3.775 \(\int (c x^2)^{5/2} (a+b x) \, dx\)

Optimal. Leaf size=41 \[ \frac{1}{6} a c^2 x^5 \sqrt{c x^2}+\frac{1}{7} b c^2 x^6 \sqrt{c x^2} \]

[Out]

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

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Rubi [A]  time = 0.0128006, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {15, 43} \[ \frac{1}{6} a c^2 x^5 \sqrt{c x^2}+\frac{1}{7} b c^2 x^6 \sqrt{c x^2} \]

Antiderivative was successfully verified.

[In]

Int[(c*x^2)^(5/2)*(a + b*x),x]

[Out]

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

Mathematica [A]  time = 0.0060522, size = 22, normalized size = 0.54 \[ \frac{1}{42} x \left (c x^2\right )^{5/2} (7 a+6 b x) \]

Antiderivative was successfully verified.

[In]

Integrate[(c*x^2)^(5/2)*(a + b*x),x]

[Out]

(x*(c*x^2)^(5/2)*(7*a + 6*b*x))/42

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Maple [A]  time = 0.002, size = 19, normalized size = 0.5 \begin{align*}{\frac{x \left ( 6\,bx+7\,a \right ) }{42} \left ( c{x}^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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((c*x^2)^(5/2)*(b*x+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.73336, size = 62, normalized size = 1.51 \begin{align*} \frac{1}{42} \,{\left (6 \, b c^{2} x^{6} + 7 \, a c^{2} x^{5}\right )} \sqrt{c x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.3229, size = 34, normalized size = 0.83 \begin{align*} \frac{a c^{\frac{5}{2}} x \left (x^{2}\right )^{\frac{5}{2}}}{6} + \frac{b c^{\frac{5}{2}} x^{2} \left (x^{2}\right )^{\frac{5}{2}}}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2)**(5/2)*(b*x+a),x)

[Out]

a*c**(5/2)*x*(x**2)**(5/2)/6 + b*c**(5/2)*x**2*(x**2)**(5/2)/7

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Giac [A]  time = 1.06079, size = 38, normalized size = 0.93 \begin{align*} \frac{1}{42} \,{\left (6 \, b c^{2} x^{7} \mathrm{sgn}\left (x\right ) + 7 \, a c^{2} x^{6} \mathrm{sgn}\left (x\right )\right )} \sqrt{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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