3.436 \(\int \sqrt{x^4 (a+b x^3)} \, dx\)

Optimal. Leaf size=25 \[ \frac{2 \left (a x^4+b x^7\right )^{3/2}}{9 b x^6} \]

[Out]

(2*(a*x^4 + b*x^7)^(3/2))/(9*b*x^6)

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Rubi [A]  time = 0.0078477, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {1979, 2000} \[ \frac{2 \left (a x^4+b x^7\right )^{3/2}}{9 b x^6} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x^4*(a + b*x^3)],x]

[Out]

(2*(a*x^4 + b*x^7)^(3/2))/(9*b*x^6)

Rule 1979

Int[(u_)^(p_), x_Symbol] :> Int[ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && GeneralizedBinomialQ[u, x] &&  !Gene
ralizedBinomialMatchQ[u, x]

Rule 2000

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(a*x^j + b*x^n)^(p + 1)/(b*(n - j)*(p + 1)*x
^(n - 1)), x] /; FreeQ[{a, b, j, n, p}, x] &&  !IntegerQ[p] && NeQ[n, j] && EqQ[j*p - n + j + 1, 0]

Rubi steps

\begin{align*} \int \sqrt{x^4 \left (a+b x^3\right )} \, dx &=\int \sqrt{a x^4+b x^7} \, dx\\ &=\frac{2 \left (a x^4+b x^7\right )^{3/2}}{9 b x^6}\\ \end{align*}

Mathematica [A]  time = 0.010894, size = 25, normalized size = 1. \[ \frac{2 \left (x^4 \left (a+b x^3\right )\right )^{3/2}}{9 b x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x^4*(a + b*x^3)],x]

[Out]

(2*(x^4*(a + b*x^3))^(3/2))/(9*b*x^6)

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Maple [A]  time = 0.004, size = 29, normalized size = 1.2 \begin{align*}{\frac{2\,b{x}^{3}+2\,a}{9\,b{x}^{2}}\sqrt{{x}^{4} \left ( b{x}^{3}+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4*(b*x^3+a))^(1/2),x)

[Out]

2/9*(b*x^3+a)*(x^4*(b*x^3+a))^(1/2)/b/x^2

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Maxima [A]  time = 1.1652, size = 19, normalized size = 0.76 \begin{align*} \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}}}{9 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4*(b*x^3+a))^(1/2),x, algorithm="maxima")

[Out]

2/9*(b*x^3 + a)^(3/2)/b

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Fricas [A]  time = 0.918432, size = 61, normalized size = 2.44 \begin{align*} \frac{2 \, \sqrt{b x^{7} + a x^{4}}{\left (b x^{3} + a\right )}}{9 \, b x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4*(b*x^3+a))^(1/2),x, algorithm="fricas")

[Out]

2/9*sqrt(b*x^7 + a*x^4)*(b*x^3 + a)/(b*x^2)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4*(b*x**3+a))**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 1.24237, size = 19, normalized size = 0.76 \begin{align*} \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}}}{9 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4*(b*x^3+a))^(1/2),x, algorithm="giac")

[Out]

2/9*(b*x^3 + a)^(3/2)/b