3.917 \(\int \frac{1}{x^{13} (a+b x^8)^2 \sqrt{c+d x^8}} \, dx\)

Optimal. Leaf size=208 \[ \frac{\sqrt{c+d x^8} \left (-4 a^2 d^2-8 a b c d+15 b^2 c^2\right )}{24 a^3 c^2 x^4 (b c-a d)}+\frac{b^2 (5 b c-6 a d) \tan ^{-1}\left (\frac{x^4 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^8}}\right )}{8 a^{7/2} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x^8} (5 b c-2 a d)}{24 a^2 c x^{12} (b c-a d)}+\frac{b \sqrt{c+d x^8}}{8 a x^{12} \left (a+b x^8\right ) (b c-a d)} \]

[Out]

-((5*b*c - 2*a*d)*Sqrt[c + d*x^8])/(24*a^2*c*(b*c - a*d)*x^12) + ((15*b^2*c^2 - 8*a*b*c*d - 4*a^2*d^2)*Sqrt[c
+ d*x^8])/(24*a^3*c^2*(b*c - a*d)*x^4) + (b*Sqrt[c + d*x^8])/(8*a*(b*c - a*d)*x^12*(a + b*x^8)) + (b^2*(5*b*c
- 6*a*d)*ArcTan[(Sqrt[b*c - a*d]*x^4)/(Sqrt[a]*Sqrt[c + d*x^8])])/(8*a^(7/2)*(b*c - a*d)^(3/2))

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Rubi [A]  time = 0.297487, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {465, 472, 583, 12, 377, 205} \[ \frac{\sqrt{c+d x^8} \left (-4 a^2 d^2-8 a b c d+15 b^2 c^2\right )}{24 a^3 c^2 x^4 (b c-a d)}+\frac{b^2 (5 b c-6 a d) \tan ^{-1}\left (\frac{x^4 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^8}}\right )}{8 a^{7/2} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x^8} (5 b c-2 a d)}{24 a^2 c x^{12} (b c-a d)}+\frac{b \sqrt{c+d x^8}}{8 a x^{12} \left (a+b x^8\right ) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^13*(a + b*x^8)^2*Sqrt[c + d*x^8]),x]

[Out]

-((5*b*c - 2*a*d)*Sqrt[c + d*x^8])/(24*a^2*c*(b*c - a*d)*x^12) + ((15*b^2*c^2 - 8*a*b*c*d - 4*a^2*d^2)*Sqrt[c
+ d*x^8])/(24*a^3*c^2*(b*c - a*d)*x^4) + (b*Sqrt[c + d*x^8])/(8*a*(b*c - a*d)*x^12*(a + b*x^8)) + (b^2*(5*b*c
- 6*a*d)*ArcTan[(Sqrt[b*c - a*d]*x^4)/(Sqrt[a]*Sqrt[c + d*x^8])])/(8*a^(7/2)*(b*c - a*d)^(3/2))

