\(\int \frac {x^7 (a+b x^3)^{2/3}}{a d-b d x^3} \, dx\) [814]

Optimal result

Integrand size = 28, antiderivative size = 512 \[ \int \frac {x^7 \left (a+b x^3\right )^{2/3}}{a d-b d x^3} \, dx=-\frac {9 a x^2 \left (a+b x^3\right )^{2/3}}{28 b^2 d}-\frac {x^5 \left (a+b x^3\right )^{2/3}}{7 b d}+\frac {2^{2/3} a^{7/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} b^{8/3} d}+\frac {a^{7/3} \arctan \left (\frac {1+\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3} b^{8/3} d}-\frac {19 a^2 x^2 \sqrt [3]{1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},-\frac {b x^3}{a}\right )}{28 b^2 d \sqrt [3]{a+b x^3}}+\frac {a^{7/3} \log \left (\frac {\left (\sqrt [3]{a}-\sqrt [3]{b} x\right )^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a}\right )}{6 \sqrt [3]{2} b^{8/3} d}+\frac {a^{7/3} \log \left (1+\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}\right )}{3 \sqrt [3]{2} b^{8/3} d}-\frac {2^{2/3} a^{7/3} \log \left (1+\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}\right )}{3 b^{8/3} d}-\frac {a^{7/3} \log \left (\frac {\sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a}}-\frac {2^{2/3} \sqrt [3]{b} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}\right )}{2 \sqrt [3]{2} b^{8/3} d} \] Output:

-9/28*a*x^2*(b*x^3+a)^(2/3)/b^2/d-1/7*x^5*(b*x^3+a)^(2/3)/b/d+1/3*2^(2/3)* 
a^(7/3)*arctan(1/3*(1-2*2^(1/3)*(a^(1/3)+b^(1/3)*x)/(b*x^3+a)^(1/3))*3^(1/ 
2))*3^(1/2)/b^(8/3)/d+1/6*a^(7/3)*arctan(1/3*(1+2^(1/3)*(a^(1/3)+b^(1/3)*x 
)/(b*x^3+a)^(1/3))*3^(1/2))*2^(2/3)*3^(1/2)/b^(8/3)/d-19/28*a^2*x^2*(1+b*x 
^3/a)^(1/3)*hypergeom([1/3, 2/3],[5/3],-b*x^3/a)/b^2/d/(b*x^3+a)^(1/3)+1/1 
2*a^(7/3)*ln((a^(1/3)-b^(1/3)*x)^2*(a^(1/3)+b^(1/3)*x)/a)*2^(2/3)/b^(8/3)/ 
d+1/6*a^(7/3)*ln(1+2^(2/3)*(a^(1/3)+b^(1/3)*x)^2/(b*x^3+a)^(2/3)-2^(1/3)*( 
a^(1/3)+b^(1/3)*x)/(b*x^3+a)^(1/3))*2^(2/3)/b^(8/3)/d-1/3*2^(2/3)*a^(7/3)* 
ln(1+2^(1/3)*(a^(1/3)+b^(1/3)*x)/(b*x^3+a)^(1/3))/b^(8/3)/d-1/4*a^(7/3)*ln 
(b^(1/3)*(a^(1/3)+b^(1/3)*x)/a^(1/3)-2^(2/3)*b^(1/3)*(b*x^3+a)^(1/3)/a^(1/ 
3))*2^(2/3)/b^(8/3)/d
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.

Time = 7.44 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.29 \[ \int \frac {x^7 \left (a+b x^3\right )^{2/3}}{a d-b d x^3} \, dx=\frac {-5 \left (9 a^2 x^2+13 a b x^5+4 b^2 x^8\right )+45 a^2 x^2 \sqrt [3]{1+\frac {b x^3}{a}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-\frac {b x^3}{a},\frac {b x^3}{a}\right )+38 a b x^5 \sqrt [3]{1+\frac {b x^3}{a}} \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{3},1,\frac {8}{3},-\frac {b x^3}{a},\frac {b x^3}{a}\right )}{140 b^2 d \sqrt [3]{a+b x^3}} \] Input:

Integrate[(x^7*(a + b*x^3)^(2/3))/(a*d - b*d*x^3),x]
 

Output:

(-5*(9*a^2*x^2 + 13*a*b*x^5 + 4*b^2*x^8) + 45*a^2*x^2*(1 + (b*x^3)/a)^(1/3 
)*AppellF1[2/3, 1/3, 1, 5/3, -((b*x^3)/a), (b*x^3)/a] + 38*a*b*x^5*(1 + (b 
*x^3)/a)^(1/3)*AppellF1[5/3, 1/3, 1, 8/3, -((b*x^3)/a), (b*x^3)/a])/(140*b 
^2*d*(a + b*x^3)^(1/3))
 

