3.9.10 \(\int \frac {\sqrt {c x^2} (a+b x)^2}{x^3} \, dx\) [810]

Optimal. Leaf size=49 \[ b^2 \sqrt {c x^2}-\frac {a^2 \sqrt {c x^2}}{x^2}+\frac {2 a b \sqrt {c x^2} \log (x)}{x} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 49, 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, 45} \begin {gather*} -\frac {a^2 \sqrt {c x^2}}{x^2}+\frac {2 a b \sqrt {c x^2} \log (x)}{x}+b^2 \sqrt {c x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Sqrt[c*x^2]*(a + b*x)^2)/x^3,x]

[Out]

b^2*Sqrt[c*x^2] - (a^2*Sqrt[c*x^2])/x^2 + (2*a*b*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 45

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

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Mathematica [A]
time = 0.01, size = 31, normalized size = 0.63 \begin {gather*} \frac {c \left (-a^2+b^2 x^2+2 a b x \log (x)\right )}{\sqrt {c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

cought exception: maximum recursion depth exceeded

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Maple [A]
time = 0.12, size = 32, normalized size = 0.65

method	result	size
default	\(\frac {\sqrt {c \,x^{2}}\, \left (2 a b \ln \left (x \right ) x +x^{2} b^{2}-a^{2}\right )}{x^{2}}\)	\(32\)
risch	\(b^{2} \sqrt {c \,x^{2}}-\frac {a^{2} \sqrt {c \,x^{2}}}{x^{2}}+\frac {2 a b \ln \left (x \right ) \sqrt {c \,x^{2}}}{x}\)	\(44\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^2*(c*x^2)^(1/2)/x^3,x,method=_RETURNVERBOSE)

[Out]

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

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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^3,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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Fricas [A]
time = 0.30, size = 31, normalized size = 0.63 \begin {gather*} \frac {{\left (b^{2} x^{2} + 2 \, a b x \log \left (x\right ) - a^{2}\right )} \sqrt {c x^{2}}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(c*x^2)^(1/2)/x^3,x, algorithm="fricas")

[Out]

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

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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^{3}}\, 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**3,x)

[Out]

Integral(sqrt(c*x**2)*(a + b*x)**2/x**3, x)

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Giac [A]
time = 0.00, size = 32, normalized size = 0.65 \begin {gather*} \sqrt {c} \left (-\frac {a^{2} \mathrm {sign}\left (x\right )}{x}+b^{2} x \mathrm {sign}\left (x\right )+2 a b \mathrm {sign}\left (x\right ) \ln \left |x\right |\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(c*x^2)^(1/2)/x^3,x)

[Out]

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

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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^3} \,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^3,x)

[Out]

int(((c*x^2)^(1/2)*(a + b*x)^2)/x^3, x)

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