3.956 \(\int \frac{x^3 (a+b x)^n}{(c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=31 \[ \frac{x (a+b x)^{n+1}}{b c (n+1) \sqrt{c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0067218, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {15, 32} \[ \frac{x (a+b x)^{n+1}}{b c (n+1) \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin{align*} \int \frac{x^3 (a+b x)^n}{\left (c x^2\right )^{3/2}} \, dx &=\frac{x \int (a+b x)^n \, dx}{c \sqrt{c x^2}}\\ &=\frac{x (a+b x)^{1+n}}{b c (1+n) \sqrt{c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0141891, size = 30, normalized size = 0.97 \[ \frac{x^3 (a+b x)^{n+1}}{b (n+1) \left (c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^3*(a + b*x)^(1 + n))/(b*(1 + n)*(c*x^2)^(3/2))

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Maple [A]  time = 0.001, size = 29, normalized size = 0.9 \begin{align*}{\frac{ \left ( bx+a \right ) ^{1+n}{x}^{3}}{b \left ( 1+n \right ) } \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x+a)^(1+n)/b/(1+n)*x^3/(c*x^2)^(3/2)

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Maxima [A]  time = 1.02099, size = 42, normalized size = 1.35 \begin{align*} \frac{{\left (b \sqrt{c} x + a \sqrt{c}\right )}{\left (b x + a\right )}^{n}}{b c^{2}{\left (n + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*sqrt(c)*x + a*sqrt(c))*(b*x + a)^n/(b*c^2*(n + 1))

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Fricas [A]  time = 1.28886, size = 77, normalized size = 2.48 \begin{align*} \frac{\sqrt{c x^{2}}{\left (b x + a\right )}{\left (b x + a\right )}^{n}}{{\left (b c^{2} n + b c^{2}\right )} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(c*x^2)*(b*x + a)*(b*x + a)^n/((b*c^2*n + b*c^2)*x)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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