Integrand size = 22, antiderivative size = 303 \[ \int x^3 \sqrt {a+b x^3} \left (A+B x^3\right ) \, dx=\frac {6 a (17 A b-8 a B) x \sqrt {a+b x^3}}{935 b^2}+\frac {2 (17 A b-8 a B) x^4 \sqrt {a+b x^3}}{187 b}+\frac {2 B x^4 \left (a+b x^3\right )^{3/2}}{17 b}-\frac {4\ 3^{3/4} \sqrt {2+\sqrt {3}} a^2 (17 A b-8 a B) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{935 b^{7/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \] Output:
6/935*a*(17*A*b-8*B*a)*x*(b*x^3+a)^(1/2)/b^2+2/187*(17*A*b-8*B*a)*x^4*(b*x ^3+a)^(1/2)/b+2/17*B*x^4*(b*x^3+a)^(3/2)/b-4/935*3^(3/4)*(1/2*6^(1/2)+1/2* 2^(1/2))*a^2*(17*A*b-8*B*a)*(a^(1/3)+b^(1/3)*x)*((a^(2/3)-a^(1/3)*b^(1/3)* x+b^(2/3)*x^2)/((1+3^(1/2))*a^(1/3)+b^(1/3)*x)^2)^(1/2)*EllipticF(((1-3^(1 /2))*a^(1/3)+b^(1/3)*x)/((1+3^(1/2))*a^(1/3)+b^(1/3)*x),I*3^(1/2)+2*I)/b^( 7/3)/(a^(1/3)*(a^(1/3)+b^(1/3)*x)/((1+3^(1/2))*a^(1/3)+b^(1/3)*x)^2)^(1/2) /(b*x^3+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 5.76 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.29 \[ \int x^3 \sqrt {a+b x^3} \left (A+B x^3\right ) \, dx=\frac {2 x \sqrt {a+b x^3} \left (-\left (\left (a+b x^3\right ) \left (-17 A b+8 a B-11 b B x^3\right )\right )+\frac {a (-17 A b+8 a B) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{\sqrt {1+\frac {b x^3}{a}}}\right )}{187 b^2} \] Input:
Integrate[x^3*Sqrt[a + b*x^3]*(A + B*x^3),x]
Output:
(2*x*Sqrt[a + b*x^3]*(-((a + b*x^3)*(-17*A*b + 8*a*B - 11*b*B*x^3)) + (a*( -17*A*b + 8*a*B)*Hypergeometric2F1[-1/2, 1/3, 4/3, -((b*x^3)/a)])/Sqrt[1 + (b*x^3)/a]))/(187*b^2)
Time = 0.59 (sec) , antiderivative size = 293, normalized size of antiderivative = 0.97, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {959, 811, 843, 759}
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.
| \(\displaystyle \int x^3 \sqrt {a+b x^3} \left (A+B x^3\right ) \, dx\) |
| \(\Big \downarrow \) 959 |
| \(\displaystyle \frac {(17 A b-8 a B) \int x^3 \sqrt {b x^3+a}dx}{17 b}+\frac {2 B x^4 \left (a+b x^3\right )^{3/2}}{17 b}\) |
| \(\Big \downarrow \) 811 |
| \(\displaystyle \frac {(17 A b-8 a B) \left (\frac {3}{11} a \int \frac {x^3}{\sqrt {b x^3+a}}dx+\frac {2}{11} x^4 \sqrt {a+b x^3}\right )}{17 b}+\frac {2 B x^4 \left (a+b x^3\right )^{3/2}}{17 b}\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle \frac {(17 A b-8 a B) \left (\frac {3}{11} a \left (\frac {2 x \sqrt {a+b x^3}}{5 b}-\frac {2 a \int \frac {1}{\sqrt {b x^3+a}}dx}{5 b}\right )+\frac {2}{11} x^4 \sqrt {a+b x^3}\right )}{17 b}+\frac {2 B x^4 \left (a+b x^3\right )^{3/2}}{17 b}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {(17 A b-8 a B) \left (\frac {3}{11} a \left (\frac {2 x \sqrt {a+b x^3}}{5 b}-\frac {4 \sqrt {2+\sqrt {3}} a \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{5 \sqrt [4]{3} b^{4/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}\right )+\frac {2}{11} x^4 \sqrt {a+b x^3}\right )}{17 b}+\frac {2 B x^4 \left (a+b x^3\right )^{3/2}}{17 b}\) |
Input:
Int[x^3*Sqrt[a + b*x^3]*(A + B*x^3),x]
