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

Optimal. Leaf size=60 \[ \frac{1}{5} a^2 c x^4 \sqrt{c x^2}+\frac{1}{3} a b c x^5 \sqrt{c x^2}+\frac{1}{7} b^2 c x^6 \sqrt{c x^2} \]

[Out]

(a^2*c*x^4*Sqrt[c*x^2])/5 + (a*b*c*x^5*Sqrt[c*x^2])/3 + (b^2*c*x^6*Sqrt[c*x^2])/7

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Rubi [A]  time = 0.0161078, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {15, 43} \[ \frac{1}{5} a^2 c x^4 \sqrt{c x^2}+\frac{1}{3} a b c x^5 \sqrt{c x^2}+\frac{1}{7} b^2 c x^6 \sqrt{c x^2} \]

Antiderivative was successfully verified.

[In]

Int[x*(c*x^2)^(3/2)*(a + b*x)^2,x]

[Out]

(a^2*c*x^4*Sqrt[c*x^2])/5 + (a*b*c*x^5*Sqrt[c*x^2])/3 + (b^2*c*x^6*Sqrt[c*x^2])/7

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)^2 \, dx &=\frac{\left (c \sqrt{c x^2}\right ) \int x^4 (a+b x)^2 \, dx}{x}\\ &=\frac{\left (c \sqrt{c x^2}\right ) \int \left (a^2 x^4+2 a b x^5+b^2 x^6\right ) \, dx}{x}\\ &=\frac{1}{5} a^2 c x^4 \sqrt{c x^2}+\frac{1}{3} a b c x^5 \sqrt{c x^2}+\frac{1}{7} b^2 c x^6 \sqrt{c x^2}\\ \end{align*}

Mathematica [A]  time = 0.0075748, size = 35, normalized size = 0.58 \[ \frac{1}{105} x^2 \left (c x^2\right )^{3/2} \left (21 a^2+35 a b x+15 b^2 x^2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*(c*x^2)^(3/2)*(21*a^2 + 35*a*b*x + 15*b^2*x^2))/105

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Maple [A]  time = 0.004, size = 32, normalized size = 0.5 \begin{align*}{\frac{{x}^{2} \left ( 15\,{b}^{2}{x}^{2}+35\,abx+21\,{a}^{2} \right ) }{105} \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)^2,x)

[Out]

1/105*x^2*(15*b^2*x^2+35*a*b*x+21*a^2)*(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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.42549, size = 86, normalized size = 1.43 \begin{align*} \frac{1}{105} \,{\left (15 \, b^{2} c x^{6} + 35 \, a b c x^{5} + 21 \, a^{2} 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)^2,x, algorithm="fricas")

[Out]

1/105*(15*b^2*c*x^6 + 35*a*b*c*x^5 + 21*a^2*c*x^4)*sqrt(c*x^2)

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Sympy [A]  time = 0.877928, size = 60, normalized size = 1. \begin{align*} \frac{a^{2} c^{\frac{3}{2}} x^{2} \left (x^{2}\right )^{\frac{3}{2}}}{5} + \frac{a b c^{\frac{3}{2}} x^{3} \left (x^{2}\right )^{\frac{3}{2}}}{3} + \frac{b^{2} c^{\frac{3}{2}} x^{4} \left (x^{2}\right )^{\frac{3}{2}}}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*c**(3/2)*x**2*(x**2)**(3/2)/5 + a*b*c**(3/2)*x**3*(x**2)**(3/2)/3 + b**2*c**(3/2)*x**4*(x**2)**(3/2)/7

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Giac [A]  time = 1.0684, size = 47, normalized size = 0.78 \begin{align*} \frac{1}{105} \,{\left (15 \, b^{2} x^{7} \mathrm{sgn}\left (x\right ) + 35 \, a b x^{6} \mathrm{sgn}\left (x\right ) + 21 \, a^{2} 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)^2,x, algorithm="giac")

[Out]

1/105*(15*b^2*x^7*sgn(x) + 35*a*b*x^6*sgn(x) + 21*a^2*x^5*sgn(x))*c^(3/2)