Rule 465

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

Rule 472

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

Rule 583

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 377

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

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{1}{x^{13} \left (a+b x^8\right )^2 \sqrt{c+d x^8}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{x^4 \left (a+b x^2\right )^2 \sqrt{c+d x^2}} \, dx,x,x^4\right )\\ &=\frac{b \sqrt{c+d x^8}}{8 a (b c-a d) x^{12} \left (a+b x^8\right )}-\frac{\operatorname{Subst}\left (\int \frac{-5 b c+2 a d-4 b d x^2}{x^4 \left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx,x,x^4\right )}{8 a (b c-a d)}\\ &=-\frac{(5 b c-2 a d) \sqrt{c+d x^8}}{24 a^2 c (b c-a d) x^{12}}+\frac{b \sqrt{c+d x^8}}{8 a (b c-a d) x^{12} \left (a+b x^8\right )}+\frac{\operatorname{Subst}\left (\int \frac{-15 b^2 c^2+8 a b c d+4 a^2 d^2-2 b d (5 b c-2 a d) x^2}{x^2 \left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx,x,x^4\right )}{24 a^2 c (b c-a d)}\\ &=-\frac{(5 b c-2 a d) \sqrt{c+d x^8}}{24 a^2 c (b c-a d) x^{12}}+\frac{\left (15 b^2 c^2-8 a b c d-4 a^2 d^2\right ) \sqrt{c+d x^8}}{24 a^3 c^2 (b c-a d) x^4}+\frac{b \sqrt{c+d x^8}}{8 a (b c-a d) x^{12} \left (a+b x^8\right )}-\frac{\operatorname{Subst}\left (\int -\frac{3 b^2 c^2 (5 b c-6 a d)}{\left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx,x,x^4\right )}{24 a^3 c^2 (b c-a d)}\\ &=-\frac{(5 b c-2 a d) \sqrt{c+d x^8}}{24 a^2 c (b c-a d) x^{12}}+\frac{\left (15 b^2 c^2-8 a b c d-4 a^2 d^2\right ) \sqrt{c+d x^8}}{24 a^3 c^2 (b c-a d) x^4}+\frac{b \sqrt{c+d x^8}}{8 a (b c-a d) x^{12} \left (a+b x^8\right )}+\frac{\left (b^2 (5 b c-6 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{\left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx,x,x^4\right )}{8 a^3 (b c-a d)}\\ &=-\frac{(5 b c-2 a d) \sqrt{c+d x^8}}{24 a^2 c (b c-a d) x^{12}}+\frac{\left (15 b^2 c^2-8 a b c d-4 a^2 d^2\right ) \sqrt{c+d x^8}}{24 a^3 c^2 (b c-a d) x^4}+\frac{b \sqrt{c+d x^8}}{8 a (b c-a d) x^{12} \left (a+b x^8\right )}+\frac{\left (b^2 (5 b c-6 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{a-(-b c+a d) x^2} \, dx,x,\frac{x^4}{\sqrt{c+d x^8}}\right )}{8 a^3 (b c-a d)}\\ &=-\frac{(5 b c-2 a d) \sqrt{c+d x^8}}{24 a^2 c (b c-a d) x^{12}}+\frac{\left (15 b^2 c^2-8 a b c d-4 a^2 d^2\right ) \sqrt{c+d x^8}}{24 a^3 c^2 (b c-a d) x^4}+\frac{b \sqrt{c+d x^8}}{8 a (b c-a d) x^{12} \left (a+b x^8\right )}+\frac{b^2 (5 b c-6 a d) \tan ^{-1}\left (\frac{\sqrt{b c-a d} x^4}{\sqrt{a} \sqrt{c+d x^8}}\right )}{8 a^{7/2} (b c-a d)^{3/2}}\\ \end{align*}

Mathematica [C]  time = 2.75445, size = 1535, normalized size = 7.38 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(x^13*(a + b*x^8)^2*Sqrt[c + d*x^8]),x]

[Out]

