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

Optimal. Leaf size=107 \[ \frac {a^4 c \sqrt {c x^2} \log (a+b x)}{b^5 x}-\frac {a^3 c \sqrt {c x^2}}{b^4}+\frac {a^2 c x \sqrt {c x^2}}{2 b^3}-\frac {a c x^2 \sqrt {c x^2}}{3 b^2}+\frac {c x^3 \sqrt {c x^2}}{4 b} \]

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Rubi [A]  time = 0.03, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {15, 43} \begin {gather*} -\frac {a^3 c \sqrt {c x^2}}{b^4}+\frac {a^2 c x \sqrt {c x^2}}{2 b^3}+\frac {a^4 c \sqrt {c x^2} \log (a+b x)}{b^5 x}-\frac {a c x^2 \sqrt {c x^2}}{3 b^2}+\frac {c x^3 \sqrt {c x^2}}{4 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x*(c*x^2)^(3/2))/(a + b*x),x]

[Out]

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

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Mathematica [A]  time = 0.01, size = 64, normalized size = 0.60 \begin {gather*} \frac {\left (c x^2\right )^{3/2} \left (12 a^4 \log (a+b x)+b x \left (-12 a^3+6 a^2 b x-4 a b^2 x^2+3 b^3 x^3\right )\right )}{12 b^5 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.05, size = 67, normalized size = 0.63 \begin {gather*} \left (c x^2\right )^{3/2} \left (\frac {a^4 \log (a+b x)}{b^5 x^3}+\frac {-12 a^3+6 a^2 b x-4 a b^2 x^2+3 b^3 x^3}{12 b^4 x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.04, size = 67, normalized size = 0.63 \begin {gather*} \frac {{\left (3 \, b^{4} c x^{4} - 4 \, a b^{3} c x^{3} + 6 \, a^{2} b^{2} c x^{2} - 12 \, a^{3} b c x + 12 \, a^{4} c \log \left (b x + a\right )\right )} \sqrt {c x^{2}}}{12 \, b^{5} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 1.00, size = 81, normalized size = 0.76 \begin {gather*} \frac {1}{12} \, c^{\frac {3}{2}} {\left (\frac {12 \, a^{4} \log \left ({\left | b x + a \right |}\right ) \mathrm {sgn}\relax (x)}{b^{5}} - \frac {12 \, a^{4} \log \left ({\left | a \right |}\right ) \mathrm {sgn}\relax (x)}{b^{5}} + \frac {3 \, b^{3} x^{4} \mathrm {sgn}\relax (x) - 4 \, a b^{2} x^{3} \mathrm {sgn}\relax (x) + 6 \, a^{2} b x^{2} \mathrm {sgn}\relax (x) - 12 \, a^{3} x \mathrm {sgn}\relax (x)}{b^{4}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 63, normalized size = 0.59 \begin {gather*} \frac {\left (c \,x^{2}\right )^{\frac {3}{2}} \left (3 b^{4} x^{4}-4 a \,b^{3} x^{3}+6 a^{2} b^{2} x^{2}+12 a^{4} \ln \left (b x +a \right )-12 a^{3} b x \right )}{12 b^{5} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.62, size = 124, normalized size = 1.16 \begin {gather*} \frac {\left (-1\right )^{\frac {2 \, c x}{b}} a^{4} c^{\frac {3}{2}} \log \left (\frac {2 \, c x}{b}\right )}{b^{5}} + \frac {\left (-1\right )^{\frac {2 \, a c x}{b}} a^{4} c^{\frac {3}{2}} \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{b^{5}} + \frac {\left (c x^{2}\right )^{\frac {3}{2}} x}{4 \, b} + \frac {\sqrt {c x^{2}} a^{2} c x}{2 \, b^{3}} - \frac {\left (c x^{2}\right )^{\frac {3}{2}} a}{3 \, b^{2}} - \frac {\sqrt {c x^{2}} a^{3} c}{b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*(c*x^2)^(3/2))/(a + b*x),x)

[Out]

int((x*(c*x^2)^(3/2))/(a + b*x), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (c x^{2}\right )^{\frac {3}{2}}}{a + b x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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