3.8.69 \(\int \frac {\sqrt {c x^2} (a+b x)^2}{x^4} \, dx\)

Optimal. Leaf size=54 \[ -\frac {a^2 \sqrt {c x^2}}{2 x^3}-\frac {2 a b \sqrt {c x^2}}{x^2}+\frac {b^2 \sqrt {c x^2} \log (x)}{x} \]

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Rubi [A]  time = 0.01, antiderivative size = 54, 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}}{2 x^3}-\frac {2 a b \sqrt {c x^2}}{x^2}+\frac {b^2 \sqrt {c x^2} \log (x)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Sqrt[c*x^2]*(a + b*x)^2)/x^4,x]

[Out]

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

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

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Mathematica [A]  time = 0.01, size = 36, normalized size = 0.67 \begin {gather*} \frac {\sqrt {c x^2} \left (2 b^2 x^2 \log (x)-a (a+4 b x)\right )}{2 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[c*x^2]*(a + b*x)^2)/x^4,x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.04, size = 38, normalized size = 0.70 \begin {gather*} \sqrt {c x^2} \left (\frac {-a^2-4 a b x}{2 x^3}+\frac {b^2 \log (x)}{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(Sqrt[c*x^2]*(a + b*x)^2)/x^4,x]

[Out]

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

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fricas [A]  time = 1.36, size = 33, normalized size = 0.61 \begin {gather*} \frac {{\left (2 \, b^{2} x^{2} \log \relax (x) - 4 \, a b x - a^{2}\right )} \sqrt {c x^{2}}}{2 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(c*x^2)^(1/2)/x^4,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.99, size = 35, normalized size = 0.65 \begin {gather*} \frac {1}{2} \, {\left (2 \, b^{2} \log \left ({\left | x \right |}\right ) \mathrm {sgn}\relax (x) - \frac {4 \, a b x \mathrm {sgn}\relax (x) + a^{2} \mathrm {sgn}\relax (x)}{x^{2}}\right )} \sqrt {c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(c*x^2)^(1/2)/x^4,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.01, size = 34, normalized size = 0.63 \begin {gather*} \frac {\sqrt {c \,x^{2}}\, \left (2 b^{2} x^{2} \ln \relax (x )-4 a b x -a^{2}\right )}{2 x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^2*(c*x^2)^(1/2)/x^4,x)

[Out]

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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(c*x^2)^(1/2)/x^4,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sqrt {c\,x^2}\,{\left (a+b\,x\right )}^2}{x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((c*x^2)^(1/2)*(a + b*x)^2)/x^4,x)

[Out]

int(((c*x^2)^(1/2)*(a + b*x)^2)/x^4, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {c x^{2}} \left (a + b x\right )^{2}}{x^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**2*(c*x**2)**(1/2)/x**4,x)

[Out]

Integral(sqrt(c*x**2)*(a + b*x)**2/x**4, x)

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