3.373 \(\int (c x)^{-1-\frac{3 j}{2}} (a x^j+b x^n)^{3/2} \, dx\)

Optimal. Leaf size=141 \[ \frac{2 a^{3/2} x^{3 j/2} (c x)^{-3 j/2} \tanh ^{-1}\left (\frac{\sqrt{a} x^{j/2}}{\sqrt{a x^j+b x^n}}\right )}{c (j-n)}-\frac{2 (c x)^{-3 j/2} \left (a x^j+b x^n\right )^{3/2}}{3 c (j-n)}-\frac{2 a x^j (c x)^{-3 j/2} \sqrt{a x^j+b x^n}}{c (j-n)} \]

[Out]

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

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Rubi [A]  time = 0.225356, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {2031, 2028, 2029, 206} \[ \frac{2 a^{3/2} x^{3 j/2} (c x)^{-3 j/2} \tanh ^{-1}\left (\frac{\sqrt{a} x^{j/2}}{\sqrt{a x^j+b x^n}}\right )}{c (j-n)}-\frac{2 (c x)^{-3 j/2} \left (a x^j+b x^n\right )^{3/2}}{3 c (j-n)}-\frac{2 a x^j (c x)^{-3 j/2} \sqrt{a x^j+b x^n}}{c (j-n)} \]

Antiderivative was successfully verified.

[In]

Int[(c*x)^(-1 - (3*j)/2)*(a*x^j + b*x^n)^(3/2),x]

[Out]

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

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

Mathematica [A]  time = 0.233284, size = 131, normalized size = 0.93 \[ -\frac{2 (c x)^{-3 j/2} \left (-3 a^{3/2} \sqrt{b} x^{\frac{1}{2} (3 j+n)} \sqrt{\frac{a x^{j-n}}{b}+1} \sinh ^{-1}\left (\frac{\sqrt{a} x^{\frac{j-n}{2}}}{\sqrt{b}}\right )+4 a^2 x^{2 j}+5 a b x^{j+n}+b^2 x^{2 n}\right )}{3 c (j-n) \sqrt{a x^j+b x^n}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*x)^(-1 - (3*j)/2)*(a*x^j + b*x^n)^(3/2),x]

[Out]

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

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Maple [F]  time = 0.61, size = 0, normalized size = 0. \begin{align*} \int \left ( cx \right ) ^{-1-{\frac{3\,j}{2}}} \left ( a{x}^{j}+b{x}^{n} \right ) ^{{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a x^{j} + b x^{n}\right )}^{\frac{3}{2}} \left (c x\right )^{-\frac{3}{2} \, j - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*x^j + b*x^n)^(3/2)*(c*x)^(-3/2*j - 1), 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((c*x)^(-1-3/2*j)*(a*x^j+b*x^n)^(3/2),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((c*x)**(-1-3/2*j)*(a*x**j+b*x**n)**(3/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a x^{j} + b x^{n}\right )}^{\frac{3}{2}} \left (c x\right )^{-\frac{3}{2} \, j - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*x^j + b*x^n)^(3/2)*(c*x)^(-3/2*j - 1), x)