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

Optimal. Leaf size=39 \[ \frac {x^2}{b \sqrt {c x^2}}-\frac {a x \log (a+b x)}{b^2 \sqrt {c x^2}} \]

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Rubi [A]  time = 0.01, antiderivative size = 39, 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 {x^2}{b \sqrt {c x^2}}-\frac {a x \log (a+b x)}{b^2 \sqrt {c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/(Sqrt[c*x^2]*(a + b*x)),x]

[Out]

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

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Mathematica [A]  time = 0.01, size = 27, normalized size = 0.69 \begin {gather*} \frac {x (b x-a \log (a+b x))}{b^2 \sqrt {c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.03, size = 36, normalized size = 0.92 \begin {gather*} \sqrt {c x^2} \left (\frac {1}{b c}-\frac {a \log (a+b x)}{b^2 c x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.95, size = 30, normalized size = 0.77 \begin {gather*} \frac {\sqrt {c x^{2}} {\left (b x - a \log \left (b x + a\right )\right )}}{b^{2} c x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 1.19, size = 51, normalized size = 1.31 \begin {gather*} \frac {a \log \left ({\left | -{\left (\sqrt {c} x - \sqrt {c x^{2}}\right )} b \sqrt {c} - 2 \, a c \right |}\right )}{b^{2} \sqrt {c}} + \frac {\sqrt {c x^{2}}}{b c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 27, normalized size = 0.69 \begin {gather*} -\frac {\left (a \ln \left (b x +a \right )-b x \right ) x}{\sqrt {c \,x^{2}}\, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.47, size = 64, normalized size = 1.64 \begin {gather*} -\frac {\left (-1\right )^{\frac {2 \, a c x}{b}} a \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{b^{2} \sqrt {c}} - \frac {a \log \left (b x\right )}{b^{2} \sqrt {c}} + \frac {\sqrt {c x^{2}}}{b c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {x^2}{\sqrt {c\,x^2}\,\left (a+b\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/((c*x^2)^(1/2)*(a + b*x)),x)

[Out]

int(x^2/((c*x^2)^(1/2)*(a + b*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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