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

Optimal. Leaf size=27 \[ \frac {x (a+b x)^3}{3 b c \sqrt {c x^2}} \]

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Rubi [A]  time = 0.00, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 32} \begin {gather*} \frac {x (a+b x)^3}{3 b c \sqrt {c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(a + b*x)^3)/(3*b*c*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 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 26, normalized size = 0.96 \begin {gather*} \frac {x^3 (a+b x)^3}{3 b \left (c x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x^3*(a + b*x)^3)/(3*b*(c*x^2)^(3/2))

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IntegrateAlgebraic [A]  time = 0.03, size = 34, normalized size = 1.26 \begin {gather*} \frac {x^4 \left (3 a^2+3 a b x+b^2 x^2\right )}{3 \left (c x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.06, size = 30, normalized size = 1.11 \begin {gather*} \frac {{\left (b^{2} x^{2} + 3 \, a b x + 3 \, a^{2}\right )} \sqrt {c x^{2}}}{3 \, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 1.05, size = 39, normalized size = 1.44 \begin {gather*} \frac {\sqrt {c x^{2}} {\left ({\left (\frac {b^{2} x}{c} + \frac {3 \, a b}{c}\right )} x + \frac {3 \, a^{2}}{c}\right )}}{3 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 31, normalized size = 1.15 \begin {gather*} \frac {\left (b^{2} x^{2}+3 a b x +3 a^{2}\right ) x^{4}}{3 \left (c \,x^{2}\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.36, size = 52, normalized size = 1.93 \begin {gather*} \frac {b^{2} x^{4}}{3 \, \sqrt {c x^{2}} c} + \frac {a b x^{3}}{\sqrt {c x^{2}} c} + \frac {a^{2} x^{2}}{\sqrt {c x^{2}} c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {x^3\,{\left (a+b\,x\right )}^2}{{\left (c\,x^2\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 0.80, size = 56, normalized size = 2.07 \begin {gather*} \frac {a^{2} x^{4}}{c^{\frac {3}{2}} \left (x^{2}\right )^{\frac {3}{2}}} + \frac {a b x^{5}}{c^{\frac {3}{2}} \left (x^{2}\right )^{\frac {3}{2}}} + \frac {b^{2} x^{6}}{3 c^{\frac {3}{2}} \left (x^{2}\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*x**4/(c**(3/2)*(x**2)**(3/2)) + a*b*x**5/(c**(3/2)*(x**2)**(3/2)) + b**2*x**6/(3*c**(3/2)*(x**2)**(3/2))

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