3.9.51 \(\int \frac {x^3 \sqrt {c x^2}}{(a+b x)^2} \, dx\)

Optimal. Leaf size=106 \[ -\frac {a^4 \sqrt {c x^2}}{b^5 x (a+b x)}-\frac {4 a^3 \sqrt {c x^2} \log (a+b x)}{b^5 x}+\frac {3 a^2 \sqrt {c x^2}}{b^4}-\frac {a x \sqrt {c x^2}}{b^3}+\frac {x^2 \sqrt {c x^2}}{3 b^2} \]

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Rubi [A]  time = 0.04, antiderivative size = 106, 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^4 \sqrt {c x^2}}{b^5 x (a+b x)}+\frac {3 a^2 \sqrt {c x^2}}{b^4}-\frac {4 a^3 \sqrt {c x^2} \log (a+b x)}{b^5 x}-\frac {a x \sqrt {c x^2}}{b^3}+\frac {x^2 \sqrt {c x^2}}{3 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^3*Sqrt[c*x^2])/(a + b*x)^2,x]

[Out]

(3*a^2*Sqrt[c*x^2])/b^4 - (a*x*Sqrt[c*x^2])/b^3 + (x^2*Sqrt[c*x^2])/(3*b^2) - (a^4*Sqrt[c*x^2])/(b^5*x*(a + b*
x)) - (4*a^3*Sqrt[c*x^2]*Log[a + b*x])/(b^5*x)

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^3 \sqrt {c x^2}}{(a+b x)^2} \, dx &=\frac {\sqrt {c x^2} \int \frac {x^4}{(a+b x)^2} \, dx}{x}\\ &=\frac {\sqrt {c x^2} \int \left (\frac {3 a^2}{b^4}-\frac {2 a x}{b^3}+\frac {x^2}{b^2}+\frac {a^4}{b^4 (a+b x)^2}-\frac {4 a^3}{b^4 (a+b x)}\right ) \, dx}{x}\\ &=\frac {3 a^2 \sqrt {c x^2}}{b^4}-\frac {a x \sqrt {c x^2}}{b^3}+\frac {x^2 \sqrt {c x^2}}{3 b^2}-\frac {a^4 \sqrt {c x^2}}{b^5 x (a+b x)}-\frac {4 a^3 \sqrt {c x^2} \log (a+b x)}{b^5 x}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 81, normalized size = 0.76 \begin {gather*} \frac {c x \left (-3 a^4+9 a^3 b x-12 a^3 (a+b x) \log (a+b x)+6 a^2 b^2 x^2-2 a b^3 x^3+b^4 x^4\right )}{3 b^5 \sqrt {c x^2} (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c*x*(-3*a^4 + 9*a^3*b*x + 6*a^2*b^2*x^2 - 2*a*b^3*x^3 + b^4*x^4 - 12*a^3*(a + b*x)*Log[a + b*x]))/(3*b^5*Sqrt
[c*x^2]*(a + b*x))

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IntegrateAlgebraic [A]  time = 0.08, size = 85, normalized size = 0.80 \begin {gather*} \sqrt {c x^2} \left (\frac {-3 a^4+9 a^3 b x+6 a^2 b^2 x^2-2 a b^3 x^3+b^4 x^4}{3 b^5 x (a+b x)}-\frac {4 a^3 \log (a+b x)}{b^5 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[c*x^2]*((-3*a^4 + 9*a^3*b*x + 6*a^2*b^2*x^2 - 2*a*b^3*x^3 + b^4*x^4)/(3*b^5*x*(a + b*x)) - (4*a^3*Log[a +
b*x])/(b^5*x))

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fricas [A]  time = 1.14, size = 83, normalized size = 0.78 \begin {gather*} \frac {{\left (b^{4} x^{4} - 2 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 9 \, a^{3} b x - 3 \, a^{4} - 12 \, {\left (a^{3} b x + a^{4}\right )} \log \left (b x + a\right )\right )} \sqrt {c x^{2}}}{3 \, {\left (b^{6} x^{2} + a b^{5} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(b^4*x^4 - 2*a*b^3*x^3 + 6*a^2*b^2*x^2 + 9*a^3*b*x - 3*a^4 - 12*(a^3*b*x + a^4)*log(b*x + a))*sqrt(c*x^2)/
(b^6*x^2 + a*b^5*x)

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giac [A]  time = 1.00, size = 96, normalized size = 0.91 \begin {gather*} -\frac {1}{3} \, \sqrt {c} {\left (\frac {12 \, a^{3} \log \left ({\left | b x + a \right |}\right ) \mathrm {sgn}\relax (x)}{b^{5}} + \frac {3 \, a^{4} \mathrm {sgn}\relax (x)}{{\left (b x + a\right )} b^{5}} - \frac {3 \, {\left (4 \, a^{3} \log \left ({\left | a \right |}\right ) + a^{3}\right )} \mathrm {sgn}\relax (x)}{b^{5}} - \frac {b^{4} x^{3} \mathrm {sgn}\relax (x) - 3 \, a b^{3} x^{2} \mathrm {sgn}\relax (x) + 9 \, a^{2} b^{2} x \mathrm {sgn}\relax (x)}{b^{6}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*sqrt(c)*(12*a^3*log(abs(b*x + a))*sgn(x)/b^5 + 3*a^4*sgn(x)/((b*x + a)*b^5) - 3*(4*a^3*log(abs(a)) + a^3)
*sgn(x)/b^5 - (b^4*x^3*sgn(x) - 3*a*b^3*x^2*sgn(x) + 9*a^2*b^2*x*sgn(x))/b^6)

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maple [A]  time = 0.01, size = 88, normalized size = 0.83 \begin {gather*} -\frac {\sqrt {c \,x^{2}}\, \left (-b^{4} x^{4}+2 a \,b^{3} x^{3}+12 a^{3} b x \ln \left (b x +a \right )-6 a^{2} b^{2} x^{2}+12 a^{4} \ln \left (b x +a \right )-9 a^{3} b x +3 a^{4}\right )}{3 \left (b x +a \right ) b^{5} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(c*x^2)^(1/2)*(-b^4*x^4+2*a*b^3*x^3+12*ln(b*x+a)*x*a^3*b-6*a^2*b^2*x^2+12*a^4*ln(b*x+a)-9*a^3*b*x+3*a^4)/
x/b^5/(b*x+a)

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maxima [A]  time = 1.52, size = 135, normalized size = 1.27 \begin {gather*} \frac {\sqrt {c x^{2}} a^{3}}{b^{5} x + a b^{4}} - \frac {4 \, \left (-1\right )^{\frac {2 \, c x}{b}} a^{3} \sqrt {c} \log \left (\frac {2 \, c x}{b}\right )}{b^{5}} - \frac {4 \, \left (-1\right )^{\frac {2 \, a c x}{b}} a^{3} \sqrt {c} \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{b^{5}} - \frac {\sqrt {c x^{2}} a x}{b^{3}} + \frac {3 \, \sqrt {c x^{2}} a^{2}}{b^{4}} + \frac {\left (c x^{2}\right )^{\frac {3}{2}}}{3 \, b^{2} c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^3\,\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^3*(c*x^2)^(1/2))/(a + b*x)^2,x)

[Out]

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

[Out]

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

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