3.9.67 \(\int \frac {x^5}{\sqrt {c x^2} (a+b x)^2} \, dx\)

Optimal. Leaf size=107 \[ -\frac {a^4 x}{b^5 \sqrt {c x^2} (a+b x)}-\frac {4 a^3 x \log (a+b x)}{b^5 \sqrt {c x^2}}+\frac {3 a^2 x^2}{b^4 \sqrt {c x^2}}-\frac {a x^3}{b^3 \sqrt {c x^2}}+\frac {x^4}{3 b^2 \sqrt {c x^2}} \]

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Rubi [A]  time = 0.03, antiderivative size = 107, 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^4 x}{b^5 \sqrt {c x^2} (a+b x)}+\frac {3 a^2 x^2}{b^4 \sqrt {c x^2}}-\frac {4 a^3 x \log (a+b x)}{b^5 \sqrt {c x^2}}-\frac {a x^3}{b^3 \sqrt {c x^2}}+\frac {x^4}{3 b^2 \sqrt {c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5/(Sqrt[c*x^2]*(a + b*x)^2),x]

[Out]

(3*a^2*x^2)/(b^4*Sqrt[c*x^2]) - (a*x^3)/(b^3*Sqrt[c*x^2]) + x^4/(3*b^2*Sqrt[c*x^2]) - (a^4*x)/(b^5*Sqrt[c*x^2]
*(a + b*x)) - (4*a^3*x*Log[a + b*x])/(b^5*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^5}{\sqrt {c x^2} (a+b x)^2} \, dx &=\frac {x \int \frac {x^4}{(a+b x)^2} \, dx}{\sqrt {c x^2}}\\ &=\frac {x \int \left (\frac {3 a^2}{b^4}-\frac {2 a x}{b^3}+\frac {x^2}{b^2}+\frac {a^4}{b^4 (a+b x)^2}-\frac {4 a^3}{b^4 (a+b x)}\right ) \, dx}{\sqrt {c x^2}}\\ &=\frac {3 a^2 x^2}{b^4 \sqrt {c x^2}}-\frac {a x^3}{b^3 \sqrt {c x^2}}+\frac {x^4}{3 b^2 \sqrt {c x^2}}-\frac {a^4 x}{b^5 \sqrt {c x^2} (a+b x)}-\frac {4 a^3 x \log (a+b x)}{b^5 \sqrt {c x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 80, normalized size = 0.75 \begin {gather*} \frac {x \left (-3 a^4+9 a^3 b x-12 a^3 (a+b x) \log (a+b x)+6 a^2 b^2 x^2-2 a b^3 x^3+b^4 x^4\right )}{3 b^5 \sqrt {c x^2} (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(-3*a^4 + 9*a^3*b*x + 6*a^2*b^2*x^2 - 2*a*b^3*x^3 + b^4*x^4 - 12*a^3*(a + b*x)*Log[a + b*x]))/(3*b^5*Sqrt[c
*x^2]*(a + b*x))

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IntegrateAlgebraic [A]  time = 0.07, size = 91, normalized size = 0.85 \begin {gather*} \sqrt {c x^2} \left (\frac {-3 a^4+9 a^3 b x+6 a^2 b^2 x^2-2 a b^3 x^3+b^4 x^4}{3 b^5 c x (a+b x)}-\frac {4 a^3 \log (a+b x)}{b^5 c x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[c*x^2]*((-3*a^4 + 9*a^3*b*x + 6*a^2*b^2*x^2 - 2*a*b^3*x^3 + b^4*x^4)/(3*b^5*c*x*(a + b*x)) - (4*a^3*Log[a
+ b*x])/(b^5*c*x))

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fricas [A]  time = 0.97, size = 85, normalized size = 0.79 \begin {gather*} \frac {{\left (b^{4} x^{4} - 2 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 9 \, a^{3} b x - 3 \, a^{4} - 12 \, {\left (a^{3} b x + a^{4}\right )} \log \left (b x + a\right )\right )} \sqrt {c x^{2}}}{3 \, {\left (b^{6} c x^{2} + a b^{5} c x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x+a)^2/(c*x^2)^(1/2),x, algorithm="fricas")

[Out]

1/3*(b^4*x^4 - 2*a*b^3*x^3 + 6*a^2*b^2*x^2 + 9*a^3*b*x - 3*a^4 - 12*(a^3*b*x + a^4)*log(b*x + a))*sqrt(c*x^2)/
(b^6*c*x^2 + a*b^5*c*x)

