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

Optimal. Leaf size=90 \[ -\frac{b x}{a^2 c \sqrt{c x^2} (a+b x)}-\frac{2 b x \log (x)}{a^3 c \sqrt{c x^2}}+\frac{2 b x \log (a+b x)}{a^3 c \sqrt{c x^2}}-\frac{1}{a^2 c \sqrt{c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0236142, antiderivative size = 90, normalized size of antiderivative = 1., 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, 44} \[ -\frac{b x}{a^2 c \sqrt{c x^2} (a+b x)}-\frac{2 b x \log (x)}{a^3 c \sqrt{c x^2}}+\frac{2 b x \log (a+b x)}{a^3 c \sqrt{c x^2}}-\frac{1}{a^2 c \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[x/((c*x^2)^(3/2)*(a + b*x)^2),x]

[Out]

-(1/(a^2*c*Sqrt[c*x^2])) - (b*x)/(a^2*c*Sqrt[c*x^2]*(a + b*x)) - (2*b*x*Log[x])/(a^3*c*Sqrt[c*x^2]) + (2*b*x*L
og[a + b*x])/(a^3*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{x}{\left (c x^2\right )^{3/2} (a+b x)^2} \, dx &=\frac{x \int \frac{1}{x^2 (a+b x)^2} \, dx}{c \sqrt{c x^2}}\\ &=\frac{x \int \left (\frac{1}{a^2 x^2}-\frac{2 b}{a^3 x}+\frac{b^2}{a^2 (a+b x)^2}+\frac{2 b^2}{a^3 (a+b x)}\right ) \, dx}{c \sqrt{c x^2}}\\ &=-\frac{1}{a^2 c \sqrt{c x^2}}-\frac{b x}{a^2 c \sqrt{c x^2} (a+b x)}-\frac{2 b x \log (x)}{a^3 c \sqrt{c x^2}}+\frac{2 b x \log (a+b x)}{a^3 c \sqrt{c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0105107, size = 59, normalized size = 0.66 \[ \frac{x^2 (-a (a+2 b x)-2 b x \log (x) (a+b x)+2 b x (a+b x) \log (a+b x))}{a^3 \left (c x^2\right )^{3/2} (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.004, size = 74, normalized size = 0.8 \begin{align*} -{\frac{{x}^{2} \left ( 2\,{b}^{2}\ln \left ( x \right ){x}^{2}-2\,{b}^{2}\ln \left ( bx+a \right ){x}^{2}+2\,ab\ln \left ( x \right ) x-2\,\ln \left ( bx+a \right ) xab+2\,abx+{a}^{2} \right ) }{{a}^{3} \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(x/(c*x^2)^(3/2)/(b*x+a)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.16488, size = 134, normalized size = 1.49 \begin{align*} -\frac{{\left (2 \, a b x + a^{2} - 2 \,{\left (b^{2} x^{2} + a b x\right )} \log \left (\frac{b x + a}{x}\right )\right )} \sqrt{c x^{2}}}{a^{3} b c^{2} x^{3} + a^{4} c^{2} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

[Out]

Exception raised: RuntimeError