3.9.38 \(\int \frac {x (a+b x)^2}{(c x^2)^{3/2}} \, dx\) [838]

Optimal. Leaf size=56 \[ -\frac {a^2}{c \sqrt {c x^2}}+\frac {b^2 x^2}{c \sqrt {c x^2}}+\frac {2 a b x \log (x)}{c \sqrt {c x^2}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 56, normalized size of antiderivative = 1.00, 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, 45} \begin {gather*} -\frac {a^2}{c \sqrt {c x^2}}+\frac {2 a b x \log (x)}{c \sqrt {c x^2}}+\frac {b^2 x^2}{c \sqrt {c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x*(a + b*x)^2)/(c*x^2)^(3/2),x]

[Out]

-(a^2/(c*Sqrt[c*x^2])) + (b^2*x^2)/(c*Sqrt[c*x^2]) + (2*a*b*x*Log[x])/(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 45

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

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Mathematica [A]
time = 0.00, size = 33, normalized size = 0.59 \begin {gather*} \frac {x^2 \left (-a^2+b^2 x^2+2 a b x \log (x)\right )}{\left (c x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*(-a^2 + b^2*x^2 + 2*a*b*x*Log[x]))/(c*x^2)^(3/2)

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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded while calling a Python object} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[(x^1*(a + b*x)^2)/(c*x^2)^(3/2),x]')

[Out]

cought exception: maximum recursion depth exceeded while calling a Python object

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Maple [A]
time = 0.13, size = 32, normalized size = 0.57

method	result	size
default	\(\frac {x^{2} \left (2 a b \ln \left (x \right ) x +x^{2} b^{2}-a^{2}\right )}{\left (c \,x^{2}\right )^{\frac {3}{2}}}\)	\(32\)
risch	\(-\frac {a^{2}}{c \sqrt {c \,x^{2}}}+\frac {b^{2} x^{2}}{c \sqrt {c \,x^{2}}}+\frac {2 a b x \ln \left (x \right )}{c \sqrt {c \,x^{2}}}\)	\(51\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.26, size = 42, normalized size = 0.75 \begin {gather*} \frac {b^{2} x^{2}}{\sqrt {c x^{2}} c} + \frac {2 \, a b \log \left (x\right )}{c^{\frac {3}{2}}} - \frac {a^{2}}{\sqrt {c x^{2}} c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.30, size = 34, normalized size = 0.61 \begin {gather*} \frac {{\left (b^{2} x^{2} + 2 \, a b x \log \left (x\right ) - a^{2}\right )} \sqrt {c x^{2}}}{c^{2} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^2*x^2 + 2*a*b*x*log(x) - a^2)*sqrt(c*x^2)/(c^2*x^2)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (a + b x\right )^{2}}{\left (c x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.00, size = 38, normalized size = 0.68 \begin {gather*} \frac {-\frac {a^{2}}{x \mathrm {sign}\left (x\right )}+\frac {b^{2} x}{\mathrm {sign}\left (x\right )}+\frac {2 a b \ln \left |x\right |}{\mathrm {sign}\left (x\right )}}{\sqrt {c} c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^2*x/sgn(x) + 2*a*b*log(abs(x))/sgn(x) - a^2/(x*sgn(x)))/c^(3/2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x\,{\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*(a + b*x)^2)/(c*x^2)^(3/2),x)

[Out]

int((x*(a + b*x)^2)/(c*x^2)^(3/2), x)

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