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giac [A]  time = 1.13, size = 155, normalized size = 1.45 \begin {gather*} \frac {\frac {{\left (b x + a\right )}^{3} {\left (\frac {6 \, a}{b x + a} - \frac {18 \, a^{2}}{{\left (b x + a\right )}^{2}} - 1\right )}}{b^{5} \mathrm {sgn}\left (-\frac {b}{b x + a} + \frac {a b}{{\left (b x + a\right )}^{2}}\right )} - \frac {12 \, a^{3} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{5} \mathrm {sgn}\left (-\frac {b}{b x + a} + \frac {a b}{{\left (b x + a\right )}^{2}}\right )} + \frac {3 \, a^{4}}{{\left (b x + a\right )} b^{5} \mathrm {sgn}\left (-\frac {b}{b x + a} + \frac {a b}{{\left (b x + a\right )}^{2}}\right )}}{3 \, \sqrt {c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x+a)^2/(c*x^2)^(1/2),x, algorithm="giac")

[Out]

1/3*((b*x + a)^3*(6*a/(b*x + a) - 18*a^2/(b*x + a)^2 - 1)/(b^5*sgn(-b/(b*x + a) + a*b/(b*x + a)^2)) - 12*a^3*l
og(abs(b*x + a)/((b*x + a)^2*abs(b)))/(b^5*sgn(-b/(b*x + a) + a*b/(b*x + a)^2)) + 3*a^4/((b*x + a)*b^5*sgn(-b/
(b*x + a) + a*b/(b*x + a)^2)))/sqrt(c)

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maple [A]  time = 0.01, size = 86, normalized size = 0.80 \begin {gather*} -\frac {\left (-b^{4} x^{4}+2 a \,b^{3} x^{3}+12 a^{3} b x \ln \left (b x +a \right )-6 a^{2} b^{2} x^{2}+12 a^{4} \ln \left (b x +a \right )-9 a^{3} b x +3 a^{4}\right ) x}{3 \sqrt {c \,x^{2}}\, \left (b x +a \right ) b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(b*x+a)^2/(c*x^2)^(1/2),x)

[Out]

-1/3*x*(-b^4*x^4+2*a*b^3*x^3+12*a^3*b*x*ln(b*x+a)-6*a^2*b^2*x^2+12*a^4*ln(b*x+a)-9*a^3*b*x+3*a^4)/(c*x^2)^(1/2
)/b^5/(b*x+a)

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maxima [A]  time = 1.56, size = 168, normalized size = 1.57 \begin {gather*} \frac {\sqrt {c x^{2}} a^{3}}{b^{5} c x + a b^{4} c} + \frac {\sqrt {c x^{2}} x^{2}}{3 \, b^{2} c} - \frac {5 \, a x^{2}}{3 \, b^{3} \sqrt {c}} - \frac {4 \, \left (-1\right )^{\frac {2 \, a c x}{b}} a^{3} \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{b^{5} \sqrt {c}} + \frac {2 \, \sqrt {c x^{2}} a x}{3 \, b^{3} c} - \frac {20 \, a^{2} x}{3 \, b^{4} \sqrt {c}} - \frac {4 \, a^{3} \log \left (b x\right )}{b^{5} \sqrt {c}} + \frac {29 \, \sqrt {c x^{2}} a^{2}}{3 \, b^{4} c} - \frac {5 \, a^{3}}{b^{5} \sqrt {c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x+a)^2/(c*x^2)^(1/2),x, algorithm="maxima")

[Out]

sqrt(c*x^2)*a^3/(b^5*c*x + a*b^4*c) + 1/3*sqrt(c*x^2)*x^2/(b^2*c) - 5/3*a*x^2/(b^3*sqrt(c)) - 4*(-1)^(2*a*c*x/
b)*a^3*log(-2*a*c*x/(b*abs(b*x + a)))/(b^5*sqrt(c)) + 2/3*sqrt(c*x^2)*a*x/(b^3*c) - 20/3*a^2*x/(b^4*sqrt(c)) -
4*a^3*log(b*x)/(b^5*sqrt(c)) + 29/3*sqrt(c*x^2)*a^2/(b^4*c) - 5*a^3/(b^5*sqrt(c))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^5}{\sqrt {c\,x^2}\,{\left (a+b\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/((c*x^2)^(1/2)*(a + b*x)^2),x)

[Out]

int(x^5/((c*x^2)^(1/2)*(a + b*x)^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(b*x+a)**2/(c*x**2)**(1/2),x)

[Out]

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

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