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

Optimal. Leaf size=37 \[ \frac{1}{5} a c x^4 \sqrt{c x^2}+\frac{1}{6} b c x^5 \sqrt{c x^2} \]

[Out]

(a*c*x^4*Sqrt[c*x^2])/5 + (b*c*x^5*Sqrt[c*x^2])/6

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Rubi [A]  time = 0.0108016, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {15, 43} \[ \frac{1}{5} a c x^4 \sqrt{c x^2}+\frac{1}{6} b c x^5 \sqrt{c x^2} \]

Antiderivative was successfully verified.

[In]

Int[x*(c*x^2)^(3/2)*(a + b*x),x]

[Out]

(a*c*x^4*Sqrt[c*x^2])/5 + (b*c*x^5*Sqrt[c*x^2])/6

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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int x \left (c x^2\right )^{3/2} (a+b x) \, dx &=\frac{\left (c \sqrt{c x^2}\right ) \int x^4 (a+b x) \, dx}{x}\\ &=\frac{\left (c \sqrt{c x^2}\right ) \int \left (a x^4+b x^5\right ) \, dx}{x}\\ &=\frac{1}{5} a c x^4 \sqrt{c x^2}+\frac{1}{6} b c x^5 \sqrt{c x^2}\\ \end{align*}

Mathematica [A]  time = 0.005597, size = 24, normalized size = 0.65 \[ \frac{1}{30} x^2 \left (c x^2\right )^{3/2} (6 a+5 b x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*(c*x^2)^(3/2)*(6*a + 5*b*x))/30

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Maple [A]  time = 0.002, size = 21, normalized size = 0.6 \begin{align*}{\frac{{x}^{2} \left ( 5\,bx+6\,a \right ) }{30} \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/30*x^2*(5*b*x+6*a)*(c*x^2)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.51571, size = 57, normalized size = 1.54 \begin{align*} \frac{1}{30} \,{\left (5 \, b c x^{5} + 6 \, a c x^{4}\right )} \sqrt{c x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/30*(5*b*c*x^5 + 6*a*c*x^4)*sqrt(c*x^2)

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Sympy [A]  time = 0.651173, size = 36, normalized size = 0.97 \begin{align*} \frac{a c^{\frac{3}{2}} x^{2} \left (x^{2}\right )^{\frac{3}{2}}}{5} + \frac{b c^{\frac{3}{2}} x^{3} \left (x^{2}\right )^{\frac{3}{2}}}{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*c**(3/2)*x**2*(x**2)**(3/2)/5 + b*c**(3/2)*x**3*(x**2)**(3/2)/6

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Giac [A]  time = 1.06546, size = 30, normalized size = 0.81 \begin{align*} \frac{1}{30} \,{\left (5 \, b x^{6} \mathrm{sgn}\left (x\right ) + 6 \, a x^{5} \mathrm{sgn}\left (x\right )\right )} c^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/30*(5*b*x^6*sgn(x) + 6*a*x^5*sgn(x))*c^(3/2)