3.1066 \(\int \frac{x^{-1+2 n} (a+b x^n)^{5/2}}{\sqrt{c+d x^n}} \, dx\)

Optimal. Leaf size=252 \[ \frac{5 (b c-a d)^3 (a d+7 b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b} \sqrt{c+d x^n}}\right )}{64 b^{3/2} d^{9/2} n}-\frac{(a d+7 b c) \left (a+b x^n\right )^{5/2} \sqrt{c+d x^n}}{24 b d^2 n}+\frac{5 (b c-a d) (a d+7 b c) \left (a+b x^n\right )^{3/2} \sqrt{c+d x^n}}{96 b d^3 n}-\frac{5 (b c-a d)^2 (a d+7 b c) \sqrt{a+b x^n} \sqrt{c+d x^n}}{64 b d^4 n}+\frac{\left (a+b x^n\right )^{7/2} \sqrt{c+d x^n}}{4 b d n} \]

[Out]

(-5*(b*c - a*d)^2*(7*b*c + a*d)*Sqrt[a + b*x^n]*Sqrt[c + d*x^n])/(64*b*d^4*n) + (5*(b*c - a*d)*(7*b*c + a*d)*(
a + b*x^n)^(3/2)*Sqrt[c + d*x^n])/(96*b*d^3*n) - ((7*b*c + a*d)*(a + b*x^n)^(5/2)*Sqrt[c + d*x^n])/(24*b*d^2*n
) + ((a + b*x^n)^(7/2)*Sqrt[c + d*x^n])/(4*b*d*n) + (5*(b*c - a*d)^3*(7*b*c + a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b
*x^n])/(Sqrt[b]*Sqrt[c + d*x^n])])/(64*b^(3/2)*d^(9/2)*n)

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Rubi [A]  time = 0.233938, antiderivative size = 252, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {446, 80, 50, 63, 217, 206} \[ \frac{5 (b c-a d)^3 (a d+7 b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b} \sqrt{c+d x^n}}\right )}{64 b^{3/2} d^{9/2} n}-\frac{(a d+7 b c) \left (a+b x^n\right )^{5/2} \sqrt{c+d x^n}}{24 b d^2 n}+\frac{5 (b c-a d) (a d+7 b c) \left (a+b x^n\right )^{3/2} \sqrt{c+d x^n}}{96 b d^3 n}-\frac{5 (b c-a d)^2 (a d+7 b c) \sqrt{a+b x^n} \sqrt{c+d x^n}}{64 b d^4 n}+\frac{\left (a+b x^n\right )^{7/2} \sqrt{c+d x^n}}{4 b d n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*(b*c - a*d)^2*(7*b*c + a*d)*Sqrt[a + b*x^n]*Sqrt[c + d*x^n])/(64*b*d^4*n) + (5*(b*c - a*d)*(7*b*c + a*d)*(
a + b*x^n)^(3/2)*Sqrt[c + d*x^n])/(96*b*d^3*n) - ((7*b*c + a*d)*(a + b*x^n)^(5/2)*Sqrt[c + d*x^n])/(24*b*d^2*n
) + ((a + b*x^n)^(7/2)*Sqrt[c + d*x^n])/(4*b*d*n) + (5*(b*c - a*d)^3*(7*b*c + a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b
*x^n])/(Sqrt[b]*Sqrt[c + d*x^n])])/(64*b^(3/2)*d^(9/2)*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 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 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

