3.2829 \(\int \frac{1}{\left (\frac{c}{(a+b x)^3}\right )^{5/2}} \, dx\)

Optimal. Leaf size=30 \[ \frac{2 (a+b x)^7}{17 b c^2 \sqrt{\frac{c}{(a+b x)^3}}} \]

[Out]

(2*(a + b*x)^7)/(17*b*c^2*Sqrt[c/(a + b*x)^3])

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Rubi [A]  time = 0.0270123, 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 \[ \frac{2 (a+b x)^7}{17 b c^2 \sqrt{\frac{c}{(a+b x)^3}}} \]

Antiderivative was successfully verified.

[In]  Int[(c/(a + b*x)^3)^(-5/2),x]

[Out]

(2*(a + b*x)^7)/(17*b*c^2*Sqrt[c/(a + b*x)^3])

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Rubi in Sympy [A]  time = 2.60042, size = 26, normalized size = 0.87 \[ \frac{2 \sqrt{\frac{c}{\left (a + b x\right )^{3}}} \left (a + b x\right )^{10}}{17 b c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c/(b*x+a)**3)**(5/2),x)

[Out]

2*sqrt(c/(a + b*x)**3)*(a + b*x)**10/(17*b*c**3)

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Mathematica [A]  time = 0.0241552, size = 25, normalized size = 0.83 \[ \frac{2 (a+b x)}{17 b \left (\frac{c}{(a+b x)^3}\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c/(a + b*x)^3)^(-5/2),x]

[Out]

(2*(a + b*x))/(17*b*(c/(a + b*x)^3)^(5/2))

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Maple [A]  time = 0.004, size = 22, normalized size = 0.7 \[{\frac{2\,bx+2\,a}{17\,b} \left ({\frac{c}{ \left ( bx+a \right ) ^{3}}} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c/(b*x+a)^3)^(5/2),x)

[Out]

2/17*(b*x+a)/b/(c/(b*x+a)^3)^(5/2)

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Maxima [A]  time = 1.44559, size = 36, normalized size = 1.2 \[ \frac{2 \,{\left (b \sqrt{c} x + a \sqrt{c}\right )}{\left (b x + a\right )}^{\frac{15}{2}}}{17 \, b c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c/(b*x + a)^3)^(-5/2),x, algorithm="maxima")

[Out]

2/17*(b*sqrt(c)*x + a*sqrt(c))*(b*x + a)^(15/2)/(b*c^3)

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Fricas [A]  time = 0.219366, size = 196, normalized size = 6.53 \[ \frac{2 \,{\left (b^{10} x^{10} + 10 \, a b^{9} x^{9} + 45 \, a^{2} b^{8} x^{8} + 120 \, a^{3} b^{7} x^{7} + 210 \, a^{4} b^{6} x^{6} + 252 \, a^{5} b^{5} x^{5} + 210 \, a^{6} b^{4} x^{4} + 120 \, a^{7} b^{3} x^{3} + 45 \, a^{8} b^{2} x^{2} + 10 \, a^{9} b x + a^{10}\right )} \sqrt{\frac{c}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}}}{17 \, b c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c/(b*x + a)^3)^(-5/2),x, algorithm="fricas")

[Out]

2/17*(b^10*x^10 + 10*a*b^9*x^9 + 45*a^2*b^8*x^8 + 120*a^3*b^7*x^7 + 210*a^4*b^6*
x^6 + 252*a^5*b^5*x^5 + 210*a^6*b^4*x^4 + 120*a^7*b^3*x^3 + 45*a^8*b^2*x^2 + 10*
a^9*b*x + a^10)*sqrt(c/(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3))/(b*c^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (\frac{c}{\left (a + b x\right )^{3}}\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(c/(b*x+a)**3)**(5/2),x)

[Out]

Integral((c/(a + b*x)**3)**(-5/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (\frac{c}{{\left (b x + a\right )}^{3}}\right )^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c/(b*x + a)^3)^(-5/2),x, algorithm="giac")

[Out]

integrate((c/(b*x + a)^3)^(-5/2), x)