3.1072 \(\int \frac{x^{-1+3 n} (a+b x^n)^{5/2}}{\sqrt{c+d x^n}} \, dx\)

Optimal. Leaf size=358 \[ \frac{\left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \left (a+b x^n\right )^{5/2} \sqrt{c+d x^n}}{240 b^2 d^3 n}-\frac{(b c-a d) \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \left (a+b x^n\right )^{3/2} \sqrt{c+d x^n}}{192 b^2 d^4 n}+\frac{(b c-a d)^2 \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \sqrt{a+b x^n} \sqrt{c+d x^n}}{128 b^2 d^5 n}-\frac{(b c-a d)^3 \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b} \sqrt{c+d x^n}}\right )}{128 b^{5/2} d^{11/2} n}-\frac{3 (a d+3 b c) \left (a+b x^n\right )^{7/2} \sqrt{c+d x^n}}{40 b^2 d^2 n}+\frac{x^n \left (a+b x^n\right )^{7/2} \sqrt{c+d x^n}}{5 b d n} \]

[Out]

((b*c - a*d)^2*(63*b^2*c^2 + 14*a*b*c*d + 3*a^2*d^2)*Sqrt[a + b*x^n]*Sqrt[c + d*x^n])/(128*b^2*d^5*n) - ((b*c
- a*d)*(63*b^2*c^2 + 14*a*b*c*d + 3*a^2*d^2)*(a + b*x^n)^(3/2)*Sqrt[c + d*x^n])/(192*b^2*d^4*n) + ((63*b^2*c^2
+ 14*a*b*c*d + 3*a^2*d^2)*(a + b*x^n)^(5/2)*Sqrt[c + d*x^n])/(240*b^2*d^3*n) - (3*(3*b*c + a*d)*(a + b*x^n)^(
7/2)*Sqrt[c + d*x^n])/(40*b^2*d^2*n) + (x^n*(a + b*x^n)^(7/2)*Sqrt[c + d*x^n])/(5*b*d*n) - ((b*c - a*d)^3*(63*
b^2*c^2 + 14*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x^n])/(Sqrt[b]*Sqrt[c + d*x^n])])/(128*b^(5/2)*d
^(11/2)*n)

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Rubi [A]  time = 0.404693, antiderivative size = 358, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.233, Rules used = {446, 90, 80, 50, 63, 217, 206} \[ \frac{\left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \left (a+b x^n\right )^{5/2} \sqrt{c+d x^n}}{240 b^2 d^3 n}-\frac{(b c-a d) \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \left (a+b x^n\right )^{3/2} \sqrt{c+d x^n}}{192 b^2 d^4 n}+\frac{(b c-a d)^2 \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \sqrt{a+b x^n} \sqrt{c+d x^n}}{128 b^2 d^5 n}-\frac{(b c-a d)^3 \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b} \sqrt{c+d x^n}}\right )}{128 b^{5/2} d^{11/2} n}-\frac{3 (a d+3 b c) \left (a+b x^n\right )^{7/2} \sqrt{c+d x^n}}{40 b^2 d^2 n}+\frac{x^n \left (a+b x^n\right )^{7/2} \sqrt{c+d x^n}}{5 b d n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b*c - a*d)^2*(63*b^2*c^2 + 14*a*b*c*d + 3*a^2*d^2)*Sqrt[a + b*x^n]*Sqrt[c + d*x^n])/(128*b^2*d^5*n) - ((b*c
- a*d)*(63*b^2*c^2 + 14*a*b*c*d + 3*a^2*d^2)*(a + b*x^n)^(3/2)*Sqrt[c + d*x^n])/(192*b^2*d^4*n) + ((63*b^2*c^2
+ 14*a*b*c*d + 3*a^2*d^2)*(a + b*x^n)^(5/2)*Sqrt[c + d*x^n])/(240*b^2*d^3*n) - (3*(3*b*c + a*d)*(a + b*x^n)^(
7/2)*Sqrt[c + d*x^n])/(40*b^2*d^2*n) + (x^n*(a + b*x^n)^(7/2)*Sqrt[c + d*x^n])/(5*b*d*n) - ((b*c - a*d)^3*(63*
b^2*c^2 + 14*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x^n])/(Sqrt[b]*Sqrt[c + d*x^n])])/(128*b^(5/2)*d
^(11/2)*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 90

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a + b*
x)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 3)), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

