3.9.68 \(\int \frac {x^4}{\sqrt {c x^2} (a+b x)^2} \, dx\)

Optimal. Leaf size=86 \[ \frac {a^3 x}{b^4 \sqrt {c x^2} (a+b x)}+\frac {3 a^2 x \log (a+b x)}{b^4 \sqrt {c x^2}}-\frac {2 a x^2}{b^3 \sqrt {c x^2}}+\frac {x^3}{2 b^2 \sqrt {c x^2}} \]

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Rubi [A]  time = 0.03, antiderivative size = 86, 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, 43} \begin {gather*} \frac {a^3 x}{b^4 \sqrt {c x^2} (a+b x)}+\frac {3 a^2 x \log (a+b x)}{b^4 \sqrt {c x^2}}-\frac {2 a x^2}{b^3 \sqrt {c x^2}}+\frac {x^3}{2 b^2 \sqrt {c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4/(Sqrt[c*x^2]*(a + b*x)^2),x]

[Out]

(-2*a*x^2)/(b^3*Sqrt[c*x^2]) + x^3/(2*b^2*Sqrt[c*x^2]) + (a^3*x)/(b^4*Sqrt[c*x^2]*(a + b*x)) + (3*a^2*x*Log[a
+ b*x])/(b^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 43

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

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Mathematica [A]  time = 0.01, size = 69, normalized size = 0.80 \begin {gather*} \frac {x \left (2 a^3-4 a^2 b x+6 a^2 (a+b x) \log (a+b x)-3 a b^2 x^2+b^3 x^3\right )}{2 b^4 \sqrt {c x^2} (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.06, size = 80, normalized size = 0.93 \begin {gather*} \sqrt {c x^2} \left (\frac {3 a^2 \log (a+b x)}{b^4 c x}+\frac {2 a^3-4 a^2 b x-3 a b^2 x^2+b^3 x^3}{2 b^4 c x (a+b x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.99, size = 74, normalized size = 0.86 \begin {gather*} \frac {{\left (b^{3} x^{3} - 3 \, a b^{2} x^{2} - 4 \, a^{2} b x + 2 \, a^{3} + 6 \, {\left (a^{2} b x + a^{3}\right )} \log \left (b x + a\right )\right )} \sqrt {c x^{2}}}{2 \, {\left (b^{5} c x^{2} + a b^{4} c x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b^3*x^3 - 3*a*b^2*x^2 - 4*a^2*b*x + 2*a^3 + 6*(a^2*b*x + a^3)*log(b*x + a))*sqrt(c*x^2)/(b^5*c*x^2 + a*b^
4*c*x)

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giac [A]  time = 1.04, size = 143, normalized size = 1.66 \begin {gather*} \frac {\frac {{\left (b x + a\right )}^{2} {\left (\frac {6 \, a}{b x + a} - 1\right )}}{b^{4} \mathrm {sgn}\left (-\frac {b}{b x + a} + \frac {a b}{{\left (b x + a\right )}^{2}}\right )} + \frac {6 \, a^{2} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{4} \mathrm {sgn}\left (-\frac {b}{b x + a} + \frac {a b}{{\left (b x + a\right )}^{2}}\right )} - \frac {2 \, a^{3}}{{\left (b x + a\right )} b^{4} \mathrm {sgn}\left (-\frac {b}{b x + a} + \frac {a b}{{\left (b x + a\right )}^{2}}\right )}}{2 \, \sqrt {c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 74, normalized size = 0.86 \begin {gather*} \frac {\left (b^{3} x^{3}+6 a^{2} b x \ln \left (b x +a \right )-3 a \,b^{2} x^{2}+6 a^{3} \ln \left (b x +a \right )-4 a^{2} b x +2 a^{3}\right ) x}{2 \sqrt {c \,x^{2}}\, \left (b x +a \right ) b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.55, size = 129, normalized size = 1.50 \begin {gather*} -\frac {\sqrt {c x^{2}} a^{2}}{b^{4} c x + a b^{3} c} + \frac {x^{2}}{2 \, b^{2} \sqrt {c}} + \frac {3 \, \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^{4} \sqrt {c}} + \frac {2 \, a x}{b^{3} \sqrt {c}} + \frac {3 \, a^{2} \log \left (b x\right )}{b^{4} \sqrt {c}} - \frac {4 \, \sqrt {c x^{2}} a}{b^{3} c} + \frac {3 \, a^{2}}{2 \, b^{4} \sqrt {c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^4}{\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^4/((c*x^2)^(1/2)*(a + b*x)^2),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4}}{\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**4/(b*x+a)**2/(c*x**2)**(1/2),x)

[Out]

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

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