Integrand size = 24, antiderivative size = 564 \[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^{9/2}} \, dx=-\frac {2 (2 A b+7 a B) \sqrt {a+b x^3}}{7 a \sqrt {x}}+\frac {3 \left (1+\sqrt {3}\right ) \sqrt [3]{b} (2 A b+7 a B) \sqrt {x} \sqrt {a+b x^3}}{7 a \left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )}-\frac {2 A \left (a+b x^3\right )^{3/2}}{7 a x^{7/2}}-\frac {3 \sqrt [4]{3} \sqrt [3]{b} (2 A b+7 a B) \sqrt {x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} E\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{7 a^{2/3} \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {3^{3/4} \left (1-\sqrt {3}\right ) \sqrt [3]{b} (2 A b+7 a B) \sqrt {x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{14 a^{2/3} \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \]
-2/7*A*(b*x^3+a)^(3/2)/a/x^(7/2)-2/7*(2*A*b+7*B*a)*(b*x^3+a)^(1/2)/a/x^(1/ 2)+3/7*b^(1/3)*(2*A*b+7*B*a)*(1+3^(1/2))*x^(1/2)*(b*x^3+a)^(1/2)/a/(a^(1/3 )+b^(1/3)*x*(1+3^(1/2)))-3/7*3^(1/4)*b^(1/3)*(2*A*b+7*B*a)*(a^(1/3)+b^(1/3 )*x)*((a^(1/3)+b^(1/3)*x*(1-3^(1/2)))^2/(a^(1/3)+b^(1/3)*x*(1+3^(1/2)))^2) ^(1/2)/(a^(1/3)+b^(1/3)*x*(1-3^(1/2)))*(a^(1/3)+b^(1/3)*x*(1+3^(1/2)))*Ell ipticE((1-(a^(1/3)+b^(1/3)*x*(1-3^(1/2)))^2/(a^(1/3)+b^(1/3)*x*(1+3^(1/2)) )^2)^(1/2),1/4*6^(1/2)+1/4*2^(1/2))*x^(1/2)*((a^(2/3)-a^(1/3)*b^(1/3)*x+b^ (2/3)*x^2)/(a^(1/3)+b^(1/3)*x*(1+3^(1/2)))^2)^(1/2)/a^(2/3)/(b*x^3+a)^(1/2 )/(b^(1/3)*x*(a^(1/3)+b^(1/3)*x)/(a^(1/3)+b^(1/3)*x*(1+3^(1/2)))^2)^(1/2)- 1/14*3^(3/4)*b^(1/3)*(2*A*b+7*B*a)*(a^(1/3)+b^(1/3)*x)*((a^(1/3)+b^(1/3)*x *(1-3^(1/2)))^2/(a^(1/3)+b^(1/3)*x*(1+3^(1/2)))^2)^(1/2)/(a^(1/3)+b^(1/3)* x*(1-3^(1/2)))*(a^(1/3)+b^(1/3)*x*(1+3^(1/2)))*EllipticF((1-(a^(1/3)+b^(1/ 3)*x*(1-3^(1/2)))^2/(a^(1/3)+b^(1/3)*x*(1+3^(1/2)))^2)^(1/2),1/4*6^(1/2)+1 /4*2^(1/2))*(1-3^(1/2))*x^(1/2)*((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/( a^(1/3)+b^(1/3)*x*(1+3^(1/2)))^2)^(1/2)/a^(2/3)/(b*x^3+a)^(1/2)/(b^(1/3)*x *(a^(1/3)+b^(1/3)*x)/(a^(1/3)+b^(1/3)*x*(1+3^(1/2)))^2)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.10 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.14 \[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^{9/2}} \, dx=\frac {2 \sqrt {a+b x^3} \left (-A \left (a+b x^3\right )-\frac {(2 A b+7 a B) x^3 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {1}{6},\frac {5}{6},-\frac {b x^3}{a}\right )}{\sqrt {1+\frac {b x^3}{a}}}\right )}{7 a x^{7/2}} \]
(2*Sqrt[a + b*x^3]*(-(A*(a + b*x^3)) - ((2*A*b + 7*a*B)*x^3*Hypergeometric 2F1[-1/2, -1/6, 5/6, -((b*x^3)/a)])/Sqrt[1 + (b*x^3)/a]))/(7*a*x^(7/2))
Time = 0.63 (sec) , antiderivative size = 536, normalized size of antiderivative = 0.95, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {955, 809, 851, 837, 25, 766, 2420}
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.