Output:
(2*B*x^4*(a + b*x^3)^(3/2))/(17*b) + ((17*A*b - 8*a*B)*((2*x^4*Sqrt[a + b* x^3])/11 + (3*a*((2*x*Sqrt[a + b*x^3])/(5*b) - (4*Sqrt[2 + Sqrt[3]]*a*(a^( 1/3) + b^(1/3)*x)*Sqrt[(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 + S qrt[3])*a^(1/3) + b^(1/3)*x)^2]*EllipticF[ArcSin[((1 - Sqrt[3])*a^(1/3) + b^(1/3)*x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)], -7 - 4*Sqrt[3]])/(5*3^(1/ 4)*b^(4/3)*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)*x))/((1 + Sqrt[3])*a^(1/3) + b ^(1/3)*x)^2]*Sqrt[a + b*x^3])))/11))/(17*b)
Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 + Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s *x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[s* ((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x] & & PosQ[a]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* x)^(m + 1)*((a + b*x^n)^p/(c*(m + n*p + 1))), x] + Simp[a*n*(p/(m + n*p + 1 )) Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x] && I GtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m , p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Simp[ a*c^n*((m - n + 1)/(b*(m + n*p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^p, x] , x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n* p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(b*e*(m + n*(p + 1) + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m + n*(p + 1) + 1)) Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[m + n*(p + 1) + 1, 0]
Time = 1.27 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.15
| method | result | size |
| risch | \(\frac {2 x \left (55 b^{2} B \,x^{6}+85 A \,b^{2} x^{3}+15 B a b \,x^{3}+51 a b A -24 a^{2} B \right ) \sqrt {b \,x^{3}+a}}{935 b^{2}}+\frac {4 i a^{2} \left (17 A b -8 B a \right ) \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{935 b^{3} \sqrt {b \,x^{3}+a}}\) | \(349\) |
| elliptic | \(\frac {2 x^{7} \sqrt {b \,x^{3}+a}\, B}{17}+\frac {2 \left (A b +\frac {3 B a}{17}\right ) x^{4} \sqrt {b \,x^{3}+a}}{11 b}+\frac {2 \left (A a -\frac {8 a \left (A b +\frac {3 B a}{17}\right )}{11 b}\right ) x \sqrt {b \,x^{3}+a}}{5 b}+\frac {4 i a \left (A a -\frac {8 a \left (A b +\frac {3 B a}{17}\right )}{11 b}\right ) \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{15 b^{2} \sqrt {b \,x^{3}+a}}\) | \(376\) |
| default | \(A \left (\frac {2 x^{4} \sqrt {b \,x^{3}+a}}{11}+\frac {6 a x \sqrt {b \,x^{3}+a}}{55 b}+\frac {4 i a^{2} \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{55 b^{2} \sqrt {b \,x^{3}+a}}\right )+B \left (\frac {2 x^{7} \sqrt {b \,x^{3}+a}}{17}+\frac {6 a \,x^{4} \sqrt {b \,x^{3}+a}}{187 b}-\frac {48 a^{2} x \sqrt {b \,x^{3}+a}}{935 b^{2}}-\frac {32 i a^{3} \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{935 b^{3} \sqrt {b \,x^{3}+a}}\right )\) | \(658\) |
Input:
int(x^3*(b*x^3+a)^(1/2)*(B*x^3+A),x,method=_RETURNVERBOSE)
Output:
2/935*x*(55*B*b^2*x^6+85*A*b^2*x^3+15*B*a*b*x^3+51*A*a*b-24*B*a^2)/b^2*(b* x^3+a)^(1/2)+4/935*I*a^2*(17*A*b-8*B*a)/b^3*3^(1/2)*(-a*b^2)^(1/3)*(I*(x+1 /2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/ 3))^(1/2)*((x-1/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)*(-I*(x+1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2) ^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)/(b*x^3+a)^(1/2)*EllipticF(1/3*3^(1 /2)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*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))
Time = 0.09 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.30 \[ \int x^3 \sqrt {a+b x^3} \left (A+B x^3\right ) \, dx=\frac {2 \, {\left (6 \, {\left (8 \, B a^{3} - 17 \, A a^{2} b\right )} \sqrt {b} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + {\left (55 \, B b^{3} x^{7} + 5 \, {\left (3 \, B a b^{2} + 17 \, A b^{3}\right )} x^{4} - 3 \, {\left (8 \, B a^{2} b - 17 \, A a b^{2}\right )} x\right )} \sqrt {b x^{3} + a}\right )}}{935 \, b^{3}} \] Input:
integrate(x^3*(b*x^3+a)^(1/2)*(B*x^3+A),x, algorithm="fricas")
Output:
2/935*(6*(8*B*a^3 - 17*A*a^2*b)*sqrt(b)*weierstrassPInverse(0, -4*a/b, x) + (55*B*b^3*x^7 + 5*(3*B*a*b^2 + 17*A*b^3)*x^4 - 3*(8*B*a^2*b - 17*A*a*b^2 )*x)*sqrt(b*x^3 + a))/b^3
Time = 1.22 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.27 \[ \int x^3 \sqrt {a+b x^3} \left (A+B x^3\right ) \, dx=\frac {A \sqrt {a} x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {7}{3}\right )} + \frac {B \sqrt {a} x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {10}{3}\right )} \] Input:
integrate(x**3*(b*x**3+a)**(1/2)*(B*x**3+A),x)
Output:
A*sqrt(a)*x**4*gamma(4/3)*hyper((-1/2, 4/3), (7/3,), b*x**3*exp_polar(I*pi )/a)/(3*gamma(7/3)) + B*sqrt(a)*x**7*gamma(7/3)*hyper((-1/2, 7/3), (10/3,) , b*x**3*exp_polar(I*pi)/a)/(3*gamma(10/3))
\[ \int x^3 \sqrt {a+b x^3} \left (A+B x^3\right ) \, dx=\int { {\left (B x^{3} + A\right )} \sqrt {b x^{3} + a} x^{3} \,d x } \] Input:
integrate(x^3*(b*x^3+a)^(1/2)*(B*x^3+A),x, algorithm="maxima")
Output:
integrate((B*x^3 + A)*sqrt(b*x^3 + a)*x^3, x)
\[ \int x^3 \sqrt {a+b x^3} \left (A+B x^3\right ) \, dx=\int { {\left (B x^{3} + A\right )} \sqrt {b x^{3} + a} x^{3} \,d x } \] Input:
integrate(x^3*(b*x^3+a)^(1/2)*(B*x^3+A),x, algorithm="giac")
Output:
integrate((B*x^3 + A)*sqrt(b*x^3 + a)*x^3, x)
Timed out. \[ \int x^3 \sqrt {a+b x^3} \left (A+B x^3\right ) \, dx=\int x^3\,\left (B\,x^3+A\right )\,\sqrt {b\,x^3+a} \,d x \] Input:
int(x^3*(A + B*x^3)*(a + b*x^3)^(1/2),x)
Output:
int(x^3*(A + B*x^3)*(a + b*x^3)^(1/2), x)
\[ \int x^3 \sqrt {a+b x^3} \left (A+B x^3\right ) \, dx=\frac {\frac {54 \sqrt {b \,x^{3}+a}\, a^{2} x}{935}+\frac {40 \sqrt {b \,x^{3}+a}\, a b \,x^{4}}{187}+\frac {2 \sqrt {b \,x^{3}+a}\, b^{2} x^{7}}{17}-\frac {54 \left (\int \frac {\sqrt {b \,x^{3}+a}}{b \,x^{3}+a}d x \right ) a^{3}}{935}}{b} \] Input:
int(x^3*(b*x^3+a)^(1/2)*(B*x^3+A),x)
Output:
(2*(27*sqrt(a + b*x**3)*a**2*x + 100*sqrt(a + b*x**3)*a*b*x**4 + 55*sqrt(a + b*x**3)*b**2*x**7 - 27*int(sqrt(a + b*x**3)/(a + b*x**3),x)*a**3))/(935 *b)