3.9.2 \(\int \frac {x^3 (a+b x)^2}{(c x^2)^{5/2}} \, dx\)

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

________________________________________________________________________________________

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 = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 43} \begin {gather*} -\frac {a^2}{c^2 \sqrt {c x^2}}+\frac {2 a b x \log (x)}{c^2 \sqrt {c x^2}}+\frac {b^2 x^2}{c^2 \sqrt {c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^3*(a + b*x)^2)/(c*x^2)^(5/2),x]

[Out]

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

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 33, normalized size = 0.59 \begin {gather*} \frac {-a^2+2 a b x \log (x)+b^2 x^2}{c^2 \sqrt {c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^3*(a + b*x)^2)/(c*x^2)^(5/2),x]

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.04, size = 35, normalized size = 0.62 \begin {gather*} \frac {-a^2 x^4+2 a b x^5 \log (x)+b^2 x^6}{\left (c x^2\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x^3*(a + b*x)^2)/(c*x^2)^(5/2),x]

[Out]

(-(a^2*x^4) + b^2*x^6 + 2*a*b*x^5*Log[x])/(c*x^2)^(5/2)

________________________________________________________________________________________

fricas [A]  time = 0.97, size = 34, normalized size = 0.61 \begin {gather*} \frac {{\left (b^{2} x^{2} + 2 \, a b x \log \relax (x) - a^{2}\right )} \sqrt {c x^{2}}}{c^{3} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)^2/(c*x^2)^(5/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [A]  time = 1.07, size = 65, normalized size = 1.16 \begin {gather*} \frac {\sqrt {c x^{2}} b^{2}}{c^{3}} - \frac {2 \, {\left (a b \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2}} \right |}\right ) - \frac {a^{2} \sqrt {c}}{\sqrt {c} x - \sqrt {c x^{2}}}\right )}}{c^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)^2/(c*x^2)^(5/2),x, algorithm="giac")

[Out]

sqrt(c*x^2)*b^2/c^3 - 2*(a*b*log(abs(-sqrt(c)*x + sqrt(c*x^2))) - a^2*sqrt(c)/(sqrt(c)*x - sqrt(c*x^2)))/c^(5/
2)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 32, normalized size = 0.57 \begin {gather*} \frac {\left (2 a b x \ln \relax (x )+b^{2} x^{2}-a^{2}\right ) x^{4}}{\left (c \,x^{2}\right )^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(b*x+a)^2/(c*x^2)^(5/2),x)

[Out]

x^4*(2*a*b*x*ln(x)+b^2*x^2-a^2)/(c*x^2)^(5/2)

________________________________________________________________________________________

maxima [A]  time = 1.45, size = 45, normalized size = 0.80 \begin {gather*} \frac {b^{2} x^{4}}{\left (c x^{2}\right )^{\frac {3}{2}} c} - \frac {a^{2} x^{2}}{\left (c x^{2}\right )^{\frac {3}{2}} c} + \frac {2 \, a b \log \relax (x)}{c^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)^2/(c*x^2)^(5/2),x, algorithm="maxima")

[Out]

b^2*x^4/((c*x^2)^(3/2)*c) - a^2*x^2/((c*x^2)^(3/2)*c) + 2*a*b*log(x)/c^(5/2)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^3\,{\left (a+b\,x\right )}^2}{{\left (c\,x^2\right )}^{5/2}} \,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)^(5/2),x)

[Out]

int((x^3*(a + b*x)^2)/(c*x^2)^(5/2), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3} \left (a + b x\right )^{2}}{\left (c x^{2}\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(b*x+a)**2/(c*x**2)**(5/2),x)

[Out]

Integral(x**3*(a + b*x)**2/(c*x**2)**(5/2), x)

________________________________________________________________________________________