| \(\displaystyle \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^{9/2}} \, dx\) |
| \(\Big \downarrow \) 955 |
| \(\displaystyle \frac {(7 a B+2 A b) \int \frac {\sqrt {b x^3+a}}{x^{3/2}}dx}{7 a}-\frac {2 A \left (a+b x^3\right )^{3/2}}{7 a x^{7/2}}\) |
| \(\Big \downarrow \) 809 |
| \(\displaystyle \frac {(7 a B+2 A b) \left (3 b \int \frac {x^{3/2}}{\sqrt {b x^3+a}}dx-\frac {2 \sqrt {a+b x^3}}{\sqrt {x}}\right )}{7 a}-\frac {2 A \left (a+b x^3\right )^{3/2}}{7 a x^{7/2}}\) |
| \(\Big \downarrow \) 851 |
| \(\displaystyle \frac {(7 a B+2 A b) \left (6 b \int \frac {x^2}{\sqrt {b x^3+a}}d\sqrt {x}-\frac {2 \sqrt {a+b x^3}}{\sqrt {x}}\right )}{7 a}-\frac {2 A \left (a+b x^3\right )^{3/2}}{7 a x^{7/2}}\) |
| \(\Big \downarrow \) 837 |
| \(\displaystyle \frac {(7 a B+2 A b) \left (6 b \left (-\frac {\left (1-\sqrt {3}\right ) a^{2/3} \int \frac {1}{\sqrt {b x^3+a}}d\sqrt {x}}{2 b^{2/3}}-\frac {\int -\frac {2 b^{2/3} x^2+\left (1-\sqrt {3}\right ) a^{2/3}}{\sqrt {b x^3+a}}d\sqrt {x}}{2 b^{2/3}}\right )-\frac {2 \sqrt {a+b x^3}}{\sqrt {x}}\right )}{7 a}-\frac {2 A \left (a+b x^3\right )^{3/2}}{7 a x^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(7 a B+2 A b) \left (6 b \left (\frac {\int \frac {2 b^{2/3} x^2+\left (1-\sqrt {3}\right ) a^{2/3}}{\sqrt {b x^3+a}}d\sqrt {x}}{2 b^{2/3}}-\frac {\left (1-\sqrt {3}\right ) a^{2/3} \int \frac {1}{\sqrt {b x^3+a}}d\sqrt {x}}{2 b^{2/3}}\right )-\frac {2 \sqrt {a+b x^3}}{\sqrt {x}}\right )}{7 a}-\frac {2 A \left (a+b x^3\right )^{3/2}}{7 a x^{7/2}}\) |
| \(\Big \downarrow \) 766 |
| \(\displaystyle \frac {(7 a B+2 A b) \left (6 b \left (\frac {\int \frac {2 b^{2/3} x^2+\left (1-\sqrt {3}\right ) a^{2/3}}{\sqrt {b x^3+a}}d\sqrt {x}}{2 b^{2/3}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} \sqrt {x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 \sqrt [4]{3} b^{2/3} \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}\right )-\frac {2 \sqrt {a+b x^3}}{\sqrt {x}}\right )}{7 a}-\frac {2 A \left (a+b x^3\right )^{3/2}}{7 a x^{7/2}}\) |
| \(\Big \downarrow \) 2420 |
| \(\displaystyle \frac {(7 a B+2 A b) \left (6 b \left (\frac {\frac {\left (1+\sqrt {3}\right ) \sqrt {x} \sqrt {a+b x^3}}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x}-\frac {\sqrt [4]{3} \sqrt [3]{a} \sqrt {x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} E\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}}{2 b^{2/3}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} \sqrt {x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 \sqrt [4]{3} b^{2/3} \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}\right )-\frac {2 \sqrt {a+b x^3}}{\sqrt {x}}\right )}{7 a}-\frac {2 A \left (a+b x^3\right )^{3/2}}{7 a x^{7/2}}\) |
