3.531 \(\int \frac{A+B x}{x^{5/2} (a+b x)^{5/2}} \, dx\)

Optimal. Leaf size=113 \[ \frac{16 \sqrt{a+b x} (2 A b-a B)}{3 a^4 \sqrt{x}}-\frac{8 (2 A b-a B)}{3 a^3 \sqrt{x} \sqrt{a+b x}}-\frac{2 (2 A b-a B)}{3 a^2 \sqrt{x} (a+b x)^{3/2}}-\frac{2 A}{3 a x^{3/2} (a+b x)^{3/2}} \]

[Out]

(-2*A)/(3*a*x^(3/2)*(a + b*x)^(3/2)) - (2*(2*A*b - a*B))/(3*a^2*Sqrt[x]*(a + b*x
)^(3/2)) - (8*(2*A*b - a*B))/(3*a^3*Sqrt[x]*Sqrt[a + b*x]) + (16*(2*A*b - a*B)*S
qrt[a + b*x])/(3*a^4*Sqrt[x])

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Rubi [A]  time = 0.130915, antiderivative size = 113, 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 \[ \frac{16 \sqrt{a+b x} (2 A b-a B)}{3 a^4 \sqrt{x}}-\frac{8 (2 A b-a B)}{3 a^3 \sqrt{x} \sqrt{a+b x}}-\frac{2 (2 A b-a B)}{3 a^2 \sqrt{x} (a+b x)^{3/2}}-\frac{2 A}{3 a x^{3/2} (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*A)/(3*a*x^(3/2)*(a + b*x)^(3/2)) - (2*(2*A*b - a*B))/(3*a^2*Sqrt[x]*(a + b*x
)^(3/2)) - (8*(2*A*b - a*B))/(3*a^3*Sqrt[x]*Sqrt[a + b*x]) + (16*(2*A*b - a*B)*S
qrt[a + b*x])/(3*a^4*Sqrt[x])

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Rubi in Sympy [A]  time = 11.5106, size = 105, normalized size = 0.93 \[ - \frac{2 A}{3 a x^{\frac{3}{2}} \left (a + b x\right )^{\frac{3}{2}}} - \frac{4 \left (A b - \frac{B a}{2}\right )}{3 a^{2} \sqrt{x} \left (a + b x\right )^{\frac{3}{2}}} - \frac{16 \left (A b - \frac{B a}{2}\right )}{3 a^{3} \sqrt{x} \sqrt{a + b x}} + \frac{32 \sqrt{a + b x} \left (A b - \frac{B a}{2}\right )}{3 a^{4} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/x**(5/2)/(b*x+a)**(5/2),x)

[Out]

-2*A/(3*a*x**(3/2)*(a + b*x)**(3/2)) - 4*(A*b - B*a/2)/(3*a**2*sqrt(x)*(a + b*x)
**(3/2)) - 16*(A*b - B*a/2)/(3*a**3*sqrt(x)*sqrt(a + b*x)) + 32*sqrt(a + b*x)*(A
*b - B*a/2)/(3*a**4*sqrt(x))

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Mathematica [A]  time = 0.0934232, size = 70, normalized size = 0.62 \[ -\frac{2 \left (a^3 (A+3 B x)-6 a^2 b x (A-2 B x)+8 a b^2 x^2 (B x-3 A)-16 A b^3 x^3\right )}{3 a^4 x^{3/2} (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(-16*A*b^3*x^3 - 6*a^2*b*x*(A - 2*B*x) + 8*a*b^2*x^2*(-3*A + B*x) + a^3*(A +
3*B*x)))/(3*a^4*x^(3/2)*(a + b*x)^(3/2))

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Maple [A]  time = 0.007, size = 76, normalized size = 0.7 \[ -{\frac{-32\,A{b}^{3}{x}^{3}+16\,B{x}^{3}a{b}^{2}-48\,aA{b}^{2}{x}^{2}+24\,B{x}^{2}{a}^{2}b-12\,{a}^{2}Abx+6\,{a}^{3}Bx+2\,A{a}^{3}}{3\,{a}^{4}}{x}^{-{\frac{3}{2}}} \left ( bx+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*(-16*A*b^3*x^3+8*B*a*b^2*x^3-24*A*a*b^2*x^2+12*B*a^2*b*x^2-6*A*a^2*b*x+3*B*
a^3*x+A*a^3)/x^(3/2)/(b*x+a)^(3/2)/a^4