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+2 n} \left (a+b x^n\right )^{5/2}}{\sqrt{c+d x^n}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x (a+b x)^{5/2}}{\sqrt{c+d x}} \, dx,x,x^n\right )}{n}\\ &=\frac{\left (a+b x^n\right )^{7/2} \sqrt{c+d x^n}}{4 b d n}-\frac{(7 b c+a d) \operatorname{Subst}\left (\int \frac{(a+b x)^{5/2}}{\sqrt{c+d x}} \, dx,x,x^n\right )}{8 b d n}\\ &=-\frac{(7 b c+a d) \left (a+b x^n\right )^{5/2} \sqrt{c+d x^n}}{24 b d^2 n}+\frac{\left (a+b x^n\right )^{7/2} \sqrt{c+d x^n}}{4 b d n}+\frac{(5 (b c-a d) (7 b c+a d)) \operatorname{Subst}\left (\int \frac{(a+b x)^{3/2}}{\sqrt{c+d x}} \, dx,x,x^n\right )}{48 b d^2 n}\\ &=\frac{5 (b c-a d) (7 b c+a d) \left (a+b x^n\right )^{3/2} \sqrt{c+d x^n}}{96 b d^3 n}-\frac{(7 b c+a d) \left (a+b x^n\right )^{5/2} \sqrt{c+d x^n}}{24 b d^2 n}+\frac{\left (a+b x^n\right )^{7/2} \sqrt{c+d x^n}}{4 b d n}-\frac{\left (5 (b c-a d)^2 (7 b c+a d)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{\sqrt{c+d x}} \, dx,x,x^n\right )}{64 b d^3 n}\\ &=-\frac{5 (b c-a d)^2 (7 b c+a d) \sqrt{a+b x^n} \sqrt{c+d x^n}}{64 b d^4 n}+\frac{5 (b c-a d) (7 b c+a d) \left (a+b x^n\right )^{3/2} \sqrt{c+d x^n}}{96 b d^3 n}-\frac{(7 b c+a d) \left (a+b x^n\right )^{5/2} \sqrt{c+d x^n}}{24 b d^2 n}+\frac{\left (a+b x^n\right )^{7/2} \sqrt{c+d x^n}}{4 b d n}+\frac{\left (5 (b c-a d)^3 (7 b c+a d)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx,x,x^n\right )}{128 b d^4 n}\\ &=-\frac{5 (b c-a d)^2 (7 b c+a d) \sqrt{a+b x^n} \sqrt{c+d x^n}}{64 b d^4 n}+\frac{5 (b c-a d) (7 b c+a d) \left (a+b x^n\right )^{3/2} \sqrt{c+d x^n}}{96 b d^3 n}-\frac{(7 b c+a d) \left (a+b x^n\right )^{5/2} \sqrt{c+d x^n}}{24 b d^2 n}+\frac{\left (a+b x^n\right )^{7/2} \sqrt{c+d x^n}}{4 b d n}+\frac{\left (5 (b c-a d)^3 (7 b c+a d)\right ) \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 )}{64 b^2 d^4 n}\\ &=-\frac{5 (b c-a d)^2 (7 b c+a d) \sqrt{a+b x^n} \sqrt{c+d x^n}}{64 b d^4 n}+\frac{5 (b c-a d) (7 b c+a d) \left (a+b x^n\right )^{3/2} \sqrt{c+d x^n}}{96 b d^3 n}-\frac{(7 b c+a d) \left (a+b x^n\right )^{5/2} \sqrt{c+d x^n}}{24 b d^2 n}+\frac{\left (a+b x^n\right )^{7/2} \sqrt{c+d x^n}}{4 b d n}+\frac{\left (5 (b c-a d)^3 (7 b c+a d)\right ) \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 )}{64 b^2 d^4 n}\\ &=-\frac{5 (b c-a d)^2 (7 b c+a d) \sqrt{a+b x^n} \sqrt{c+d x^n}}{64 b d^4 n}+\frac{5 (b c-a d) (7 b c+a d) \left (a+b x^n\right )^{3/2} \sqrt{c+d x^n}}{96 b d^3 n}-\frac{(7 b c+a d) \left (a+b x^n\right )^{5/2} \sqrt{c+d x^n}}{24 b d^2 n}+\frac{\left (a+b x^n\right )^{7/2} \sqrt{c+d x^n}}{4 b d n}+\frac{5 (b c-a d)^3 (7 b c+a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b} \sqrt{c+d x^n}}\right )}{64 b^{3/2} d^{9/2} n}\\ \end{align*}

Mathematica [A]  time = 0.808172, size = 223, normalized size = 0.88 \[ \frac{b \sqrt{d} \sqrt{a+b x^n} \left (c+d x^n\right ) \left (a^2 b d^2 \left (118 d x^n-191 c\right )+15 a^3 d^3+a b^2 d \left (265 c^2-172 c d x^n+136 d^2 x^{2 n}\right )+b^3 \left (70 c^2 d x^n-105 c^3-56 c d^2 x^{2 n}+48 d^3 x^{3 n}\right )\right )+15 (a d+7 b c) (b c-a d)^{7/2} \sqrt{\frac{b \left (c+d x^n\right )}{b c-a d}} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b c-a d}}\right )}{192 b^2 d^{9/2} n \sqrt{c+d x^n}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*Sqrt[d]*Sqrt[a + b*x^n]*(c + d*x^n)*(15*a^3*d^3 + a^2*b*d^2*(-191*c + 118*d*x^n) + a*b^2*d*(265*c^2 - 172*c
*d*x^n + 136*d^2*x^(2*n)) + b^3*(-105*c^3 + 70*c^2*d*x^n - 56*c*d^2*x^(2*n) + 48*d^3*x^(3*n))) + 15*(b*c - a*d
)^(7/2)*(7*b*c + a*d)*Sqrt[(b*(c + d*x^n))/(b*c - a*d)]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^n])/Sqrt[b*c - a*d]])/(1
92*b^2*d^(9/2)*n*Sqrt[c + d*x^n])