(Sqrt[c + d*x^8]*(45*c^3*Sqrt[(a*(b*c - a*d)*x^8*(c + d*x^8))/(c^2*(a + b*x^8)^2)] - 270*c^2*d*x^8*Sqrt[(a*(b*
c - a*d)*x^8*(c + d*x^8))/(c^2*(a + b*x^8)^2)] - 1080*c*d^2*x^16*Sqrt[(a*(b*c - a*d)*x^8*(c + d*x^8))/(c^2*(a
+ b*x^8)^2)] - 720*d^3*x^24*Sqrt[(a*(b*c - a*d)*x^8*(c + d*x^8))/(c^2*(a + b*x^8)^2)] - 45*c^3*ArcSin[Sqrt[((b
*c - a*d)*x^8)/(c*(a + b*x^8))]] + 270*c^2*d*x^8*ArcSin[Sqrt[((b*c - a*d)*x^8)/(c*(a + b*x^8))]] + 1080*c*d^2*
x^16*ArcSin[Sqrt[((b*c - a*d)*x^8)/(c*(a + b*x^8))]] + 720*d^3*x^24*ArcSin[Sqrt[((b*c - a*d)*x^8)/(c*(a + b*x^
8))]] - 16*c^3*(((b*c - a*d)*x^8)/(c*(a + b*x^8)))^(5/2)*Sqrt[(a*(c + d*x^8))/(c*(a + b*x^8))]*Hypergeometric2
F1[2, 3, 7/2, ((b*c - a*d)*x^8)/(c*(a + b*x^8))] + 528*c^2*d*x^8*(((b*c - a*d)*x^8)/(c*(a + b*x^8)))^(5/2)*Sqr
t[(a*(c + d*x^8))/(c*(a + b*x^8))]*Hypergeometric2F1[2, 3, 7/2, ((b*c - a*d)*x^8)/(c*(a + b*x^8))] + 1248*c*d^
2*x^16*(((b*c - a*d)*x^8)/(c*(a + b*x^8)))^(5/2)*Sqrt[(a*(c + d*x^8))/(c*(a + b*x^8))]*Hypergeometric2F1[2, 3,
7/2, ((b*c - a*d)*x^8)/(c*(a + b*x^8))] + 704*d^3*x^24*(((b*c - a*d)*x^8)/(c*(a + b*x^8)))^(5/2)*Sqrt[(a*(c +
d*x^8))/(c*(a + b*x^8))]*Hypergeometric2F1[2, 3, 7/2, ((b*c - a*d)*x^8)/(c*(a + b*x^8))] + 96*c^3*(((b*c - a*
d)*x^8)/(c*(a + b*x^8)))^(5/2)*Sqrt[(a*(c + d*x^8))/(c*(a + b*x^8))]*HypergeometricPFQ[{2, 2, 3}, {1, 7/2}, ((
b*c - a*d)*x^8)/(c*(a + b*x^8))] + 576*c^2*d*x^8*(((b*c - a*d)*x^8)/(c*(a + b*x^8)))^(5/2)*Sqrt[(a*(c + d*x^8)
)/(c*(a + b*x^8))]*HypergeometricPFQ[{2, 2, 3}, {1, 7/2}, ((b*c - a*d)*x^8)/(c*(a + b*x^8))] + 864*c*d^2*x^16*
(((b*c - a*d)*x^8)/(c*(a + b*x^8)))^(5/2)*Sqrt[(a*(c + d*x^8))/(c*(a + b*x^8))]*HypergeometricPFQ[{2, 2, 3}, {
1, 7/2}, ((b*c - a*d)*x^8)/(c*(a + b*x^8))] + 384*d^3*x^24*(((b*c - a*d)*x^8)/(c*(a + b*x^8)))^(5/2)*Sqrt[(a*(
c + d*x^8))/(c*(a + b*x^8))]*HypergeometricPFQ[{2, 2, 3}, {1, 7/2}, ((b*c - a*d)*x^8)/(c*(a + b*x^8))] + 64*c^
3*(((b*c - a*d)*x^8)/(c*(a + b*x^8)))^(5/2)*Sqrt[(a*(c + d*x^8))/(c*(a + b*x^8))]*HypergeometricPFQ[{2, 2, 2,
3}, {1, 1, 7/2}, ((b*c - a*d)*x^8)/(c*(a + b*x^8))] + 192*c^2*d*x^8*(((b*c - a*d)*x^8)/(c*(a + b*x^8)))^(5/2)*
Sqrt[(a*(c + d*x^8))/(c*(a + b*x^8))]*HypergeometricPFQ[{2, 2, 2, 3}, {1, 1, 7/2}, ((b*c - a*d)*x^8)/(c*(a + b
*x^8))] + 192*c*d^2*x^16*(((b*c - a*d)*x^8)/(c*(a + b*x^8)))^(5/2)*Sqrt[(a*(c + d*x^8))/(c*(a + b*x^8))]*Hyper
geometricPFQ[{2, 2, 2, 3}, {1, 1, 7/2}, ((b*c - a*d)*x^8)/(c*(a + b*x^8))] + 64*d^3*x^24*(((b*c - a*d)*x^8)/(c
*(a + b*x^8)))^(5/2)*Sqrt[(a*(c + d*x^8))/(c*(a + b*x^8))]*HypergeometricPFQ[{2, 2, 2, 3}, {1, 1, 7/2}, ((b*c
- a*d)*x^8)/(c*(a + b*x^8))]))/(360*c^4*x^12*(((b*c - a*d)*x^8)/(c*(a + b*x^8)))^(3/2)*(a + b*x^8)^2*Sqrt[(a*(
c + d*x^8))/(c*(a + b*x^8))])

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Maple [F]  time = 0.063, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{13} \left ( b{x}^{8}+a \right ) ^{2}}{\frac{1}{\sqrt{d{x}^{8}+c}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^13/(b*x^8+a)^2/(d*x^8+c)^(1/2),x)

[Out]

int(1/x^13/(b*x^8+a)^2/(d*x^8+c)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{8} + a\right )}^{2} \sqrt{d x^{8} + c} x^{13}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^13/(b*x^8+a)^2/(d*x^8+c)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/((b*x^8 + a)^2*sqrt(d*x^8 + c)*x^13), x)

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Fricas [A]  time = 3.53749, size = 1561, normalized size = 7.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^13/(b*x^8+a)^2/(d*x^8+c)^(1/2),x, algorithm="fricas")