(-2*A*(a + b*x^3)^(3/2))/(7*a*x^(7/2)) + ((2*A*b + 7*a*B)*((-2*Sqrt[a + b* x^3])/Sqrt[x] + 6*b*((((1 + Sqrt[3])*Sqrt[x]*Sqrt[a + b*x^3])/(a^(1/3) + ( 1 + Sqrt[3])*b^(1/3)*x) - (3^(1/4)*a^(1/3)*Sqrt[x]*(a^(1/3) + b^(1/3)*x)*S qrt[(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/(a^(1/3) + (1 + Sqrt[3])*b ^(1/3)*x)^2]*EllipticE[ArcCos[(a^(1/3) + (1 - Sqrt[3])*b^(1/3)*x)/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)], (2 + Sqrt[3])/4])/(Sqrt[(b^(1/3)*x*(a^(1/3) + b^(1/3)*x))/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)^2]*Sqrt[a + b*x^3]))/(2* b^(2/3)) - ((1 - Sqrt[3])*a^(1/3)*Sqrt[x]*(a^(1/3) + b^(1/3)*x)*Sqrt[(a^(2 /3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x) ^2]*EllipticF[ArcCos[(a^(1/3) + (1 - Sqrt[3])*b^(1/3)*x)/(a^(1/3) + (1 + S qrt[3])*b^(1/3)*x)], (2 + Sqrt[3])/4])/(4*3^(1/4)*b^(2/3)*Sqrt[(b^(1/3)*x* (a^(1/3) + b^(1/3)*x))/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)^2]*Sqrt[a + b*x ^3]))))/(7*a)
3.6.24.3.1 Defintions of rubi rules used
Int[1/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[x*(s + r*x^2)*(Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/ (s + (1 + Sqrt[3])*r*x^2)^2]/(2*3^(1/4)*s*Sqrt[a + b*x^6]*Sqrt[r*x^2*((s + r*x^2)/(s + (1 + Sqrt[3])*r*x^2)^2)]))*EllipticF[ArcCos[(s + (1 - Sqrt[3])* r*x^2)/(s + (1 + Sqrt[3])*r*x^2)], (2 + Sqrt[3])/4], x]] /; FreeQ[{a, b}, x ]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* x)^(m + 1)*((a + b*x^n)^p/(c*(m + 1))), x] - Simp[b*n*(p/(c^n*(m + 1))) I nt[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && IGtQ [n, 0] && GtQ[p, 0] && LtQ[m, -1] && !ILtQ[(m + n*p + n + 1)/n, 0] && IntB inomialQ[a, b, c, n, m, p, x]
Int[(x_)^4/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(Sqrt[3] - 1)*(s^2/(2*r^2)) Int[1/Sqrt[ a + b*x^6], x], x] - Simp[1/(2*r^2) Int[((Sqrt[3] - 1)*s^2 - 2*r^2*x^4)/S qrt[a + b*x^6], x], x]] /; FreeQ[{a, b}, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^ n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)) Int[(e *x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b* c - a*d, 0] && (IntegerQ[n] || GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) && !ILtQ[p, -1]
Int[((c_) + (d_.)*(x_)^4)/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(1 + Sqrt[3])*d*s^3*x*(Sqr t[a + b*x^6]/(2*a*r^2*(s + (1 + Sqrt[3])*r*x^2))), x] - Simp[3^(1/4)*d*s*x* (s + r*x^2)*(Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/(s + (1 + Sqrt[3])*r*x^2)^2]/(2 *r^2*Sqrt[(r*x^2*(s + r*x^2))/(s + (1 + Sqrt[3])*r*x^2)^2]*Sqrt[a + b*x^6]) )*EllipticE[ArcCos[(s + (1 - Sqrt[3])*r*x^2)/(s + (1 + Sqrt[3])*r*x^2)], (2 + Sqrt[3])/4], x]] /; FreeQ[{a, b, c, d}, x] && EqQ[2*Rt[b/a, 3]^2*c - (1 - Sqrt[3])*d, 0]
Result contains complex when optimal does not.
