*d)^3*
Sqrt[a + b*x]*Sqrt[c + d*x]) - (16*d*(b*c + a*d)*Sqrt[a + b*x])/(3*(b*c - a*d)^4
*Sqrt[c + d*x])

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Rubi in Sympy [A]  time = 24.1799, size = 143, normalized size = 0.91 \[ - \frac{16 b \sqrt{c + d x} \left (a d + b c\right )}{3 \sqrt{a + b x} \left (a d - b c\right )^{4}} - \frac{8 b \sqrt{c + d x} \left (a d + b c\right )}{3 d \left (a + b x\right )^{\frac{3}{2}} \left (a d - b c\right )^{3}} + \frac{2 c}{3 d \left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )} - \frac{2 \left (a d + b c\right )}{d \left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-16*b*sqrt(c + d*x)*(a*d + b*c)/(3*sqrt(a + b*x)*(a*d - b*c)**4) - 8*b*sqrt(c +
d*x)*(a*d + b*c)/(3*d*(a + b*x)**(3/2)*(a*d - b*c)**3) + 2*c/(3*d*(a + b*x)**(3/
2)*(c + d*x)**(3/2)*(a*d - b*c)) - 2*(a*d + b*c)/(d*(a + b*x)**(3/2)*sqrt(c + d*
x)*(a*d - b*c)**2)

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Mathematica [A]  time = 0.198246, size = 128, normalized size = 0.81 \[ \sqrt{a+b x} \sqrt{c+d x} \left (-\frac{2 b (5 a d+3 b c)}{3 (a+b x) (b c-a d)^4}+\frac{2 a b}{3 (a+b x)^2 (b c-a d)^3}-\frac{2 d (3 a d+5 b c)}{3 (c+d x) (b c-a d)^4}-\frac{2 c d}{3 (c+d x)^2 (b c-a d)^3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.012, size = 198, normalized size = 1.3 \[ -{\frac{16\,a{b}^{2}{d}^{3}{x}^{3}+16\,{b}^{3}c{d}^{2}{x}^{3}+24\,{a}^{2}b{d}^{3}{x}^{2}+48\,a{b}^{2}c{d}^{2}{x}^{2}+24\,{b}^{3}{c}^{2}d{x}^{2}+6\,{a}^{3}{d}^{3}x+42\,{a}^{2}bc{d}^{2}x+42\,a{b}^{2}{c}^{2}dx+6\,{b}^{3}{c}^{3}x+4\,{a}^{3}c{d}^{2}+24\,{a}^{2}b{c}^{2}d+4\,a{b}^{2}{c}^{3}}{3\,{a}^{4}{d}^{4}-12\,{a}^{3}bc{d}^{3}+18\,{a}^{2}{c}^{2}{b}^{2}{d}^{2}-12\,a{b}^{3}{c}^{3}d+3\,{b}^{4}{c}^{4}} \left ( bx+a \right ) ^{-{\frac{3}{2}}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*(8*a*b^2*d^3*x^3+8*b^3*c*d^2*x^3+12*a^2*b*d^3*x^2+24*a*b^2*c*d^2*x^2+12*b^3
*c^2*d*x^2+3*a^3*d^3*x+21*a^2*b*c*d^2*x+21*a*b^2*c^2*d*x+3*b^3*c^3*x+2*a^3*c*d^2
+12*a^2*b*c^2*d+2*a*b^2*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)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.769265, size = 632, normalized size = 4. \[ -\frac{2 \,{\left (2 \, a b^{2} c^{3} + 12 \, a^{2} b c^{2} d + 2 \, a^{3} c d^{2} + 8 \,{\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{3} + 12 \,{\left (b^{3} c^{2} d + 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} + 3 \,{\left (b^{3} c^{3} + 7 \, a b^{2} c^{2} d + 7 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{3 \,{\left (a^{2} b^{4} c^{6} - 4 \, a^{3} b^{3} c^{5} d + 6 \, a^{4} b^{2} c^{4} d^{2} - 4 \, a^{5} b c^{3} d^{3} + a^{6} c^{2} d^{4} +{\left (b^{6} c^{4} d^{2} - 4 \, a b^{5} c^{3} d^{3} + 6 \, a^{2} b^{4} c^{2} d^{4} - 4 \, a^{3} b^{3} c d^{5} + a^{4} b^{2} d^{6}\right )} x^{4} + 2 \,{\left (b^{6} c^{5} d - 3 \, a b^{5} c^{4} d^{2} + 2 \, a^{2} b^{4} c^{3} d^{3} + 2 \, a^{3} b^{3} c^{2} d^{4} - 3 \, a^{4} b^{2} c d^{5} + a^{5} b d^{6}\right )} x^{3} +{\left (b^{6} c^{6} - 9 \, a^{2} b^{4} c^{4} d^{2} + 16 \, a^{3} b^{3} c^{3} d^{3} - 9 \, a^{4} b^{2} c^{2} d^{4} + a^{6} d^{6}\right )} x^{2} + 2 \,{\left (a b^{5} c^{6} - 3 \, a^{2} b^{4} c^{5} d + 2 \, a^{3} b^{3} c^{4} d^{2} + 2 \, a^{4} b^{2} c^{3} d^{3} - 3 \, a^{5} b c^{2} d^{4} + a^{6} c d^{5}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*(2*a*b^2*c^3 + 12*a^2*b*c^2*d + 2*a^3*c*d^2 + 8*(b^3*c*d^2 + a*b^2*d^3)*x^3
