3.780 \(\int \frac{x^3 (a+b x)}{\sqrt{c x^2}} \, dx\)

Optimal. Leaf size=35 \[ \frac{a x^4}{3 \sqrt{c x^2}}+\frac{b x^5}{4 \sqrt{c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.008578, antiderivative size = 35, 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} \[ \frac{a x^4}{3 \sqrt{c x^2}}+\frac{b x^5}{4 \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*(a + b*x))/Sqrt[c*x^2],x]

[Out]

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

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

Mathematica [A]  time = 0.0043804, size = 24, normalized size = 0.69 \[ \frac{x^4 (4 a+3 b x)}{12 \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^4*(4*a + 3*b*x))/(12*Sqrt[c*x^2])

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Maple [A]  time = 0.002, size = 21, normalized size = 0.6 \begin{align*}{\frac{{x}^{4} \left ( 3\,bx+4\,a \right ) }{12}{\frac{1}{\sqrt{c{x}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*x^4*(3*b*x+4*a)/(c*x^2)^(1/2)

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Maxima [A]  time = 1.04352, size = 45, normalized size = 1.29 \begin{align*} \frac{\sqrt{c x^{2}} b x^{3}}{4 \, c} + \frac{\sqrt{c x^{2}} a x^{2}}{3 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.80944, size = 54, normalized size = 1.54 \begin{align*} \frac{{\left (3 \, b x^{3} + 4 \, a x^{2}\right )} \sqrt{c x^{2}}}{12 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*b*x^3 + 4*a*x^2)*sqrt(c*x^2)/c

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Sympy [A]  time = 0.561217, size = 36, normalized size = 1.03 \begin{align*} \frac{a x^{4}}{3 \sqrt{c} \sqrt{x^{2}}} + \frac{b x^{5}}{4 \sqrt{c} \sqrt{x^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.06321, size = 35, normalized size = 1. \begin{align*} \frac{1}{12} \, \sqrt{c x^{2}}{\left (\frac{3 \, b x}{c} + \frac{4 \, a}{c}\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*sqrt(c*x^2)*(3*b*x/c + 4*a/c)*x^2