Rubi [A] (verified)

Time = 1.28 (sec) , antiderivative size = 502, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {978, 27, 1052, 27, 1054, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x^7 \left (a+b x^3\right )^{2/3}}{a d-b d x^3} \, dx\)

\(\Big \downarrow \) 978

\(\displaystyle \frac {\int \frac {a x^4 \left (9 b x^3+5 a\right )}{\left (a-b x^3\right ) \sqrt [3]{b x^3+a}}dx}{7 b d}-\frac {x^5 \left (a+b x^3\right )^{2/3}}{7 b d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {a \int \frac {x^4 \left (9 b x^3+5 a\right )}{\left (a-b x^3\right ) \sqrt [3]{b x^3+a}}dx}{7 b d}-\frac {x^5 \left (a+b x^3\right )^{2/3}}{7 b d}\)

\(\Big \downarrow \) 1052

\(\displaystyle \frac {a \left (\frac {\int \frac {2 a b x \left (19 b x^3+9 a\right )}{\left (a-b x^3\right ) \sqrt [3]{b x^3+a}}dx}{4 b^2}-\frac {9 x^2 \left (a+b x^3\right )^{2/3}}{4 b}\right )}{7 b d}-\frac {x^5 \left (a+b x^3\right )^{2/3}}{7 b d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {a \left (\frac {a \int \frac {x \left (19 b x^3+9 a\right )}{\left (a-b x^3\right ) \sqrt [3]{b x^3+a}}dx}{2 b}-\frac {9 x^2 \left (a+b x^3\right )^{2/3}}{4 b}\right )}{7 b d}-\frac {x^5 \left (a+b x^3\right )^{2/3}}{7 b d}\)

\(\Big \downarrow \) 1054

\(\displaystyle \frac {a \left (\frac {a \int \left (\frac {28 a x}{\left (a-b x^3\right ) \sqrt [3]{b x^3+a}}-\frac {19 x}{\sqrt [3]{b x^3+a}}\right )dx}{2 b}-\frac {9 x^2 \left (a+b x^3\right )^{2/3}}{4 b}\right )}{7 b d}-\frac {x^5 \left (a+b x^3\right )^{2/3}}{7 b d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {a \left (\frac {a \left (\frac {14\ 2^{2/3} \sqrt [3]{a} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} b^{2/3}}+\frac {7\ 2^{2/3} \sqrt [3]{a} \arctan \left (\frac {\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} b^{2/3}}+\frac {7\ 2^{2/3} \sqrt [3]{a} \log \left (\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{3 b^{2/3}}-\frac {14\ 2^{2/3} \sqrt [3]{a} \log \left (\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{3 b^{2/3}}-\frac {7 \sqrt [3]{a} \log \left (\frac {\sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a}}-\frac {2^{2/3} \sqrt [3]{b} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}\right )}{\sqrt [3]{2} b^{2/3}}+\frac {7 \sqrt [3]{a} \log \left (\frac {\left (\sqrt [3]{a}-\sqrt [3]{b} x\right )^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a}\right )}{3 \sqrt [3]{2} b^{2/3}}-\frac {19 x^2 \sqrt [3]{\frac {b x^3}{a}+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},-\frac {b x^3}{a}\right )}{2 \sqrt [3]{a+b x^3}}\right )}{2 b}-\frac {9 x^2 \left (a+b x^3\right )^{2/3}}{4 b}\right )}{7 b d}-\frac {x^5 \left (a+b x^3\right )^{2/3}}{7 b d}\)

Input:

Int[(x^7*(a + b*x^3)^(2/3))/(a*d - b*d*x^3),x]
 

Output:

