3.1076 \(\int \frac{x^{-1+3 n}}{(a+b x^n)^{3/2} \sqrt{c+d x^n}} \, dx\)

Optimal. Leaf size=133 \[ -\frac{2 a^2 \sqrt{c+d x^n}}{b^2 n (b c-a d) \sqrt{a+b x^n}}-\frac{(3 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b} \sqrt{c+d x^n}}\right )}{b^{5/2} d^{3/2} n}+\frac{\sqrt{a+b x^n} \sqrt{c+d x^n}}{b^2 d n} \]

[Out]

(-2*a^2*Sqrt[c + d*x^n])/(b^2*(b*c - a*d)*n*Sqrt[a + b*x^n]) + (Sqrt[a + b*x^n]*Sqrt[c + d*x^n])/(b^2*d*n) - (
(b*c + 3*a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x^n])/(Sqrt[b]*Sqrt[c + d*x^n])])/(b^(5/2)*d^(3/2)*n)

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Rubi [A]  time = 0.160575, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {446, 89, 80, 63, 217, 206} \[ -\frac{2 a^2 \sqrt{c+d x^n}}{b^2 n (b c-a d) \sqrt{a+b x^n}}-\frac{(3 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b} \sqrt{c+d x^n}}\right )}{b^{5/2} d^{3/2} n}+\frac{\sqrt{a+b x^n} \sqrt{c+d x^n}}{b^2 d n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a^2*Sqrt[c + d*x^n])/(b^2*(b*c - a*d)*n*Sqrt[a + b*x^n]) + (Sqrt[a + b*x^n]*Sqrt[c + d*x^n])/(b^2*d*n) - (
(b*c + 3*a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x^n])/(Sqrt[b]*Sqrt[c + d*x^n])])/(b^(5/2)*d^(3/2)*n)

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 89

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c - a*
d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

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

Mathematica [A]  time = 0.434388, size = 185, normalized size = 1.39 \[ \frac{\sqrt{b c-a d} \left (-3 a^2 d^2+2 a b c d+b^2 c^2\right ) \sqrt{a+b x^n} \sqrt{\frac{b \left (c+d x^n\right )}{b c-a d}} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b c-a d}}\right )-b \sqrt{d} \left (c+d x^n\right ) \left (-3 a^2 d+a b \left (c-d x^n\right )+b^2 c x^n\right )}{b^3 d^{3/2} n (a d-b c) \sqrt{a+b x^n} \sqrt{c+d x^n}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.067, size = 0, normalized size = 0. \begin{align*} \int{{x}^{-1+3\,n} \left ( a+b{x}^{n} \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{c+d{x}^{n}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.87666, size = 1143, normalized size = 8.59 \begin{align*} \left [\frac{4 \,{\left (a b^{2} c d - 3 \, a^{2} b d^{2} +{\left (b^{3} c d - a b^{2} d^{2}\right )} x^{n}\right )} \sqrt{b x^{n} + a} \sqrt{d x^{n} + c} +{\left ({\left (b^{3} c^{2} + 2 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} \sqrt{b d} x^{n} +{\left (a b^{2} c^{2} + 2 \, a^{2} b c d - 3 \, a^{3} d^{2}\right )} \sqrt{b d}\right )} \log \left (8 \, b^{2} d^{2} x^{2 \, n} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} - 4 \,{\left (2 \, \sqrt{b d} b d x^{n} +{\left (b c + a d\right )} \sqrt{b d}\right )} \sqrt{b x^{n} + a} \sqrt{d x^{n} + c} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x^{n}\right )}{4 \,{\left ({\left (b^{5} c d^{2} - a b^{4} d^{3}\right )} n x^{n} +{\left (a b^{4} c d^{2} - a^{2} b^{3} d^{3}\right )} n\right )}}, \frac{2 \,{\left (a b^{2} c d - 3 \, a^{2} b d^{2} +{\left (b^{3} c d - a b^{2} d^{2}\right )} x^{n}\right )} \sqrt{b x^{n} + a} \sqrt{d x^{n} + c} +{\left ({\left (b^{3} c^{2} + 2 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} \sqrt{-b d} x^{n} +{\left (a b^{2} c^{2} + 2 \, a^{2} b c d - 3 \, a^{3} d^{2}\right )} \sqrt{-b d}\right )} \arctan \left (\frac{{\left (2 \, \sqrt{-b d} b d x^{n} +{\left (b c + a d\right )} \sqrt{-b d}\right )} \sqrt{b x^{n} + a} \sqrt{d x^{n} + c}}{2 \,{\left (b^{2} d^{2} x^{2 \, n} + a b c d +{\left (b^{2} c d + a b d^{2}\right )} x^{n}\right )}}\right )}{2 \,{\left ({\left (b^{5} c d^{2} - a b^{4} d^{3}\right )} n x^{n} +{\left (a b^{4} c d^{2} - a^{2} b^{3} d^{3}\right )} n\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(4*(a*b^2*c*d - 3*a^2*b*d^2 + (b^3*c*d - a*b^2*d^2)*x^n)*sqrt(b*x^n + a)*sqrt(d*x^n + c) + ((b^3*c^2 + 2*
a*b^2*c*d - 3*a^2*b*d^2)*sqrt(b*d)*x^n + (a*b^2*c^2 + 2*a^2*b*c*d - 3*a^3*d^2)*sqrt(b*d))*log(8*b^2*d^2*x^(2*n
) + b^2*c^2 + 6*a*b*c*d + a^2*d^2 - 4*(2*sqrt(b*d)*b*d*x^n + (b*c + a*d)*sqrt(b*d))*sqrt(b*x^n + a)*sqrt(d*x^n
+ c) + 8*(b^2*c*d + a*b*d^2)*x^n))/((b^5*c*d^2 - a*b^4*d^3)*n*x^n + (a*b^4*c*d^2 - a^2*b^3*d^3)*n), 1/2*(2*(a
*b^2*c*d - 3*a^2*b*d^2 + (b^3*c*d - a*b^2*d^2)*x^n)*sqrt(b*x^n + a)*sqrt(d*x^n + c) + ((b^3*c^2 + 2*a*b^2*c*d
- 3*a^2*b*d^2)*sqrt(-b*d)*x^n + (a*b^2*c^2 + 2*a^2*b*c*d - 3*a^3*d^2)*sqrt(-b*d))*arctan(1/2*(2*sqrt(-b*d)*b*d
*x^n + (b*c + a*d)*sqrt(-b*d))*sqrt(b*x^n + a)*sqrt(d*x^n + c)/(b^2*d^2*x^(2*n) + a*b*c*d + (b^2*c*d + a*b*d^2
)*x^n)))/((b^5*c*d^2 - a*b^4*d^3)*n*x^n + (a*b^4*c*d^2 - a^2*b^3*d^3)*n)]

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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(x**(-1+3*n)/(a+b*x**n)**(3/2)/(c+d*x**n)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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