3.583 \(\int \frac{1}{x^{7/2} (a+b x)^{3/2}} \, dx\)

Optimal. Leaf size=87 \[ -\frac{32 b^2 \sqrt{a+b x}}{5 a^4 \sqrt{x}}+\frac{16 b \sqrt{a+b x}}{5 a^3 x^{3/2}}-\frac{12 \sqrt{a+b x}}{5 a^2 x^{5/2}}+\frac{2}{a x^{5/2} \sqrt{a+b x}} \]

[Out]

2/(a*x^(5/2)*Sqrt[a + b*x]) - (12*Sqrt[a + b*x])/(5*a^2*x^(5/2)) + (16*b*Sqrt[a
+ b*x])/(5*a^3*x^(3/2)) - (32*b^2*Sqrt[a + b*x])/(5*a^4*Sqrt[x])

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Rubi [A]  time = 0.0609161, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{32 b^2 \sqrt{a+b x}}{5 a^4 \sqrt{x}}+\frac{16 b \sqrt{a+b x}}{5 a^3 x^{3/2}}-\frac{12 \sqrt{a+b x}}{5 a^2 x^{5/2}}+\frac{2}{a x^{5/2} \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(7/2)*(a + b*x)^(3/2)),x]

[Out]

2/(a*x^(5/2)*Sqrt[a + b*x]) - (12*Sqrt[a + b*x])/(5*a^2*x^(5/2)) + (16*b*Sqrt[a
+ b*x])/(5*a^3*x^(3/2)) - (32*b^2*Sqrt[a + b*x])/(5*a^4*Sqrt[x])

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Rubi in Sympy [A]  time = 9.45717, size = 82, normalized size = 0.94 \[ \frac{2}{a x^{\frac{5}{2}} \sqrt{a + b x}} - \frac{12 \sqrt{a + b x}}{5 a^{2} x^{\frac{5}{2}}} + \frac{16 b \sqrt{a + b x}}{5 a^{3} x^{\frac{3}{2}}} - \frac{32 b^{2} \sqrt{a + b x}}{5 a^{4} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**(7/2)/(b*x+a)**(3/2),x)

[Out]

2/(a*x**(5/2)*sqrt(a + b*x)) - 12*sqrt(a + b*x)/(5*a**2*x**(5/2)) + 16*b*sqrt(a
+ b*x)/(5*a**3*x**(3/2)) - 32*b**2*sqrt(a + b*x)/(5*a**4*sqrt(x))

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Mathematica [A]  time = 0.0317292, size = 49, normalized size = 0.56 \[ -\frac{2 \left (a^3-2 a^2 b x+8 a b^2 x^2+16 b^3 x^3\right )}{5 a^4 x^{5/2} \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^(7/2)*(a + b*x)^(3/2)),x]

[Out]

(-2*(a^3 - 2*a^2*b*x + 8*a*b^2*x^2 + 16*b^3*x^3))/(5*a^4*x^(5/2)*Sqrt[a + b*x])

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Maple [A]  time = 0.006, size = 44, normalized size = 0.5 \[ -{\frac{32\,{b}^{3}{x}^{3}+16\,a{b}^{2}{x}^{2}-4\,{a}^{2}bx+2\,{a}^{3}}{5\,{a}^{4}}{x}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(7/2)/(b*x+a)^(3/2),x)

[Out]

-2/5*(16*b^3*x^3+8*a*b^2*x^2-2*a^2*b*x+a^3)/x^(5/2)/(b*x+a)^(1/2)/a^4

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Maxima [A]  time = 1.36313, size = 86, normalized size = 0.99 \[ -\frac{2 \, b^{3} \sqrt{x}}{\sqrt{b x + a} a^{4}} - \frac{2 \,{\left (\frac{15 \, \sqrt{b x + a} b^{2}}{\sqrt{x}} - \frac{5 \,{\left (b x + a\right )}^{\frac{3}{2}} b}{x^{\frac{3}{2}}} + \frac{{\left (b x + a\right )}^{\frac{5}{2}}}{x^{\frac{5}{2}}}\right )}}{5 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(3/2)*x^(7/2)),x, algorithm="maxima")

[Out]

-2*b^3*sqrt(x)/(sqrt(b*x + a)*a^4) - 2/5*(15*sqrt(b*x + a)*b^2/sqrt(x) - 5*(b*x
+ a)^(3/2)*b/x^(3/2) + (b*x + a)^(5/2)/x^(5/2))/a^4

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Fricas [A]  time = 0.21008, size = 58, normalized size = 0.67 \[ -\frac{2 \,{\left (16 \, b^{3} x^{3} + 8 \, a b^{2} x^{2} - 2 \, a^{2} b x + a^{3}\right )}}{5 \, \sqrt{b x + a} a^{4} x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(3/2)*x^(7/2)),x, algorithm="fricas")

[Out]

-2/5*(16*b^3*x^3 + 8*a*b^2*x^2 - 2*a^2*b*x + a^3)/(sqrt(b*x + a)*a^4*x^(5/2))

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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(1/x**(7/2)/(b*x+a)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.216414, size = 147, normalized size = 1.69 \[ -\frac{4 \, b^{\frac{9}{2}}}{{\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )} a^{3}{\left | b \right |}} + \frac{{\left (\frac{15 \, a^{4}}{b} +{\left (\frac{11 \,{\left (b x + a\right )} a^{2}}{b} - \frac{25 \, a^{3}}{b}\right )}{\left (b x + a\right )}\right )} \sqrt{b x + a}}{40 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(3/2)*x^(7/2)),x, algorithm="giac")

[Out]

-4*b^(9/2)/(((sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2 + a*b)*a^3*abs(
b)) + 1/40*(15*a^4/b + (11*(b*x + a)*a^2/b - 25*a^3/b)*(b*x + a))*sqrt(b*x + a)/
((b*x + a)*b - a*b)^(5/2)