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

Optimal. Leaf size=37 \[ \frac{1}{6} a c x^5 \sqrt{c x^2}+\frac{1}{7} b c x^6 \sqrt{c x^2} \]

[Out]

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

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Rubi [A]  time = 0.0124938, antiderivative size = 37, 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} \[ \frac{1}{6} a c x^5 \sqrt{c x^2}+\frac{1}{7} b c x^6 \sqrt{c x^2} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(c*x^2)^(3/2)*(a + b*x),x]

[Out]

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

Mathematica [A]  time = 0.0061477, size = 24, normalized size = 0.65 \[ \frac{1}{42} x^3 \left (c x^2\right )^{3/2} (7 a+6 b x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^3*(c*x^2)^(3/2)*(7*a + 6*b*x))/42

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Maple [A]  time = 0.003, size = 21, normalized size = 0.6 \begin{align*}{\frac{{x}^{3} \left ( 6\,bx+7\,a \right ) }{42} \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/42*x^3*(6*b*x+7*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^2*(c*x^2)^(3/2)*(b*x+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.54471, size = 57, normalized size = 1.54 \begin{align*} \frac{1}{42} \,{\left (6 \, b c x^{6} + 7 \, a c x^{5}\right )} \sqrt{c x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/42*(6*b*c*x^6 + 7*a*c*x^5)*sqrt(c*x^2)

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Sympy [A]  time = 0.832824, size = 36, normalized size = 0.97 \begin{align*} \frac{a c^{\frac{3}{2}} x^{3} \left (x^{2}\right )^{\frac{3}{2}}}{6} + \frac{b 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**2*(c*x**2)**(3/2)*(b*x+a),x)

[Out]

a*c**(3/2)*x**3*(x**2)**(3/2)/6 + b*c**(3/2)*x**4*(x**2)**(3/2)/7

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Giac [A]  time = 1.05123, size = 30, normalized size = 0.81 \begin{align*} \frac{1}{42} \,{\left (6 \, b x^{7} \mathrm{sgn}\left (x\right ) + 7 \, a x^{6} \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^2*(c*x^2)^(3/2)*(b*x+a),x, algorithm="giac")

[Out]

1/42*(6*b*x^7*sgn(x) + 7*a*x^6*sgn(x))*c^(3/2)