3.1077 \(\int \frac{x^{-1+3 n}}{(a+b x^n)^{5/2} \sqrt{c+d x^n}} \, dx\)

Optimal. Leaf size=147 \[ -\frac{2 a^2 \sqrt{c+d x^n}}{3 b^2 n (b c-a d) \left (a+b x^n\right )^{3/2}}+\frac{4 a (3 b c-2 a d) \sqrt{c+d x^n}}{3 b^2 n (b c-a d)^2 \sqrt{a+b x^n}}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b} \sqrt{c+d x^n}}\right )}{b^{5/2} \sqrt{d} n} \]

[Out]

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

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Rubi [A]  time = 0.142043, antiderivative size = 147, 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, 78, 63, 217, 206} \[ -\frac{2 a^2 \sqrt{c+d x^n}}{3 b^2 n (b c-a d) \left (a+b x^n\right )^{3/2}}+\frac{4 a (3 b c-2 a d) \sqrt{c+d x^n}}{3 b^2 n (b c-a d)^2 \sqrt{a+b x^n}}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b} \sqrt{c+d x^n}}\right )}{b^{5/2} \sqrt{d} n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 + 3*n)/((a + b*x^n)^(5/2)*Sqrt[c + d*x^n]),x]

[Out]

(-2*a^2*Sqrt[c + d*x^n])/(3*b^2*(b*c - a*d)*n*(a + b*x^n)^(3/2)) + (4*a*(3*b*c - 2*a*d)*Sqrt[c + d*x^n])/(3*b^
2*(b*c - a*d)^2*n*Sqrt[a + b*x^n]) + (2*ArcTanh[(Sqrt[d]*Sqrt[a + b*x^n])/(Sqrt[b]*Sqrt[c + d*x^n])])/(b^(5/2)
*Sqrt[d]*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 78

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

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

Mathematica [A]  time = 0.68142, size = 217, normalized size = 1.48 \[ \frac{2 \sqrt{c+d x^n} \left (\frac{\left (3 b^2 c^2-a^2 d^2\right ) \left (a+b x^n\right )}{d (b c-a d)^2}+\frac{a^2}{a d-b c}-\frac{3 \left (a+b x^n\right ) \left (\sqrt{b c-a d} \sqrt{\frac{b \left (c+d x^n\right )}{b c-a d}}-\sqrt{d} \sqrt{a+b x^n} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b c-a d}}\right )\right )}{d \sqrt{b c-a d} \sqrt{\frac{b \left (c+d x^n\right )}{b c-a d}}}\right )}{3 b^2 n \left (a+b x^n\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 + 3*n)/((a + b*x^n)^(5/2)*Sqrt[c + d*x^n]),x]

[Out]

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

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Maple [F]  time = 0.066, size = 0, normalized size = 0. \begin{align*} \int{{x}^{-1+3\,n} \left ( a+b{x}^{n} \right ) ^{-{\frac{5}{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)^(5/2)/(c+d*x^n)^(1/2),x)

[Out]

int(x^(-1+3*n)/(a+b*x^n)^(5/2)/(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{5}{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)^(5/2)/(c+d*x^n)^(1/2),x, algorithm="maxima")

[Out]

integrate(x^(3*n - 1)/((b*x^n + a)^(5/2)*sqrt(d*x^n + c)), x)

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Fricas [B]  time = 3.04903, size = 1620, normalized size = 11.02 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+3*n)/(a+b*x^n)^(5/2)/(c+d*x^n)^(1/2),x, algorithm="fricas")

[Out]

[1/6*(4*(5*a^2*b^2*c*d - 3*a^3*b*d^2 + 2*(3*a*b^3*c*d - 2*a^2*b^2*d^2)*x^n)*sqrt(b*x^n + a)*sqrt(d*x^n + c) +
3*((b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*sqrt(b*d)*x^(2*n) + 2*(a*b^3*c^2 - 2*a^2*b^2*c*d + a^3*b*d^2)*sqrt(b*
d)*x^n + (a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*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^7*c^2*d - 2*a*b^6*c*d^2 + a^2*b^5*d^3)*n*x^(2*n) + 2*(a*b^6*c^2*d - 2*a^2*b^5*c*d^2 + a^3*b^4*d^3)*n*x
^n + (a^2*b^5*c^2*d - 2*a^3*b^4*c*d^2 + a^4*b^3*d^3)*n), 1/3*(2*(5*a^2*b^2*c*d - 3*a^3*b*d^2 + 2*(3*a*b^3*c*d
- 2*a^2*b^2*d^2)*x^n)*sqrt(b*x^n + a)*sqrt(d*x^n + c) - 3*((b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*sqrt(-b*d)*x^
(2*n) + 2*(a*b^3*c^2 - 2*a^2*b^2*c*d + a^3*b*d^2)*sqrt(-b*d)*x^n + (a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*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^7*c^2*d - 2*a*b^6*c*d^2 + a^2*b^5*d^3)*n*x^(2*n) + 2*(a*b^6*c^
2*d - 2*a^2*b^5*c*d^2 + a^3*b^4*d^3)*n*x^n + (a^2*b^5*c^2*d - 2*a^3*b^4*c*d^2 + a^4*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)**(5/2)/(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{5}{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)^(5/2)/(c+d*x^n)^(1/2),x, algorithm="giac")

[Out]

integrate(x^(3*n - 1)/((b*x^n + a)^(5/2)*sqrt(d*x^n + c)), x)