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Maxima [A]  time = 1.33498, size = 176, normalized size = 1.56 \[ \frac{2 \, B x}{3 \,{\left (b x^{2} + a x\right )}^{\frac{3}{2}} a} - \frac{16 \, B b x}{3 \, \sqrt{b x^{2} + a x} a^{3}} - \frac{4 \, A b x}{3 \,{\left (b x^{2} + a x\right )}^{\frac{3}{2}} a^{2}} + \frac{32 \, A b^{2} x}{3 \, \sqrt{b x^{2} + a x} a^{4}} - \frac{8 \, B}{3 \, \sqrt{b x^{2} + a x} a^{2}} - \frac{2 \, A}{3 \,{\left (b x^{2} + a x\right )}^{\frac{3}{2}} a} + \frac{16 \, A b}{3 \, \sqrt{b x^{2} + a x} a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*B*x/((b*x^2 + a*x)^(3/2)*a) - 16/3*B*b*x/(sqrt(b*x^2 + a*x)*a^3) - 4/3*A*b*x
/((b*x^2 + a*x)^(3/2)*a^2) + 32/3*A*b^2*x/(sqrt(b*x^2 + a*x)*a^4) - 8/3*B/(sqrt(
b*x^2 + a*x)*a^2) - 2/3*A/((b*x^2 + a*x)^(3/2)*a) + 16/3*A*b/(sqrt(b*x^2 + a*x)*
a^3)

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Fricas [A]  time = 0.234425, size = 117, normalized size = 1.04 \[ -\frac{2 \,{\left (A a^{3} + 8 \,{\left (B a b^{2} - 2 \, A b^{3}\right )} x^{3} + 12 \,{\left (B a^{2} b - 2 \, A a b^{2}\right )} x^{2} + 3 \,{\left (B a^{3} - 2 \, A a^{2} b\right )} x\right )}}{3 \,{\left (a^{4} b x^{2} + a^{5} x\right )} \sqrt{b x + a} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*(A*a^3 + 8*(B*a*b^2 - 2*A*b^3)*x^3 + 12*(B*a^2*b - 2*A*a*b^2)*x^2 + 3*(B*a^
3 - 2*A*a^2*b)*x)/((a^4*b*x^2 + a^5*x)*sqrt(b*x + a)*sqrt(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((B*x+A)/x**(5/2)/(b*x+a)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.280413, size = 409, normalized size = 3.62 \[ \frac{\sqrt{b x + a}{\left (\frac{{\left (3 \, B a^{4} b^{3}{\left | b \right |} - 8 \, A a^{3} b^{4}{\left | b \right |}\right )}{\left (b x + a\right )}}{a^{2} b^{6}} - \frac{3 \,{\left (B a^{5} b^{3}{\left | b \right |} - 3 \, A a^{4} b^{4}{\left | b \right |}\right )}}{a^{2} b^{6}}\right )}}{48 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{3}{2}}} - \frac{4 \,{\left (3 \, B a{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{4} b^{\frac{5}{2}} + 12 \, B a^{2}{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac{7}{2}} - 6 \, A{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{4} b^{\frac{7}{2}} + 5 \, B a^{3} b^{\frac{9}{2}} - 18 \, A a{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac{9}{2}} - 8 \, A a^{2} b^{\frac{11}{2}}\right )}}{3 \,{\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}^{3} a^{3}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*sqrt(b*x + a)*((3*B*a^4*b^3*abs(b) - 8*A*a^3*b^4*abs(b))*(b*x + a)/(a^2*b^6
) - 3*(B*a^5*b^3*abs(b) - 3*A*a^4*b^4*abs(b))/(a^2*b^6))/((b*x + a)*b - a*b)^(3/
2) - 4/3*(3*B*a*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^4*b^(5/2) + 12
*B*a^2*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2*b^(7/2) - 6*A*(sqrt(b
*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^4*b^(7/2) + 5*B*a^3*b^(9/2) - 18*A*a*
(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2*b^(9/2) - 8*A*a^2*b^(11/2))/
(((sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2 + a*b)^3*a^3*abs(b))