-1/7*(x^5*(a + b*x^3)^(2/3))/(b*d) + (a*((-9*x^2*(a + b*x^3)^(2/3))/(4*b) 
+ (a*((14*2^(2/3)*a^(1/3)*ArcTan[(1 - (2*2^(1/3)*(a^(1/3) + b^(1/3)*x))/(a 
 + b*x^3)^(1/3))/Sqrt[3]])/(Sqrt[3]*b^(2/3)) + (7*2^(2/3)*a^(1/3)*ArcTan[( 
1 + (2^(1/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3))/Sqrt[3]])/(Sqrt[3]* 
b^(2/3)) - (19*x^2*(1 + (b*x^3)/a)^(1/3)*Hypergeometric2F1[1/3, 2/3, 5/3, 
-((b*x^3)/a)])/(2*(a + b*x^3)^(1/3)) + (7*a^(1/3)*Log[((a^(1/3) - b^(1/3)* 
x)^2*(a^(1/3) + b^(1/3)*x))/a])/(3*2^(1/3)*b^(2/3)) + (7*2^(2/3)*a^(1/3)*L 
og[1 + (2^(2/3)*(a^(1/3) + b^(1/3)*x)^2)/(a + b*x^3)^(2/3) - (2^(1/3)*(a^( 
1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3)])/(3*b^(2/3)) - (14*2^(2/3)*a^(1/3)*L 
og[1 + (2^(1/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3)])/(3*b^(2/3)) - ( 
7*a^(1/3)*Log[(b^(1/3)*(a^(1/3) + b^(1/3)*x))/a^(1/3) - (2^(2/3)*b^(1/3)*( 
a + b*x^3)^(1/3))/a^(1/3)])/(2^(1/3)*b^(2/3))))/(2*b)))/(7*b*d)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 978

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

rule 1052

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

rule 1054

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

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [F]

\[\int \frac {x^{7} \left (b \,x^{3}+a \right )^{\frac {2}{3}}}{-b d \,x^{3}+a d}d x\]

Input:

int(x^7*(b*x^3+a)^(2/3)/(-b*d*x^3+a*d),x)
 

Output:

int(x^7*(b*x^3+a)^(2/3)/(-b*d*x^3+a*d),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {x^7 \left (a+b x^3\right )^{2/3}}{a d-b d x^3} \, dx=\text {Timed out} \] Input:

integrate(x^7*(b*x^3+a)^(2/3)/(-b*d*x^3+a*d),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {x^7 \left (a+b x^3\right )^{2/3}}{a d-b d x^3} \, dx=- \frac {\int \frac {x^{7} \left (a + b x^{3}\right )^{\frac {2}{3}}}{- a + b x^{3}}\, dx}{d} \] Input:

integrate(x**7*(b*x**3+a)**(2/3)/(-b*d*x**3+a*d),x)
 

Output:

-Integral(x**7*(a + b*x**3)**(2/3)/(-a + b*x**3), x)/d
 

Maxima [F]

\[ \int \frac {x^7 \left (a+b x^3\right )^{2/3}}{a d-b d x^3} \, dx=\int { -\frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}} x^{7}}{b d x^{3} - a d} \,d x } \] Input:

integrate(x^7*(b*x^3+a)^(2/3)/(-b*d*x^3+a*d),x, algorithm="maxima")
 

Output:

-integrate((b*x^3 + a)^(2/3)*x^7/(b*d*x^3 - a*d), x)
 

Giac [F]

\[ \int \frac {x^7 \left (a+b x^3\right )^{2/3}}{a d-b d x^3} \, dx=\int { -\frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}} x^{7}}{b d x^{3} - a d} \,d x } \] Input:

integrate(x^7*(b*x^3+a)^(2/3)/(-b*d*x^3+a*d),x, algorithm="giac")
 

Output:

integrate(-(b*x^3 + a)^(2/3)*x^7/(b*d*x^3 - a*d), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^7 \left (a+b x^3\right )^{2/3}}{a d-b d x^3} \, dx=\int \frac {x^7\,{\left (b\,x^3+a\right )}^{2/3}}{a\,d-b\,d\,x^3} \,d x \] Input:

int((x^7*(a + b*x^3)^(2/3))/(a*d - b*d*x^3),x)
 

Output:

int((x^7*(a + b*x^3)^(2/3))/(a*d - b*d*x^3), x)
 

Reduce [F]

\[ \int \frac {x^7 \left (a+b x^3\right )^{2/3}}{a d-b d x^3} \, dx=\frac {-9 \left (b \,x^{3}+a \right )^{\frac {2}{3}} a \,x^{2}-4 \left (b \,x^{3}+a \right )^{\frac {2}{3}} b \,x^{5}+38 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}} x^{4}}{-b^{2} x^{6}+a^{2}}d x \right ) a^{2} b +18 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}} x}{-b^{2} x^{6}+a^{2}}d x \right ) a^{3}}{28 b^{2} d} \] Input:

int(x^7*(b*x^3+a)^(2/3)/(-b*d*x^3+a*d),x)
 

Output:

( - 9*(a + b*x**3)**(2/3)*a*x**2 - 4*(a + b*x**3)**(2/3)*b*x**5 + 38*int(( 
(a + b*x**3)**(2/3)*x**4)/(a**2 - b**2*x**6),x)*a**2*b + 18*int(((a + b*x* 
*3)**(2/3)*x)/(a**2 - b**2*x**6),x)*a**3)/(28*b**2*d)