3.10.12 \(\int \frac {x^2}{\sqrt {c x^2} (a+b x)^2} \, dx\) [912]

Optimal. Leaf size=43 \[ \frac {a x}{b^2 \sqrt {c x^2} (a+b x)}+\frac {x \log (a+b x)}{b^2 \sqrt {c x^2}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 43, 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 x}{b^2 \sqrt {c x^2} (a+b x)}+\frac {x \log (a+b x)}{b^2 \sqrt {c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/(Sqrt[c*x^2]*(a + b*x)^2),x]

[Out]

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

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Mathematica [A]
time = 0.01, size = 35, normalized size = 0.81 \begin {gather*} \frac {x (a+(a+b x) \log (a+b x))}{b^2 \sqrt {c x^2} (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.13, size = 39, normalized size = 0.91

method	result	size
default	\(\frac {x \left (b \ln \left (b x +a \right ) x +a \ln \left (b x +a \right )+a \right )}{\sqrt {c \,x^{2}}\, b^{2} \left (b x +a \right )}\)	\(39\)
risch	\(\frac {a x}{b^{2} \left (b x +a \right ) \sqrt {c \,x^{2}}}+\frac {x \ln \left (b x +a \right )}{b^{2} \sqrt {c \,x^{2}}}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.30, size = 68, normalized size = 1.58 \begin {gather*} -\frac {\sqrt {c x^{2}}}{b^{2} c x + a b c} + \frac {\left (-1\right )^{\frac {2 \, a c x}{b}} \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{b^{2} \sqrt {c}} + \frac {\log \left (b x\right )}{b^{2} \sqrt {c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.46, size = 40, normalized size = 0.93 \begin {gather*} \frac {\sqrt {c x^{2}} {\left ({\left (b x + a\right )} \log \left (b x + a\right ) + a\right )}}{b^{3} c x^{2} + a b^{2} c x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.58, size = 53, normalized size = 1.23 \begin {gather*} -\frac {{\left (\log \left ({\left | a \right |}\right ) + 1\right )} \mathrm {sgn}\left (x\right )}{b^{2} \sqrt {c}} + \frac {\log \left ({\left | b x + a \right |}\right )}{b^{2} \sqrt {c} \mathrm {sgn}\left (x\right )} + \frac {a}{{\left (b x + a\right )} b^{2} \sqrt {c} \mathrm {sgn}\left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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