3.10.20 \(\int x^3 (c x^2)^p (a+b x)^{-5-2 p} \, dx\)

Optimal. Leaf size=33 \[ \frac {x^4 \left (c x^2\right )^p (a+b x)^{-2 (p+2)}}{2 a (p+2)} \]

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Rubi [A]  time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {15, 37} \begin {gather*} \frac {x^4 \left (c x^2\right )^p (a+b x)^{-2 (p+2)}}{2 a (p+2)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*(c*x^2)^p*(a + b*x)^(-5 - 2*p),x]

[Out]

(x^4*(c*x^2)^p)/(2*a*(2 + p)*(a + b*x)^(2*(2 + p)))

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 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin {align*} \int x^3 \left (c x^2\right )^p (a+b x)^{-5-2 p} \, dx &=\left (x^{-2 p} \left (c x^2\right )^p\right ) \int x^{3+2 p} (a+b x)^{-5-2 p} \, dx\\ &=\frac {x^4 \left (c x^2\right )^p (a+b x)^{-2 (2+p)}}{2 a (2+p)}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 32, normalized size = 0.97 \begin {gather*} \frac {x^4 \left (c x^2\right )^p (a+b x)^{-2 p-4}}{a (2 p+4)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x^4*(c*x^2)^p*(a + b*x)^(-4 - 2*p))/(a*(4 + 2*p))

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IntegrateAlgebraic [F]  time = 0.09, size = 0, normalized size = 0.00 \begin {gather*} \int x^3 \left (c x^2\right )^p (a+b x)^{-5-2 p} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^3*(c*x^2)^p*(a + b*x)^(-5 - 2*p),x]

[Out]

Defer[IntegrateAlgebraic][x^3*(c*x^2)^p*(a + b*x)^(-5 - 2*p), x]

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fricas [A]  time = 1.22, size = 40, normalized size = 1.21 \begin {gather*} \frac {{\left (b x^{5} + a x^{4}\right )} \left (c x^{2}\right )^{p} {\left (b x + a\right )}^{-2 \, p - 5}}{2 \, {\left (a p + 2 \, a\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(c*x^2)^p*(b*x+a)^(-5-2*p),x, algorithm="fricas")

[Out]

1/2*(b*x^5 + a*x^4)*(c*x^2)^p*(b*x + a)^(-2*p - 5)/(a*p + 2*a)

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giac [B]  time = 1.26, size = 74, normalized size = 2.24 \begin {gather*} \frac {\left (c x^{2}\right )^{p} b x^{5} e^{\left (-2 \, p \log \left (b x + a\right ) - 5 \, \log \left (b x + a\right )\right )} + \left (c x^{2}\right )^{p} a x^{4} e^{\left (-2 \, p \log \left (b x + a\right ) - 5 \, \log \left (b x + a\right )\right )}}{2 \, {\left (a p + 2 \, a\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(c*x^2)^p*(b*x+a)^(-5-2*p),x, algorithm="giac")

[Out]

1/2*((c*x^2)^p*b*x^5*e^(-2*p*log(b*x + a) - 5*log(b*x + a)) + (c*x^2)^p*a*x^4*e^(-2*p*log(b*x + a) - 5*log(b*x
+ a)))/(a*p + 2*a)

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maple [A]  time = 0.00, size = 32, normalized size = 0.97 \begin {gather*} \frac {x^{4} \left (c \,x^{2}\right )^{p} \left (b x +a \right )^{-2 p -4}}{2 \left (p +2\right ) a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(c*x^2)^p*(b*x+a)^(-5-2*p),x)

[Out]

1/2*(b*x+a)^(-4-2*p)*x^4/a/(2+p)*(c*x^2)^p

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (c x^{2}\right )^{p} {\left (b x + a\right )}^{-2 \, p - 5} x^{3}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(c*x^2)^p*(b*x+a)^(-5-2*p),x, algorithm="maxima")

[Out]

integrate((c*x^2)^p*(b*x + a)^(-2*p - 5)*x^3, x)

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mupad [B]  time = 0.27, size = 33, normalized size = 1.00 \begin {gather*} \frac {x^4\,{\left (c\,x^2\right )}^p}{2\,a\,\left (p+2\right )\,{\left (a+b\,x\right )}^{2\,p+4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^4*(c*x^2)^p)/(2*a*(p + 2)*(a + b*x)^(2*p + 4))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(c*x**2)**p*(b*x+a)**(-5-2*p),x)

[Out]

Timed out

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