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Maple [F]  time = 0.072, size = 0, normalized size = 0. \begin{align*} \int{{x}^{-1+2\,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+2*n)*(a+b*x^n)^(5/2)/(c+d*x^n)^(1/2),x)

[Out]

int(x^(-1+2*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{{\left (b x^{n} + a\right )}^{\frac{5}{2}} x^{2 \, n - 1}}{\sqrt{d x^{n} + c}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.37345, size = 1342, normalized size = 5.33 \begin{align*} \left [-\frac{15 \,{\left (7 \, b^{4} c^{4} - 20 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} \sqrt{b d} \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 (48 \, b^{4} d^{4} x^{3 \, n} - 105 \, b^{4} c^{3} d + 265 \, a b^{3} c^{2} d^{2} - 191 \, a^{2} b^{2} c d^{3} + 15 \, a^{3} b d^{4} - 8 \,{\left (7 \, b^{4} c d^{3} - 17 \, a b^{3} d^{4}\right )} x^{2 \, n} + 2 \,{\left (35 \, b^{4} c^{2} d^{2} - 86 \, a b^{3} c d^{3} + 59 \, a^{2} b^{2} d^{4}\right )} x^{n}\right )} \sqrt{b x^{n} + a} \sqrt{d x^{n} + c}}{768 \, b^{2} d^{5} n}, -\frac{15 \,{\left (7 \, b^{4} c^{4} - 20 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} \sqrt{-b d} \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 (48 \, b^{4} d^{4} x^{3 \, n} - 105 \, b^{4} c^{3} d + 265 \, a b^{3} c^{2} d^{2} - 191 \, a^{2} b^{2} c d^{3} + 15 \, a^{3} b d^{4} - 8 \,{\left (7 \, b^{4} c d^{3} - 17 \, a b^{3} d^{4}\right )} x^{2 \, n} + 2 \,{\left (35 \, b^{4} c^{2} d^{2} - 86 \, a b^{3} c d^{3} + 59 \, a^{2} b^{2} d^{4}\right )} x^{n}\right )} \sqrt{b x^{n} + a} \sqrt{d x^{n} + c}}{384 \, b^{2} d^{5} n}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/768*(15*(7*b^4*c^4 - 20*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 - a^4*d^4)*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)*sq
rt(d*x^n + c) + 8*(b^2*c*d + a*b*d^2)*x^n) - 4*(48*b^4*d^4*x^(3*n) - 105*b^4*c^3*d + 265*a*b^3*c^2*d^2 - 191*a
^2*b^2*c*d^3 + 15*a^3*b*d^4 - 8*(7*b^4*c*d^3 - 17*a*b^3*d^4)*x^(2*n) + 2*(35*b^4*c^2*d^2 - 86*a*b^3*c*d^3 + 59
*a^2*b^2*d^4)*x^n)*sqrt(b*x^n + a)*sqrt(d*x^n + c))/(b^2*d^5*n), -1/384*(15*(7*b^4*c^4 - 20*a*b^3*c^3*d + 18*a
^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 - a^4*d^4)*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)) - 2*(48*b^4*d^4*x^(3*
n) - 105*b^4*c^3*d + 265*a*b^3*c^2*d^2 - 191*a^2*b^2*c*d^3 + 15*a^3*b*d^4 - 8*(7*b^4*c*d^3 - 17*a*b^3*d^4)*x^(
2*n) + 2*(35*b^4*c^2*d^2 - 86*a*b^3*c*d^3 + 59*a^2*b^2*d^4)*x^n)*sqrt(b*x^n + a)*sqrt(d*x^n + c))/(b^2*d^5*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+2*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{{\left (b x^{n} + a\right )}^{\frac{5}{2}} x^{2 \, n - 1}}{\sqrt{d x^{n} + c}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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