3.9.78 \(\int \frac {x^3}{\sqrt {c x^2} (a+b x)} \, dx\) [878]

Optimal. Leaf size=61 \[ -\frac {a x^2}{b^2 \sqrt {c x^2}}+\frac {x^3}{2 b \sqrt {c x^2}}+\frac {a^2 x \log (a+b x)}{b^3 \sqrt {c x^2}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 61, 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 x \log (a+b x)}{b^3 \sqrt {c x^2}}-\frac {a x^2}{b^2 \sqrt {c x^2}}+\frac {x^3}{2 b \sqrt {c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/(Sqrt[c*x^2]*(a + b*x)),x]

[Out]

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

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Mathematica [A]
time = 0.01, size = 39, normalized size = 0.64 \begin {gather*} \frac {x \left (b x (-2 a+b x)+2 a^2 \log (a+b x)\right )}{2 b^3 \sqrt {c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.11, size = 38, normalized size = 0.62

method	result	size
default	\(\frac {x \left (x^{2} b^{2}+2 a^{2} \ln \left (b x +a \right )-2 a b x \right )}{2 \sqrt {c \,x^{2}}\, b^{3}}\)	\(38\)
risch	\(\frac {x \left (\frac {1}{2} x^{2} b -a x \right )}{\sqrt {c \,x^{2}}\, b^{2}}+\frac {a^{2} x \ln \left (b x +a \right )}{b^{3} \sqrt {c \,x^{2}}}\)	\(46\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 100, normalized size = 1.64 \begin {gather*} \frac {x^{2}}{2 \, b \sqrt {c}} + \frac {\left (-1\right )^{\frac {2 \, a c x}{b}} a^{2} \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{b^{3} \sqrt {c}} + \frac {2 \, a x}{b^{2} \sqrt {c}} + \frac {a^{2} \log \left (b x\right )}{b^{3} \sqrt {c}} - \frac {3 \, \sqrt {c x^{2}} a}{b^{2} c} + \frac {3 \, a^{2}}{2 \, b^{3} \sqrt {c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.65, size = 42, normalized size = 0.69 \begin {gather*} \frac {{\left (b^{2} x^{2} - 2 \, a b x + 2 \, a^{2} \log \left (b x + a\right )\right )} \sqrt {c x^{2}}}{2 \, b^{3} c x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Giac [A]
time = 0.63, size = 66, normalized size = 1.08 \begin {gather*} -\frac {a^{2} \log \left ({\left | a \right |}\right ) \mathrm {sgn}\left (x\right )}{b^{3} \sqrt {c}} + \frac {a^{2} \log \left ({\left | b x + a \right |}\right )}{b^{3} \sqrt {c} \mathrm {sgn}\left (x\right )} + \frac {b \sqrt {c} x^{2} \mathrm {sgn}\left (x\right ) - 2 \, a \sqrt {c} x \mathrm {sgn}\left (x\right )}{2 \, b^{2} c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {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(x^3/((c*x^2)^(1/2)*(a + b*x)),x)

[Out]

int(x^3/((c*x^2)^(1/2)*(a + b*x)), x)

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