[Out]

[-1/96*(3*((5*b^4*c^3 - 6*a*b^3*c^2*d)*x^20 + (5*a*b^3*c^3 - 6*a^2*b^2*c^2*d)*x^12)*sqrt(-a*b*c + a^2*d)*log((
(b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^16 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^8 + a^2*c^2 - 4*((b*c - 2*a*d)*x^12 - a*c
*x^4)*sqrt(d*x^8 + c)*sqrt(-a*b*c + a^2*d))/(b^2*x^16 + 2*a*b*x^8 + a^2)) - 4*((15*a*b^4*c^3 - 23*a^2*b^3*c^2*
d + 4*a^3*b^2*c*d^2 + 4*a^4*b*d^3)*x^16 + 2*(5*a^2*b^3*c^3 - 8*a^3*b^2*c^2*d + a^4*b*c*d^2 + 2*a^5*d^3)*x^8 -
2*a^3*b^2*c^3 + 4*a^4*b*c^2*d - 2*a^5*c*d^2)*sqrt(d*x^8 + c))/((a^4*b^3*c^4 - 2*a^5*b^2*c^3*d + a^6*b*c^2*d^2)
*x^20 + (a^5*b^2*c^4 - 2*a^6*b*c^3*d + a^7*c^2*d^2)*x^12), 1/48*(3*((5*b^4*c^3 - 6*a*b^3*c^2*d)*x^20 + (5*a*b^
3*c^3 - 6*a^2*b^2*c^2*d)*x^12)*sqrt(a*b*c - a^2*d)*arctan(1/2*((b*c - 2*a*d)*x^8 - a*c)*sqrt(d*x^8 + c)*sqrt(a
*b*c - a^2*d)/((a*b*c*d - a^2*d^2)*x^12 + (a*b*c^2 - a^2*c*d)*x^4)) + 2*((15*a*b^4*c^3 - 23*a^2*b^3*c^2*d + 4*
a^3*b^2*c*d^2 + 4*a^4*b*d^3)*x^16 + 2*(5*a^2*b^3*c^3 - 8*a^3*b^2*c^2*d + a^4*b*c*d^2 + 2*a^5*d^3)*x^8 - 2*a^3*
b^2*c^3 + 4*a^4*b*c^2*d - 2*a^5*c*d^2)*sqrt(d*x^8 + c))/((a^4*b^3*c^4 - 2*a^5*b^2*c^3*d + a^6*b*c^2*d^2)*x^20
+ (a^5*b^2*c^4 - 2*a^6*b*c^3*d + a^7*c^2*d^2)*x^12)]

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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(1/x**13/(b*x**8+a)**2/(d*x**8+c)**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 1.18832, size = 244, normalized size = 1.17 \begin{align*} \frac{b^{3} c \sqrt{d + \frac{c}{x^{8}}}}{8 \,{\left (a^{3} b c - a^{4} d\right )}{\left (b c + a{\left (d + \frac{c}{x^{8}}\right )} - a d\right )}} - \frac{{\left (5 \, b^{3} c - 6 \, a b^{2} d\right )} \arctan \left (\frac{a \sqrt{d + \frac{c}{x^{8}}}}{\sqrt{a b c - a^{2} d}}\right )}{8 \,{\left (a^{3} b c - a^{4} d\right )} \sqrt{a b c - a^{2} d}} + \frac{6 \, a^{3} b c^{5} \sqrt{d + \frac{c}{x^{8}}} - a^{4} c^{4}{\left (d + \frac{c}{x^{8}}\right )}^{\frac{3}{2}} + 3 \, a^{4} c^{4} \sqrt{d + \frac{c}{x^{8}}} d}{12 \, a^{6} c^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^13/(b*x^8+a)^2/(d*x^8+c)^(1/2),x, algorithm="giac")

[Out]

1/8*b^3*c*sqrt(d + c/x^8)/((a^3*b*c - a^4*d)*(b*c + a*(d + c/x^8) - a*d)) - 1/8*(5*b^3*c - 6*a*b^2*d)*arctan(a
*sqrt(d + c/x^8)/sqrt(a*b*c - a^2*d))/((a^3*b*c - a^4*d)*sqrt(a*b*c - a^2*d)) + 1/12*(6*a^3*b*c^5*sqrt(d + c/x
^8) - a^4*c^4*(d + c/x^8)^(3/2) + 3*a^4*c^4*sqrt(d + c/x^8)*d)/(a^6*c^6)