3.11.76 \(\int \frac {x^{-1+3 n}}{(a+b x^n)^{3/2} \sqrt {c+d x^n}} \, dx\) [1076]

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

[Out]

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

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Rubi [A]
time = 0.10, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {457, 91, 81, 65, 223, 212} \begin {gather*} -\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} \end {gather*}

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 65

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 81

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 91

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 212

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

Rule 223

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 457

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]]

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 {\text {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 \text {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) \text {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) \text {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) \text {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*}

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Mathematica [A]
time = 0.50, size = 185, normalized size = 1.39 \begin {gather*} \frac {-b \sqrt {d} \left (c+d x^n\right ) \left (-3 a^2 d+b^2 c x^n+a b \left (c-d x^n\right )\right )+\sqrt {b c-a d} \left (b^2 c^2+2 a b c d-3 a^2 d^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^3 d^{3/2} (-b c+a d) n \sqrt {a+b x^n} \sqrt {c+d x^n}} \end {gather*}

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 = 0.11, size = 0, normalized size = 0.00 \[\int \frac {x^{-1+3 n}}{\left (a +b \,x^{n}\right )^{\frac {3}{2}} \sqrt {c +d \,x^{n}}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 264 vs. \(2 (113) = 226\).
time = 2.98, size = 540, normalized size = 4.06 \begin {gather*} \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 {gather*}

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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

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]

Exception raised: SystemError >> excessive stack use: stack is 3006 deep

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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