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

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

[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])

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Rubi [A]  time = 0.0323105, antiderivative size = 118, normalized size of antiderivative = 1., 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} \[ \frac{b^2 x}{a^3 c \sqrt{c x^2} (a+b x)}+\frac{3 b^2 x \log (x)}{a^4 c \sqrt{c x^2}}-\frac{3 b^2 x \log (a+b x)}{a^4 c \sqrt{c x^2}}+\frac{2 b}{a^3 c \sqrt{c x^2}}-\frac{1}{2 a^2 c x \sqrt{c x^2}} \]

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*}

Mathematica [A]  time = 0.0086627, size = 80, normalized size = 0.68 \[ \frac{x \left (a \left (-a^2+3 a b x+6 b^2 x^2\right )+6 b^2 x^2 \log (x) (a+b x)-6 b^2 x^2 (a+b x) \log (a+b x)\right )}{2 a^4 \left (c x^2\right )^{3/2} (a+b x)} \]

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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Maple [A]  time = 0.006, size = 93, normalized size = 0.8 \begin{align*}{\frac{x \left ( 6\,{b}^{3}\ln \left ( x \right ){x}^{3}-6\,{b}^{3}\ln \left ( bx+a \right ){x}^{3}+6\,\ln \left ( x \right ){x}^{2}a{b}^{2}-6\,\ln \left ( bx+a \right ){x}^{2}a{b}^{2}+6\,a{b}^{2}{x}^{2}+3\,{a}^{2}bx-{a}^{3} \right ) }{2\,{a}^{4} \left ( bx+a \right ) } \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}

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*ln(x)*x^3-6*b^3*ln(b*x+a)*x^3+6*ln(x)*x^2*a*b^2-6*ln(b*x+a)*x^2*a*b^2+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 [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Fricas [A]  time = 1.24315, size = 165, normalized size = 1.4 \begin{align*} \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{align*}

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

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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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

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]

Exception raised: RuntimeError