3.9.59 \(\int \frac {\sqrt {c x^2}}{x^4 (a+b x)} \, dx\) [859]

Optimal. Leaf size=84 \[ -\frac {\sqrt {c x^2}}{2 a x^3}+\frac {b \sqrt {c x^2}}{a^2 x^2}+\frac {b^2 \sqrt {c x^2} \log (x)}{a^3 x}-\frac {b^2 \sqrt {c x^2} \log (a+b x)}{a^3 x} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 84, 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, 46} \begin {gather*} \frac {b^2 \sqrt {c x^2} \log (x)}{a^3 x}-\frac {b^2 \sqrt {c x^2} \log (a+b x)}{a^3 x}+\frac {b \sqrt {c x^2}}{a^2 x^2}-\frac {\sqrt {c x^2}}{2 a x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[c*x^2]/(x^4*(a + b*x)),x]

[Out]

-1/2*Sqrt[c*x^2]/(a*x^3) + (b*Sqrt[c*x^2])/(a^2*x^2) + (b^2*Sqrt[c*x^2]*Log[x])/(a^3*x) - (b^2*Sqrt[c*x^2]*Log
[a + b*x])/(a^3*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 46

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] && Lt
Q[m + n + 2, 0])

Rubi steps

\begin {align*} \int \frac {\sqrt {c x^2}}{x^4 (a+b x)} \, dx &=\frac {\sqrt {c x^2} \int \frac {1}{x^3 (a+b x)} \, dx}{x}\\ &=\frac {\sqrt {c x^2} \int \left (\frac {1}{a x^3}-\frac {b}{a^2 x^2}+\frac {b^2}{a^3 x}-\frac {b^3}{a^3 (a+b x)}\right ) \, dx}{x}\\ &=-\frac {\sqrt {c x^2}}{2 a x^3}+\frac {b \sqrt {c x^2}}{a^2 x^2}+\frac {b^2 \sqrt {c x^2} \log (x)}{a^3 x}-\frac {b^2 \sqrt {c x^2} \log (a+b x)}{a^3 x}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 53, normalized size = 0.63 \begin {gather*} \frac {\sqrt {c x^2} \left (-a (a-2 b x)+2 b^2 x^2 \log (x)-2 b^2 x^2 \log (a+b x)\right )}{2 a^3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.14, size = 51, normalized size = 0.61

method	result	size
default	\(\frac {\sqrt {c \,x^{2}}\, \left (2 b^{2} \ln \left (x \right ) x^{2}-2 b^{2} \ln \left (b x +a \right ) x^{2}+2 a b x -a^{2}\right )}{2 a^{3} x^{3}}\)	\(51\)
risch	\(\frac {\sqrt {c \,x^{2}}\, \left (\frac {b x}{a^{2}}-\frac {1}{2 a}\right )}{x^{3}}+\frac {\sqrt {c \,x^{2}}\, b^{2} \ln \left (-x \right )}{x \,a^{3}}-\frac {b^{2} \ln \left (b x +a \right ) \sqrt {c \,x^{2}}}{a^{3} x}\)	\(70\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.26, size = 52, normalized size = 0.62 \begin {gather*} -\frac {b^{2} \sqrt {c} \log \left (b x + a\right )}{a^{3}} + \frac {b^{2} \sqrt {c} \log \left (x\right )}{a^{3}} + \frac {2 \, b \sqrt {c} x - a \sqrt {c}}{2 \, a^{2} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.40, size = 44, normalized size = 0.52 \begin {gather*} \frac {{\left (2 \, b^{2} x^{2} \log \left (\frac {x}{b x + a}\right ) + 2 \, a b x - a^{2}\right )} \sqrt {c x^{2}}}{2 \, a^{3} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Ch
eck [abs(sa

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {c\,x^2}}{x^4\,\left (a+b\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2)^(1/2)/(x^4*(a + b*x)),x)

[Out]

int((c*x^2)^(1/2)/(x^4*(a + b*x)), x)

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