3.9.76 \(\int \frac {x^4}{(c x^2)^{3/2} (a+b x)^2} \, dx\)

Optimal. Leaf size=49 \[ \frac {a x}{b^2 c \sqrt {c x^2} (a+b x)}+\frac {x \log (a+b x)}{b^2 c \sqrt {c x^2}} \]

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Rubi [A]  time = 0.01, antiderivative size = 49, 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 x}{b^2 c \sqrt {c x^2} (a+b x)}+\frac {x \log (a+b x)}{b^2 c \sqrt {c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4/((c*x^2)^(3/2)*(a + b*x)^2),x]

[Out]

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

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Mathematica [A]  time = 0.01, size = 37, normalized size = 0.76 \begin {gather*} \frac {x^3 ((a+b x) \log (a+b x)+a)}{b^2 \left (c x^2\right )^{3/2} (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4/((c*x^2)^(3/2)*(a + b*x)^2),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.05, size = 39, normalized size = 0.80 \begin {gather*} \frac {\frac {a x^3}{b^2 (a+b x)}+\frac {x^3 \log (a+b x)}{b^2}}{\left (c x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^4/((c*x^2)^(3/2)*(a + b*x)^2),x]

[Out]

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

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fricas [A]  time = 1.12, size = 44, normalized size = 0.90 \begin {gather*} \frac {\sqrt {c x^{2}} {\left ({\left (b x + a\right )} \log \left (b x + a\right ) + a\right )}}{b^{3} c^{2} x^{2} + a b^{2} c^{2} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(c*x^2)^(3/2)/(b*x+a)^2,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 1.03, size = 89, normalized size = 1.82 \begin {gather*} \frac {\frac {\log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{2} \mathrm {sgn}\left (-\frac {b}{b x + a} + \frac {a b}{{\left (b x + a\right )}^{2}}\right )} - \frac {a}{{\left (b x + a\right )} b^{2} \mathrm {sgn}\left (-\frac {b}{b x + a} + \frac {a b}{{\left (b x + a\right )}^{2}}\right )}}{c^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(c*x^2)^(3/2)/(b*x+a)^2,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.00, size = 41, normalized size = 0.84 \begin {gather*} \frac {\left (b x \ln \left (b x +a \right )+a \ln \left (b x +a \right )+a \right ) x^{3}}{\left (c \,x^{2}\right )^{\frac {3}{2}} \left (b x +a \right ) b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(c*x^2)^(3/2)/(b*x+a)^2,x)

[Out]

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

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maxima [B]  time = 1.56, size = 108, normalized size = 2.20 \begin {gather*} -\frac {a^{2}}{\sqrt {c x^{2}} b^{4} c x + \sqrt {c x^{2}} a b^{3} 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} c^{\frac {3}{2}}} + \frac {\log \left (b x\right )}{b^{2} c^{\frac {3}{2}}} + \frac {3 \, a}{\sqrt {c x^{2}} b^{3} c} - \frac {2 \, a}{b^{3} c^{\frac {3}{2}} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(c*x^2)^(3/2)/(b*x+a)^2,x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^4}{{\left (c\,x^2\right )}^{3/2}\,{\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)^(3/2)*(a + b*x)^2),x)

[Out]

int(x^4/((c*x^2)^(3/2)*(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}}{\left (c x^{2}\right )^{\frac {3}{2}} \left (a + b x\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(c*x**2)**(3/2)/(b*x+a)**2,x)

[Out]

Integral(x**4/((c*x**2)**(3/2)*(a + b*x)**2), x)

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