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+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{x^n \left (a+b x^n\right )^{7/2} \sqrt{c+d x^n}}{5 b d n}+\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^{5/2} \left (-a c-\frac{3}{2} (3 b c+a d) x\right )}{\sqrt{c+d x}} \, dx,x,x^n\right )}{5 b d n}\\ &=-\frac{3 (3 b c+a d) \left (a+b x^n\right )^{7/2} \sqrt{c+d x^n}}{40 b^2 d^2 n}+\frac{x^n \left (a+b x^n\right )^{7/2} \sqrt{c+d x^n}}{5 b d n}+\frac{\left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^{5/2}}{\sqrt{c+d x}} \, dx,x,x^n\right )}{80 b^2 d^2 n}\\ &=\frac{\left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (a+b x^n\right )^{5/2} \sqrt{c+d x^n}}{240 b^2 d^3 n}-\frac{3 (3 b c+a d) \left (a+b x^n\right )^{7/2} \sqrt{c+d x^n}}{40 b^2 d^2 n}+\frac{x^n \left (a+b x^n\right )^{7/2} \sqrt{c+d x^n}}{5 b d n}-\frac{\left ((b c-a d) \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right )\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^{3/2}}{\sqrt{c+d x}} \, dx,x,x^n\right )}{96 b^2 d^3 n}\\ &=-\frac{(b c-a d) \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (a+b x^n\right )^{3/2} \sqrt{c+d x^n}}{192 b^2 d^4 n}+\frac{\left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (a+b x^n\right )^{5/2} \sqrt{c+d x^n}}{240 b^2 d^3 n}-\frac{3 (3 b c+a d) \left (a+b x^n\right )^{7/2} \sqrt{c+d x^n}}{40 b^2 d^2 n}+\frac{x^n \left (a+b x^n\right )^{7/2} \sqrt{c+d x^n}}{5 b d n}+\frac{\left ((b c-a d)^2 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right )\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{\sqrt{c+d x}} \, dx,x,x^n\right )}{128 b^2 d^4 n}\\ &=\frac{(b c-a d)^2 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \sqrt{a+b x^n} \sqrt{c+d x^n}}{128 b^2 d^5 n}-\frac{(b c-a d) \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (a+b x^n\right )^{3/2} \sqrt{c+d x^n}}{192 b^2 d^4 n}+\frac{\left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (a+b x^n\right )^{5/2} \sqrt{c+d x^n}}{240 b^2 d^3 n}-\frac{3 (3 b c+a d) \left (a+b x^n\right )^{7/2} \sqrt{c+d x^n}}{40 b^2 d^2 n}+\frac{x^n \left (a+b x^n\right )^{7/2} \sqrt{c+d x^n}}{5 b d n}-\frac{\left ((b c-a d)^3 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx,x,x^n\right )}{256 b^2 d^5 n}\\ &=\frac{(b c-a d)^2 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \sqrt{a+b x^n} \sqrt{c+d x^n}}{128 b^2 d^5 n}-\frac{(b c-a d) \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (a+b x^n\right )^{3/2} \sqrt{c+d x^n}}{192 b^2 d^4 n}+\frac{\left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (a+b x^n\right )^{5/2} \sqrt{c+d x^n}}{240 b^2 d^3 n}-\frac{3 (3 b c+a d) \left (a+b x^n\right )^{7/2} \sqrt{c+d x^n}}{40 b^2 d^2 n}+\frac{x^n \left (a+b x^n\right )^{7/2} \sqrt{c+d x^n}}{5 b d n}-\frac{\left ((b c-a d)^3 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right )\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 )}{128 b^3 d^5 n}\\ &=\frac{(b c-a d)^2 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \sqrt{a+b x^n} \sqrt{c+d x^n}}{128 b^2 d^5 n}-\frac{(b c-a d) \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (a+b x^n\right )^{3/2} \sqrt{c+d x^n}}{192 b^2 d^4 n}+\frac{\left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (a+b x^n\right )^{5/2} \sqrt{c+d x^n}}{240 b^2 d^3 n}-\frac{3 (3 b c+a d) \left (a+b x^n\right )^{7/2} \sqrt{c+d x^n}}{40 b^2 d^2 n}+\frac{x^n \left (a+b x^n\right )^{7/2} \sqrt{c+d x^n}}{5 b d n}-\frac{\left ((b c-a d)^3 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right )\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 )}{128 b^3 d^5 n}\\ &=\frac{(b c-a d)^2 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \sqrt{a+b x^n} \sqrt{c+d x^n}}{128 b^2 d^5 n}-\frac{(b c-a d) \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (a+b x^n\right )^{3/2} \sqrt{c+d x^n}}{192 b^2 d^4 n}+\frac{\left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (a+b x^n\right )^{5/2} \sqrt{c+d x^n}}{240 b^2 d^3 n}-\frac{3 (3 b c+a d) \left (a+b x^n\right )^{7/2} \sqrt{c+d x^n}}{40 b^2 d^2 n}+\frac{x^n \left (a+b x^n\right )^{7/2} \sqrt{c+d x^n}}{5 b d n}-\frac{(b c-a d)^3 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b} \sqrt{c+d x^n}}\right )}{128 b^{5/2} d^{11/2} n}\\ \end{align*}