Time = 4.98 (sec) , antiderivative size = 1127, normalized size of antiderivative = 2.00
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(1127\) |
| elliptic | \(\text {Expression too large to display}\) | \(1177\) |
| default | \(\text {Expression too large to display}\) | \(5911\) |
-2/7*(b*x^3+a)^(1/2)*(3*A*b*x^3+7*B*a*x^3+A*a)/x^(7/2)/a+3/7*b*(2*A*b+7*B* a)/a*(x*(x+1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*(x+1/2/b*( -a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))+(1/2/b*(-a*b^2)^(1/3)-1/2*I* 3^(1/2)/b*(-a*b^2)^(1/3))*((-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2) ^(1/3))*x/(-1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(x-1/b*(- a*b^2)^(1/3)))^(1/2)*(x-1/b*(-a*b^2)^(1/3))^2*(1/b*(-a*b^2)^(1/3)*(x+1/2/b *(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(-1/2/b*(-a*b^2)^(1/3)-1/2 *I*3^(1/2)/b*(-a*b^2)^(1/3))/(x-1/b*(-a*b^2)^(1/3)))^(1/2)*(1/b*(-a*b^2)^( 1/3)*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(-1/2/b*(-a*b ^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(x-1/b*(-a*b^2)^(1/3)))^(1/2)*(( (-1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/b*(-a*b^2)^(1/3)+1/ b^2*(-a*b^2)^(2/3))/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)) *b/(-a*b^2)^(1/3)*EllipticF(((-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^ 2)^(1/3))*x/(-1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(x-1/b* (-a*b^2)^(1/3)))^(1/2),((3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/ 3))*(1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(1/2/b*(-a*b^2)^ (1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(3/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/ b*(-a*b^2)^(1/3)))^(1/2))+(1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^( 1/3))*EllipticE(((-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*x/ (-1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(x-1/b*(-a*b^2)^...
\[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^{9/2}} \, dx=\int { \frac {{\left (B x^{3} + A\right )} \sqrt {b x^{3} + a}}{x^{\frac {9}{2}}} \,d x } \]
Result contains complex when optimal does not.
Time = 9.85 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.17 \[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^{9/2}} \, dx=\frac {A \sqrt {a} \Gamma \left (- \frac {7}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{6}, - \frac {1}{2} \\ - \frac {1}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{\frac {7}{2}} \Gamma \left (- \frac {1}{6}\right )} + \frac {B \sqrt {a} \Gamma \left (- \frac {1}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{6} \\ \frac {5}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt {x} \Gamma \left (\frac {5}{6}\right )} \]
A*sqrt(a)*gamma(-7/6)*hyper((-7/6, -1/2), (-1/6,), b*x**3*exp_polar(I*pi)/ a)/(3*x**(7/2)*gamma(-1/6)) + B*sqrt(a)*gamma(-1/6)*hyper((-1/2, -1/6), (5 /6,), b*x**3*exp_polar(I*pi)/a)/(3*sqrt(x)*gamma(5/6))
\[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^{9/2}} \, dx=\int { \frac {{\left (B x^{3} + A\right )} \sqrt {b x^{3} + a}}{x^{\frac {9}{2}}} \,d x } \]
\[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^{9/2}} \, dx=\int { \frac {{\left (B x^{3} + A\right )} \sqrt {b x^{3} + a}}{x^{\frac {9}{2}}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^{9/2}} \, dx=\int \frac {\left (B\,x^3+A\right )\,\sqrt {b\,x^3+a}}{x^{9/2}} \,d x \]