3.9.36 \(\int \frac {x^3 (a+b x)^2}{(c x^2)^{3/2}} \, dx\) [836]

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

[Out]

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

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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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Mathics [A]
time = 1.90, size = 27, normalized size = 1.00 \begin {gather*} \frac {x^4 \left (a^2+a b x+\frac {b^2 x^2}{3}\right )}{{\left (c x^2\right )}^{\frac {3}{2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.11, size = 23, normalized size = 0.85

method	result	size
default	\(\frac {\left (b x +a \right )^{3} x^{3}}{3 \left (c \,x^{2}\right )^{\frac {3}{2}} b}\)	\(23\)
risch	\(\frac {x \left (b x +a \right )^{3}}{3 b c \sqrt {c \,x^{2}}}\)	\(24\)
gosper	\(\frac {x^{4} \left (x^{2} b^{2}+3 a b x +3 a^{2}\right )}{3 \left (c \,x^{2}\right )^{\frac {3}{2}}}\)	\(31\)
trager	\(\frac {\left (x^{2} b^{2}+3 a b x +b^{2} x +3 a^{2}+3 a b +b^{2}\right ) \left (-1+x \right ) \sqrt {c \,x^{2}}}{3 c^{2} x}\)	\(49\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (23) = 46\).
time = 0.26, 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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Fricas [A]
time = 0.29, 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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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (20) = 40\).
time = 0.28, size = 46, normalized size = 1.70 \begin {gather*} \frac {a^{2} x^{4}}{\left (c x^{2}\right )^{\frac {3}{2}}} + \frac {a b x^{5}}{\left (c x^{2}\right )^{\frac {3}{2}}} + \frac {b^{2} x^{6}}{3 \left (c 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*x**2)**(3/2) + a*b*x**5/(c*x**2)**(3/2) + b**2*x**6/(3*(c*x**2)**(3/2))

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Giac [A]
time = 0.00, size = 33, normalized size = 1.22 \begin {gather*} \frac {\frac {1}{3} b^{2} x^{3}+a b x^{2}+a^{2} x}{\sqrt {c} c \mathrm {sign}\left (x\right )} \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]

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

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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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