3.2823 \(\int \frac{1}{\left (\frac{c}{(a+b x)^2}\right )^{5/2}} \, dx\)

Optimal. Leaf size=30 \[ \frac{(a+b x)^5}{6 b c^2 \sqrt{\frac{c}{(a+b x)^2}}} \]

[Out]

(a + b*x)^5/(6*b*c^2*Sqrt[c/(a + b*x)^2])

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Rubi [A]  time = 0.0260773, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{(a+b x)^5}{6 b c^2 \sqrt{\frac{c}{(a+b x)^2}}} \]

Antiderivative was successfully verified.

[In]  Int[(c/(a + b*x)^2)^(-5/2),x]

[Out]

(a + b*x)^5/(6*b*c^2*Sqrt[c/(a + b*x)^2])

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Rubi in Sympy [A]  time = 2.55311, size = 24, normalized size = 0.8 \[ \frac{\sqrt{\frac{c}{\left (a + b x\right )^{2}}} \left (a + b x\right )^{7}}{6 b c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(c/(a + b*x)**2)*(a + b*x)**7/(6*b*c**3)

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Mathematica [A]  time = 0.0289677, size = 25, normalized size = 0.83 \[ \frac{a+b x}{6 b \left (\frac{c}{(a+b x)^2}\right )^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(a + b*x)/(6*b*(c/(a + b*x)^2)^(5/2))

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Maple [B]  time = 0.005, size = 73, normalized size = 2.4 \[{\frac{x \left ({b}^{5}{x}^{5}+6\,a{b}^{4}{x}^{4}+15\,{a}^{2}{b}^{3}{x}^{3}+20\,{a}^{3}{b}^{2}{x}^{2}+15\,{a}^{4}bx+6\,{a}^{5} \right ) }{6\, \left ( bx+a \right ) ^{5}} \left ({\frac{c}{ \left ( bx+a \right ) ^{2}}} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c/(b*x+a)^2)^(5/2),x)

[Out]

1/6*x*(b^5*x^5+6*a*b^4*x^4+15*a^2*b^3*x^3+20*a^3*b^2*x^2+15*a^4*b*x+6*a^5)/(b*x+
a)^5/(c/(b*x+a)^2)^(5/2)

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Maxima [A]  time = 1.40529, size = 80, normalized size = 2.67 \[ \frac{b^{5} x^{6} + 6 \, a b^{4} x^{5} + 15 \, a^{2} b^{3} x^{4} + 20 \, a^{3} b^{2} x^{3} + 15 \, a^{4} b x^{2} + 6 \, a^{5} x}{6 \, c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(b^5*x^6 + 6*a*b^4*x^5 + 15*a^2*b^3*x^4 + 20*a^3*b^2*x^3 + 15*a^4*b*x^2 + 6*
a^5*x)/c^(5/2)

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Fricas [A]  time = 0.218509, size = 124, normalized size = 4.13 \[ \frac{{\left (b^{6} x^{7} + 7 \, a b^{5} x^{6} + 21 \, a^{2} b^{4} x^{5} + 35 \, a^{3} b^{3} x^{4} + 35 \, a^{4} b^{2} x^{3} + 21 \, a^{5} b x^{2} + 6 \, a^{6} x\right )} \sqrt{\frac{c}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{6 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(b^6*x^7 + 7*a*b^5*x^6 + 21*a^2*b^4*x^5 + 35*a^3*b^3*x^4 + 35*a^4*b^2*x^3 +
21*a^5*b*x^2 + 6*a^6*x)*sqrt(c/(b^2*x^2 + 2*a*b*x + a^2))/c^3

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((c/(a + b*x)**2)**(-5/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c/(b*x + a)^2)^(-5/2), x)