Mathematica [A]  time = 1.59239, size = 274, normalized size = 0.77 \[ \frac{\sqrt{c+d x^n} \left (\frac{5 (b c-a d)^3 \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \left (-\frac{16 d^3 \left (a+b x^n\right )^3}{15 (a d-b c)^3}-\frac{4 d^2 \left (a+b x^n\right )^2}{3 (b c-a d)^2}-\frac{2 d \left (a+b x^n\right )}{a d-b c}-\frac{2 \sqrt{d} \sqrt{a+b x^n} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b c-a d}}\right )}{\sqrt{b c-a d} \sqrt{\frac{b \left (c+d x^n\right )}{b c-a d}}}\right )}{4 b d^5}-\frac{24 (a d+3 b c) \left (a+b x^n\right )^4}{b d}+64 x^n \left (a+b x^n\right )^4\right )}{320 b d n \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c + d*x^n]*((-24*(3*b*c + a*d)*(a + b*x^n)^4)/(b*d) + 64*x^n*(a + b*x^n)^4 + (5*(b*c - a*d)^3*(63*b^2*c^
2 + 14*a*b*c*d + 3*a^2*d^2)*((-2*d*(a + b*x^n))/(-(b*c) + a*d) - (4*d^2*(a + b*x^n)^2)/(3*(b*c - a*d)^2) - (16
*d^3*(a + b*x^n)^3)/(15*(-(b*c) + a*d)^3) - (2*Sqrt[d]*Sqrt[a + b*x^n]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^n])/Sqrt[
b*c - a*d]])/(Sqrt[b*c - a*d]*Sqrt[(b*(c + d*x^n))/(b*c - a*d)])))/(4*b*d^5)))/(320*b*d*n*Sqrt[a + b*x^n])

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Maple [F]  time = 0.065, 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{{\left (b x^{n} + a\right )}^{\frac{5}{2}} x^{3 \, 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+3*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^(3*n - 1)/sqrt(d*x^n + c), x)

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Fricas [A]  time = 1.64667, size = 1719, normalized size = 4.8 \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]

[-1/7680*(15*(63*b^5*c^5 - 175*a*b^4*c^4*d + 150*a^2*b^3*c^3*d^2 - 30*a^3*b^2*c^2*d^3 - 5*a^4*b*c*d^4 - 3*a^5*
d^5)*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)*sq
rt(b*d))*sqrt(b*x^n + a)*sqrt(d*x^n + c) + 8*(b^2*c*d + a*b*d^2)*x^n) - 4*(384*b^5*d^5*x^(4*n) + 945*b^5*c^4*d
- 2310*a*b^4*c^3*d^2 + 1564*a^2*b^3*c^2*d^3 - 90*a^3*b^2*c*d^4 - 45*a^4*b*d^5 - 144*(3*b^5*c*d^4 - 7*a*b^4*d^
5)*x^(3*n) + 8*(63*b^5*c^2*d^3 - 148*a*b^4*c*d^4 + 93*a^2*b^3*d^5)*x^(2*n) - 2*(315*b^5*c^3*d^2 - 749*a*b^4*c^
2*d^3 + 481*a^2*b^3*c*d^4 - 15*a^3*b^2*d^5)*x^n)*sqrt(b*x^n + a)*sqrt(d*x^n + c))/(b^3*d^6*n), 1/3840*(15*(63*
b^5*c^5 - 175*a*b^4*c^4*d + 150*a^2*b^3*c^3*d^2 - 30*a^3*b^2*c^2*d^3 - 5*a^4*b*c*d^4 - 3*a^5*d^5)*sqrt(-b*d)*a
rctan(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*(384*b^5*d^5*x^(4*n) + 945*b^5*c^4*d - 2310*a*b^4*c^3*d^2 + 1564*a^2*b^
3*c^2*d^3 - 90*a^3*b^2*c*d^4 - 45*a^4*b*d^5 - 144*(3*b^5*c*d^4 - 7*a*b^4*d^5)*x^(3*n) + 8*(63*b^5*c^2*d^3 - 14
8*a*b^4*c*d^4 + 93*a^2*b^3*d^5)*x^(2*n) - 2*(315*b^5*c^3*d^2 - 749*a*b^4*c^2*d^3 + 481*a^2*b^3*c*d^4 - 15*a^3*
b^2*d^5)*x^n)*sqrt(b*x^n + a)*sqrt(d*x^n + c))/(b^3*d^6*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{{\left (b x^{n} + a\right )}^{\frac{5}{2}} x^{3 \, 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+3*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^(3*n - 1)/sqrt(d*x^n + c), x)