3.9.84 \(\int \frac {1}{x^3 \sqrt {c x^2} (a+b x)} \, dx\) [884]

Optimal. Leaf size=100 \[ -\frac {b^2}{a^3 \sqrt {c x^2}}-\frac {1}{3 a x^2 \sqrt {c x^2}}+\frac {b}{2 a^2 x \sqrt {c x^2}}-\frac {b^3 x \log (x)}{a^4 \sqrt {c x^2}}+\frac {b^3 x \log (a+b x)}{a^4 \sqrt {c x^2}} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 100, 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^3 x \log (x)}{a^4 \sqrt {c x^2}}+\frac {b^3 x \log (a+b x)}{a^4 \sqrt {c x^2}}-\frac {b^2}{a^3 \sqrt {c x^2}}+\frac {b}{2 a^2 x \sqrt {c x^2}}-\frac {1}{3 a x^2 \sqrt {c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^3*Sqrt[c*x^2]*(a + b*x)),x]

[Out]

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

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Mathematica [A]
time = 0.01, size = 63, normalized size = 0.63 \begin {gather*} \frac {c \left (a \left (-2 a^2+3 a b x-6 b^2 x^2\right )-6 b^3 x^3 \log (x)+6 b^3 x^3 \log (a+b x)\right )}{6 a^4 \left (c x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*Sqrt[c*x^2]*(a + b*x)),x]

[Out]

(c*(a*(-2*a^2 + 3*a*b*x - 6*b^2*x^2) - 6*b^3*x^3*Log[x] + 6*b^3*x^3*Log[a + b*x]))/(6*a^4*(c*x^2)^(3/2))

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Maple [A]
time = 0.14, size = 62, normalized size = 0.62

method	result	size
default	\(-\frac {6 b^{3} \ln \left (x \right ) x^{3}-6 b^{3} \ln \left (b x +a \right ) x^{3}+6 a \,b^{2} x^{2}-3 a^{2} b x +2 a^{3}}{6 x^{2} \sqrt {c \,x^{2}}\, a^{4}}\)	\(62\)
risch	\(\frac {-\frac {1}{3 a}+\frac {b x}{2 a^{2}}-\frac {b^{2} x^{2}}{a^{3}}}{\sqrt {c \,x^{2}}\, x^{2}}-\frac {b^{3} x \ln \left (x \right )}{a^{4} \sqrt {c \,x^{2}}}+\frac {x \,b^{3} \ln \left (-b x -a \right )}{\sqrt {c \,x^{2}}\, a^{4}}\)	\(79\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6/x^2*(6*b^3*ln(x)*x^3-6*b^3*ln(b*x+a)*x^3+6*a*b^2*x^2-3*a^2*b*x+2*a^3)/(c*x^2)^(1/2)/a^4

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Maxima [A]
time = 0.27, size = 69, normalized size = 0.69 \begin {gather*} \frac {b^{3} \log \left (b x + a\right )}{a^{4} \sqrt {c}} - \frac {b^{3} \log \left (x\right )}{a^{4} \sqrt {c}} - \frac {6 \, b^{2} \sqrt {c} x^{2} - 3 \, a b \sqrt {c} x + 2 \, a^{2} \sqrt {c}}{6 \, a^{3} c x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^3*log(b*x + a)/(a^4*sqrt(c)) - b^3*log(x)/(a^4*sqrt(c)) - 1/6*(6*b^2*sqrt(c)*x^2 - 3*a*b*sqrt(c)*x + 2*a^2*s
qrt(c))/(a^3*c*x^3)

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Fricas [A]
time = 0.43, size = 58, normalized size = 0.58 \begin {gather*} \frac {{\left (6 \, b^{3} x^{3} \log \left (\frac {b x + a}{x}\right ) - 6 \, a b^{2} x^{2} + 3 \, a^{2} b x - 2 \, a^{3}\right )} \sqrt {c x^{2}}}{6 \, a^{4} c x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(6*b^3*x^3*log((b*x + a)/x) - 6*a*b^2*x^2 + 3*a^2*b*x - 2*a^3)*sqrt(c*x^2)/(a^4*c*x^4)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{3} \sqrt {c x^{2}} \left (a + b x\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(x**3*sqrt(c*x**2)*(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(1/x^3/(b*x+a)/(c*x^2)^(1/2),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 {1}{x^3\,\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(1/(x^3*(c*x^2)^(1/2)*(a + b*x)),x)

[Out]

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

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