+ 12*(b^3*c^2*d + 2*a*b^2*c*d^2 + a^2*b*d^3)*x^2 + 3*(b^3*c^3 + 7*a*b^2*c^2*d +
7*a^2*b*c*d^2 + a^3*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)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.49531, size = 933, normalized size = 5.91 \[ \frac{\frac{\sqrt{b x + a}{\left (\frac{{\left (5 \, b^{8} c^{4} d^{3}{\left | b \right |} - 12 \, a b^{7} c^{3} d^{4}{\left | b \right |} + 6 \, a^{2} b^{6} c^{2} d^{5}{\left | b \right |} + 4 \, a^{3} b^{5} c d^{6}{\left | b \right |} - 3 \, a^{4} b^{4} d^{7}{\left | b \right |}\right )}{\left (b x + a\right )}}{b^{8} c^{2} d^{4} - 2 \, a b^{7} c d^{5} + a^{2} b^{6} d^{6}} + \frac{3 \,{\left (2 \, b^{9} c^{5} d^{2}{\left | b \right |} - 7 \, a b^{8} c^{4} d^{3}{\left | b \right |} + 8 \, a^{2} b^{7} c^{3} d^{4}{\left | b \right |} - 2 \, a^{3} b^{6} c^{2} d^{5}{\left | b \right |} - 2 \, a^{4} b^{5} c d^{6}{\left | b \right |} + a^{5} b^{4} d^{7}{\left | b \right |}\right )}}{b^{8} c^{2} d^{4} - 2 \, a b^{7} c d^{5} + a^{2} b^{6} d^{6}}\right )}}{{\left (b^{2} c +{\left (b x + a\right )} b d - a b d\right )}^{\frac{3}{2}}} - \frac{16 \,{\left (3 \, \sqrt{b d} b^{8} c^{3} - \sqrt{b d} a b^{7} c^{2} d - 7 \, \sqrt{b d} a^{2} b^{6} c d^{2} + 5 \, \sqrt{b d} a^{3} b^{5} d^{3} - 6 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} b^{6} c^{2} - 6 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{5} c d + 12 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b^{4} d^{2} + 3 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{4} b^{4} c + 3 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{4} a b^{3} d\right )}}{{\left (b^{3} c^{3}{\left | b \right |} - 3 \, a b^{2} c^{2} d{\left | b \right |} + 3 \, a^{2} b c d^{2}{\left | b \right |} - a^{3} d^{3}{\left | b \right |}\right )}{\left (b^{2} c - a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}^{3}}}{12 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(sqrt(b*x + a)*((5*b^8*c^4*d^3*abs(b) - 12*a*b^7*c^3*d^4*abs(b) + 6*a^2*b^6
*c^2*d^5*abs(b) + 4*a^3*b^5*c*d^6*abs(b) - 3*a^4*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) + 3*(2*b^9*c^5*d^2*abs(b) - 7*a*b^8*c^4*d
^3*abs(b) + 8*a^2*b^7*c^3*d^4*abs(b) - 2*a^3*b^6*c^2*d^5*abs(b) - 2*a^4*b^5*c*d^
6*abs(b) + a^5*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) - 16*(3*sqrt(b*d)*b^8*c^3 - sqrt(b*d)*a*b^7*c^
2*d - 7*sqrt(b*d)*a^2*b^6*c*d^2 + 5*sqrt(b*d)*a^3*b^5*d^3 - 6*sqrt(b*d)*(sqrt(b*
d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^6*c^2 - 6*sqrt(b*d)*
(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^5*c*d + 12
*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^2
*b^4*d^2 + 3*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a
*b*d))^4*b^4*c + 3*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b
*d - a*b*d))^4*a*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))/b