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

Optimal. Leaf size=122 \[ \frac{2 a^{3/2} \sqrt{x} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b x^n}}\right )}{c^2 (1-n) \sqrt{c x}}-\frac{2 a \sqrt{a x+b x^n}}{c^2 (1-n) \sqrt{c x}}-\frac{2 \left (a x+b x^n\right )^{3/2}}{3 c (1-n) (c x)^{3/2}} \]

[Out]

(-2*a*Sqrt[a*x + b*x^n])/(c^2*(1 - n)*Sqrt[c*x]) - (2*(a*x + b*x^n)^(3/2))/(3*c*
(1 - n)*(c*x)^(3/2)) + (2*a^(3/2)*Sqrt[x]*ArcTanh[(Sqrt[a]*Sqrt[x])/Sqrt[a*x + b
*x^n]])/(c^2*(1 - n)*Sqrt[c*x])

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Rubi [A]  time = 0.303265, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ \frac{2 a^{3/2} \sqrt{x} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b x^n}}\right )}{c^2 (1-n) \sqrt{c x}}-\frac{2 a \sqrt{a x+b x^n}}{c^2 (1-n) \sqrt{c x}}-\frac{2 \left (a x+b x^n\right )^{3/2}}{3 c (1-n) (c x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*a*Sqrt[a*x + b*x^n])/(c^2*(1 - n)*Sqrt[c*x]) - (2*(a*x + b*x^n)^(3/2))/(3*c*
(1 - n)*(c*x)^(3/2)) + (2*a^(3/2)*Sqrt[x]*ArcTanh[(Sqrt[a]*Sqrt[x])/Sqrt[a*x + b
*x^n]])/(c^2*(1 - n)*Sqrt[c*x])

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Rubi in Sympy [A]  time = 32.6313, size = 104, normalized size = 0.85 \[ \frac{2 a^{\frac{3}{2}} \sqrt{c x} \operatorname{atanh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x + b x^{n}}} \right )}}{c^{3} \sqrt{x} \left (- n + 1\right )} - \frac{2 a \sqrt{a x + b x^{n}}}{c^{2} \sqrt{c x} \left (- n + 1\right )} - \frac{2 \left (a x + b x^{n}\right )^{\frac{3}{2}}}{3 c \left (c x\right )^{\frac{3}{2}} \left (- n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**(3/2)*sqrt(c*x)*atanh(sqrt(a)*sqrt(x)/sqrt(a*x + b*x**n))/(c**3*sqrt(x)*(-n
+ 1)) - 2*a*sqrt(a*x + b*x**n)/(c**2*sqrt(c*x)*(-n + 1)) - 2*(a*x + b*x**n)**(3
/2)/(3*c*(c*x)**(3/2)*(-n + 1))

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Mathematica [A]  time = 0.326149, size = 120, normalized size = 0.98 \[ \frac{x \left (-6 a^{3/2} \sqrt{b} x^{\frac{n+3}{2}} \sqrt{\frac{a x^{1-n}}{b}+1} \sinh ^{-1}\left (\frac{\sqrt{a} x^{\frac{1}{2}-\frac{n}{2}}}{\sqrt{b}}\right )+8 a^2 x^2+10 a b x^{n+1}+2 b^2 x^{2 n}\right )}{3 (n-1) (c x)^{5/2} \sqrt{a x+b x^n}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(8*a^2*x^2 + 2*b^2*x^(2*n) + 10*a*b*x^(1 + n) - 6*a^(3/2)*Sqrt[b]*x^((3 + n)/
2)*Sqrt[1 + (a*x^(1 - n))/b]*ArcSinh[(Sqrt[a]*x^(1/2 - n/2))/Sqrt[b]]))/(3*(-1 +
n)*(c*x)^(5/2)*Sqrt[a*x + b*x^n])

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Maple [F]  time = 0.054, size = 0, normalized size = 0. \[ \int{1 \left ( ax+b{x}^{n} \right ) ^{{\frac{3}{2}}} \left ( cx \right ) ^{-{\frac{5}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((a*x+b*x^n)^(3/2)/(c*x)^(5/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((a*x + b*x^n)^(3/2)/(c*x)^(5/2), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

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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((a*x+b*x**n)**(3/2)/(c*x)**(5/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((a*x + b*x^n)^(3/2)/(c*x)^(5/2), x)