3.374 \(\int \frac{(a x^3+b x^n)^{3/2}}{(c x)^{11/2}} \, dx\)

Optimal. Leaf size=128 \[ \frac{2 a^{3/2} \sqrt{c x} \tanh ^{-1}\left (\frac{\sqrt{a} x^{3/2}}{\sqrt{a x^3+b x^n}}\right )}{c^6 (3-n) \sqrt{x}}-\frac{2 a \sqrt{a x^3+b x^n}}{c^4 (3-n) (c x)^{3/2}}-\frac{2 \left (a x^3+b x^n\right )^{3/2}}{3 c (3-n) (c x)^{9/2}} \]

[Out]

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

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Rubi [A]  time = 0.208872, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {2028, 2031, 2029, 206} \[ \frac{2 a^{3/2} \sqrt{c x} \tanh ^{-1}\left (\frac{\sqrt{a} x^{3/2}}{\sqrt{a x^3+b x^n}}\right )}{c^6 (3-n) \sqrt{x}}-\frac{2 a \sqrt{a x^3+b x^n}}{c^4 (3-n) (c x)^{3/2}}-\frac{2 \left (a x^3+b x^n\right )^{3/2}}{3 c (3-n) (c x)^{9/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a*x^3 + b*x^n)^(3/2)/(c*x)^(11/2),x]

[Out]

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

Rule 2028

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a*x^j + b
*x^n)^p)/(c*p*(n - j)), x] + Dist[a/c^j, Int[(c*x)^(m + j)*(a*x^j + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c,
j, m, n}, x] && IGtQ[p + 1/2, 0] && NeQ[n, j] && EqQ[Simplify[m + j*p + 1], 0] && (IntegerQ[j] || GtQ[c, 0])

Rule 2031

Int[((c_)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(c^IntPart[m]*(c*x)^FracPar
t[m])/x^FracPart[m], Int[x^m*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] && IntegerQ[p + 1/2]
&& NeQ[n, j] && EqQ[Simplify[m + j*p + 1], 0]

Rule 2029

Int[(x_)^(m_.)/Sqrt[(a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> Dist[-2/(n - j), Subst[Int[1/(1 - a*x^2
), x], x, x^(j/2)/Sqrt[a*x^j + b*x^n]], x] /; FreeQ[{a, b, j, n}, x] && EqQ[m, j/2 - 1] && NeQ[n, j]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\left (a x^3+b x^n\right )^{3/2}}{(c x)^{11/2}} \, dx &=-\frac{2 \left (a x^3+b x^n\right )^{3/2}}{3 c (3-n) (c x)^{9/2}}+\frac{a \int \frac{\sqrt{a x^3+b x^n}}{(c x)^{5/2}} \, dx}{c^3}\\ &=-\frac{2 a \sqrt{a x^3+b x^n}}{c^4 (3-n) (c x)^{3/2}}-\frac{2 \left (a x^3+b x^n\right )^{3/2}}{3 c (3-n) (c x)^{9/2}}+\frac{a^2 \int \frac{\sqrt{c x}}{\sqrt{a x^3+b x^n}} \, dx}{c^6}\\ &=-\frac{2 a \sqrt{a x^3+b x^n}}{c^4 (3-n) (c x)^{3/2}}-\frac{2 \left (a x^3+b x^n\right )^{3/2}}{3 c (3-n) (c x)^{9/2}}+\frac{\left (a^2 \sqrt{c x}\right ) \int \frac{\sqrt{x}}{\sqrt{a x^3+b x^n}} \, dx}{c^6 \sqrt{x}}\\ &=-\frac{2 a \sqrt{a x^3+b x^n}}{c^4 (3-n) (c x)^{3/2}}-\frac{2 \left (a x^3+b x^n\right )^{3/2}}{3 c (3-n) (c x)^{9/2}}+\frac{\left (2 a^2 \sqrt{c x}\right ) \operatorname{Subst}\left (\int \frac{1}{1-a x^2} \, dx,x,\frac{x^{3/2}}{\sqrt{a x^3+b x^n}}\right )}{c^6 (3-n) \sqrt{x}}\\ &=-\frac{2 a \sqrt{a x^3+b x^n}}{c^4 (3-n) (c x)^{3/2}}-\frac{2 \left (a x^3+b x^n\right )^{3/2}}{3 c (3-n) (c x)^{9/2}}+\frac{2 a^{3/2} \sqrt{c x} \tanh ^{-1}\left (\frac{\sqrt{a} x^{3/2}}{\sqrt{a x^3+b x^n}}\right )}{c^6 (3-n) \sqrt{x}}\\ \end{align*}

Mathematica [A]  time = 0.293409, size = 126, normalized size = 0.98 \[ \frac{2 \sqrt{c x} \left (-3 a^{3/2} \sqrt{b} x^{\frac{n+9}{2}} \sqrt{\frac{a x^{3-n}}{b}+1} \sinh ^{-1}\left (\frac{\sqrt{a} x^{\frac{3}{2}-\frac{n}{2}}}{\sqrt{b}}\right )+4 a^2 x^6+5 a b x^{n+3}+b^2 x^{2 n}\right )}{3 c^6 (n-3) x^5 \sqrt{a x^3+b x^n}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*x^3 + b*x^n)^(3/2)/(c*x)^(11/2),x]

[Out]

(2*Sqrt[c*x]*(4*a^2*x^6 + b^2*x^(2*n) + 5*a*b*x^(3 + n) - 3*a^(3/2)*Sqrt[b]*x^((9 + n)/2)*Sqrt[1 + (a*x^(3 - n
))/b]*ArcSinh[(Sqrt[a]*x^(3/2 - n/2))/Sqrt[b]]))/(3*c^6*(-3 + n)*x^5*Sqrt[a*x^3 + b*x^n])

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Maple [F]  time = 0.335, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a{x}^{3}+b{x}^{n} \right ) ^{{\frac{3}{2}}} \left ( cx \right ) ^{-{\frac{11}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^3+b*x^n)^(3/2)/(c*x)^(11/2),x)

[Out]

int((a*x^3+b*x^n)^(3/2)/(c*x)^(11/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a x^{3} + b x^{n}\right )}^{\frac{3}{2}}}{\left (c x\right )^{\frac{11}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^3+b*x^n)^(3/2)/(c*x)^(11/2),x, algorithm="maxima")

[Out]

integrate((a*x^3 + b*x^n)^(3/2)/(c*x)^(11/2), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^3+b*x^n)^(3/2)/(c*x)^(11/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x**3+b*x**n)**(3/2)/(c*x)**(11/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a x^{3} + b x^{n}\right )}^{\frac{3}{2}}}{\left (c x\right )^{\frac{11}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^3+b*x^n)^(3/2)/(c*x)^(11/2),x, algorithm="giac")

[Out]

integrate((a*x^3 + b*x^n)^(3/2)/(c*x)^(11/2), x)