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

Optimal. Leaf size=47 \[ -\frac {a^2}{\sqrt {c x^2}}+\frac {2 a b x \log (x)}{\sqrt {c x^2}}+\frac {b^2 x^2}{\sqrt {c x^2}} \]

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Rubi [A]  time = 0.01, antiderivative size = 47, 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^2}{\sqrt {c x^2}}+\frac {2 a b x \log (x)}{\sqrt {c x^2}}+\frac {b^2 x^2}{\sqrt {c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2/(x*Sqrt[c*x^2]),x]

[Out]

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

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Mathematica [A]  time = 0.01, size = 34, normalized size = 0.72 \begin {gather*} \frac {c x^2 \left (-a^2+2 a b x \log (x)+b^2 x^2\right )}{\left (c x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.03, size = 43, normalized size = 0.91 \begin {gather*} \sqrt {c x^2} \left (\frac {b^2 x^2-a^2}{c x^2}+\frac {2 a b \log (x)}{c x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.03, size = 34, normalized size = 0.72 \begin {gather*} \frac {{\left (b^{2} x^{2} + 2 \, a b x \log \relax (x) - a^{2}\right )} \sqrt {c x^{2}}}{c x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.94, size = 65, normalized size = 1.38 \begin {gather*} \frac {\sqrt {c x^{2}} b^{2}}{c} - \frac {2 \, {\left (a b \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2}} \right |}\right ) - \frac {a^{2} \sqrt {c}}{\sqrt {c} x - \sqrt {c x^{2}}}\right )}}{\sqrt {c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 29, normalized size = 0.62 \begin {gather*} \frac {2 a b x \ln \relax (x )+b^{2} x^{2}-a^{2}}{\sqrt {c \,x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.36, size = 35, normalized size = 0.74 \begin {gather*} \frac {2 \, a b \log \relax (x)}{\sqrt {c}} + \frac {\sqrt {c x^{2}} b^{2}}{c} - \frac {a^{2}}{\sqrt {c} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^2}{x\,\sqrt {c\,x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^2/(x*(c*x^2)^(1/2)),x)

[Out]

int((a + b*x)^2/(x*(c*x^2)^(1/2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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