3.9.80 \(\int \frac {1}{(c x^2)^{3/2} (a+b x)^2} \, dx\)

Optimal. Leaf size=118 \[ \frac {3 b^2 x \log (x)}{a^4 c \sqrt {c x^2}}-\frac {3 b^2 x \log (a+b x)}{a^4 c \sqrt {c x^2}}+\frac {b^2 x}{a^3 c \sqrt {c x^2} (a+b x)}+\frac {2 b}{a^3 c \sqrt {c x^2}}-\frac {1}{2 a^2 c x \sqrt {c x^2}} \]

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Rubi [A]  time = 0.03, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {15, 44} \begin {gather*} \frac {b^2 x}{a^3 c \sqrt {c x^2} (a+b x)}+\frac {3 b^2 x \log (x)}{a^4 c \sqrt {c x^2}}-\frac {3 b^2 x \log (a+b x)}{a^4 c \sqrt {c x^2}}+\frac {2 b}{a^3 c \sqrt {c x^2}}-\frac {1}{2 a^2 c x \sqrt {c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((c*x^2)^(3/2)*(a + b*x)^2),x]

[Out]

(2*b)/(a^3*c*Sqrt[c*x^2]) - 1/(2*a^2*c*x*Sqrt[c*x^2]) + (b^2*x)/(a^3*c*Sqrt[c*x^2]*(a + b*x)) + (3*b^2*x*Log[x
])/(a^4*c*Sqrt[c*x^2]) - (3*b^2*x*Log[a + b*x])/(a^4*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 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin {align*} \int \frac {1}{\left (c x^2\right )^{3/2} (a+b x)^2} \, dx &=\frac {x \int \frac {1}{x^3 (a+b x)^2} \, dx}{c \sqrt {c x^2}}\\ &=\frac {x \int \left (\frac {1}{a^2 x^3}-\frac {2 b}{a^3 x^2}+\frac {3 b^2}{a^4 x}-\frac {b^3}{a^3 (a+b x)^2}-\frac {3 b^3}{a^4 (a+b x)}\right ) \, dx}{c \sqrt {c x^2}}\\ &=\frac {2 b}{a^3 c \sqrt {c x^2}}-\frac {1}{2 a^2 c x \sqrt {c x^2}}+\frac {b^2 x}{a^3 c \sqrt {c x^2} (a+b x)}+\frac {3 b^2 x \log (x)}{a^4 c \sqrt {c x^2}}-\frac {3 b^2 x \log (a+b x)}{a^4 c \sqrt {c x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 80, normalized size = 0.68 \begin {gather*} \frac {x \left (a \left (-a^2+3 a b x+6 b^2 x^2\right )+6 b^2 x^2 \log (x) (a+b x)-6 b^2 x^2 (a+b x) \log (a+b x)\right )}{2 a^4 \left (c x^2\right )^{3/2} (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.07, size = 77, normalized size = 0.65 \begin {gather*} \frac {\frac {3 b^2 x^3 \log (x)}{a^4}-\frac {3 b^2 x^3 \log (a+b x)}{a^4}+\frac {-a^2 x+3 a b x^2+6 b^2 x^3}{2 a^3 (a+b x)}}{\left (c x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.98, size = 83, normalized size = 0.70 \begin {gather*} \frac {{\left (6 \, a b^{2} x^{2} + 3 \, a^{2} b x - a^{3} + 6 \, {\left (b^{3} x^{3} + a b^{2} x^{2}\right )} \log \left (\frac {x}{b x + a}\right )\right )} \sqrt {c x^{2}}}{2 \, {\left (a^{4} b c^{2} x^{4} + a^{5} c^{2} x^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^2)^(3/2)/(b*x+a)^2,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 1.03, size = 152, normalized size = 1.29 \begin {gather*} -\frac {\frac {6 \, b^{2} \log \left ({\left | -\frac {a}{b x + a} + 1 \right |}\right )}{a^{4} \mathrm {sgn}\left (-\frac {b}{b x + a} + \frac {a b}{{\left (b x + a\right )}^{2}}\right )} + \frac {2 \, b^{2}}{{\left (b x + a\right )} a^{3} \mathrm {sgn}\left (-\frac {b}{b x + a} + \frac {a b}{{\left (b x + a\right )}^{2}}\right )} - \frac {\frac {6 \, a b^{2}}{b x + a} - 5 \, b^{2}}{a^{4} {\left (\frac {a}{b x + a} - 1\right )}^{2} \mathrm {sgn}\left (-\frac {b}{b x + a} + \frac {a b}{{\left (b x + a\right )}^{2}}\right )}}{2 \, c^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^2)^(3/2)/(b*x+a)^2,x, algorithm="giac")

[Out]

-1/2*(6*b^2*log(abs(-a/(b*x + a) + 1))/(a^4*sgn(-b/(b*x + a) + a*b/(b*x + a)^2)) + 2*b^2/((b*x + a)*a^3*sgn(-b
/(b*x + a) + a*b/(b*x + a)^2)) - (6*a*b^2/(b*x + a) - 5*b^2)/(a^4*(a/(b*x + a) - 1)^2*sgn(-b/(b*x + a) + a*b/(
b*x + a)^2)))/c^(3/2)

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maple [A]  time = 0.01, size = 93, normalized size = 0.79 \begin {gather*} \frac {\left (6 b^{3} x^{3} \ln \relax (x )-6 b^{3} x^{3} \ln \left (b x +a \right )+6 a \,b^{2} x^{2} \ln \relax (x )-6 a \,b^{2} x^{2} \ln \left (b x +a \right )+6 a \,b^{2} x^{2}+3 a^{2} b x -a^{3}\right ) x}{2 \left (c \,x^{2}\right )^{\frac {3}{2}} \left (b x +a \right ) a^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.46, size = 98, normalized size = 0.83 \begin {gather*} -\frac {b}{\sqrt {c x^{2}} a^{2} b c x + \sqrt {c x^{2}} a^{3} c} - \frac {3 \, \left (-1\right )^{\frac {2 \, a c x}{b}} b^{2} \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{a^{4} c^{\frac {3}{2}}} + \frac {3 \, b}{\sqrt {c x^{2}} a^{3} c} - \frac {1}{2 \, a^{2} c^{\frac {3}{2}} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^2)^(3/2)/(b*x+a)^2,x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (c\,x^2\right )}^{3/2}\,{\left (a+b\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((c*x^2)^(3/2)*(a + b*x)^2),x)

[Out]

int(1/((c*x^2)^(3/2)*(a + b*x)^2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (c x^{2}\right )^{\frac {3}{2}} \left (a + b x\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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