Integrand size = 21, antiderivative size = 324 \[ \int \sqrt {a+b x^3} \left (c+d x^3\right )^2 \, dx=\frac {2}{935} \left (187 c^2-\frac {4 a d (17 b c-4 a d)}{b^2}\right ) x \sqrt {a+b x^3}+\frac {4 d (17 b c-4 a d) x \left (a+b x^3\right )^{3/2}}{187 b^2}+\frac {2 d^2 x^4 \left (a+b x^3\right )^{3/2}}{17 b}+\frac {2\ 3^{3/4} \sqrt {2+\sqrt {3}} a \left (187 b^2 c^2-4 a d (17 b c-4 a d)\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{935 b^{7/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \] Output:
2/935*(187*c^2-4*a*d*(-4*a*d+17*b*c)/b^2)*x*(b*x^3+a)^(1/2)+4/187*d*(-4*a* d+17*b*c)*x*(b*x^3+a)^(3/2)/b^2+2/17*d^2*x^4*(b*x^3+a)^(3/2)/b+2/935*3^(3/ 4)*(1/2*6^(1/2)+1/2*2^(1/2))*a*(187*b^2*c^2-4*a*d*(-4*a*d+17*b*c))*(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 = 7.54 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.50 \[ \int \sqrt {a+b x^3} \left (c+d x^3\right )^2 \, dx=\frac {x \sqrt {a+b x^3} \left (40 a \left (14 c^2+7 c d x^3+2 d^2 x^6\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{3},\frac {10}{3},-\frac {b x^3}{a}\right )+3 b x^3 \left (11 c^2+16 c d x^3+5 d^2 x^6\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4}{3},\frac {13}{3},-\frac {b x^3}{a}\right )+9 b x^3 \left (c+d x^3\right )^2 \, _3F_2\left (\frac {1}{2},\frac {4}{3},2;1,\frac {13}{3};-\frac {b x^3}{a}\right )\right )}{560 a \sqrt {1+\frac {b x^3}{a}}} \] Input:
Integrate[Sqrt[a + b*x^3]*(c + d*x^3)^2,x]
Output:
(x*Sqrt[a + b*x^3]*(40*a*(14*c^2 + 7*c*d*x^3 + 2*d^2*x^6)*Hypergeometric2F 1[-1/2, 1/3, 10/3, -((b*x^3)/a)] + 3*b*x^3*(11*c^2 + 16*c*d*x^3 + 5*d^2*x^ 6)*Hypergeometric2F1[1/2, 4/3, 13/3, -((b*x^3)/a)] + 9*b*x^3*(c + d*x^3)^2 *HypergeometricPFQ[{1/2, 4/3, 2}, {1, 13/3}, -((b*x^3)/a)]))/(560*a*Sqrt[1 + (b*x^3)/a])
Time = 0.65 (sec) , antiderivative size = 322, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {933, 27, 913, 748, 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.
| \(\displaystyle \int \sqrt {a+b x^3} \left (c+d x^3\right )^2 \, dx\) |
| \(\Big \downarrow \) 933 |
| \(\displaystyle \frac {2 \int \frac {1}{2} \sqrt {b x^3+a} \left (d (23 b c-8 a d) x^3+c (17 b c-2 a d)\right )dx}{17 b}+\frac {2 d x \left (a+b x^3\right )^{3/2} \left (c+d x^3\right )}{17 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \sqrt {b x^3+a} \left (d (23 b c-8 a d) x^3+c (17 b c-2 a d)\right )dx}{17 b}+\frac {2 d x \left (a+b x^3\right )^{3/2} \left (c+d x^3\right )}{17 b}\) |
| \(\Big \downarrow \) 913 |
| \(\displaystyle \frac {\frac {\left (16 a^2 d^2-68 a b c d+187 b^2 c^2\right ) \int \sqrt {b x^3+a}dx}{11 b}+\frac {2 d x \left (a+b x^3\right )^{3/2} (23 b c-8 a d)}{11 b}}{17 b}+\frac {2 d x \left (a+b x^3\right )^{3/2} \left (c+d x^3\right )}{17 b}\) |
| \(\Big \downarrow \) 748 |
| \(\displaystyle \frac {\frac {\left (16 a^2 d^2-68 a b c d+187 b^2 c^2\right ) \left (\frac {3}{5} a \int \frac {1}{\sqrt {b x^3+a}}dx+\frac {2}{5} x \sqrt {a+b x^3}\right )}{11 b}+\frac {2 d x \left (a+b x^3\right )^{3/2} (23 b c-8 a d)}{11 b}}{17 b}+\frac {2 d x \left (a+b x^3\right )^{3/2} \left (c+d x^3\right )}{17 b}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {\frac {\left (16 a^2 d^2-68 a b c d+187 b^2 c^2\right ) \left (\frac {2\ 3^{3/4} \sqrt {2+\sqrt {3}} a \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{5 \sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {2}{5} x \sqrt {a+b x^3}\right )}{11 b}+\frac {2 d x \left (a+b x^3\right )^{3/2} (23 b c-8 a d)}{11 b}}{17 b}+\frac {2 d x \left (a+b x^3\right )^{3/2} \left (c+d x^3\right )}{17 b}\) |
Input:
Int[Sqrt[a + b*x^3]*(c + d*x^3)^2,x]
Output:
(2*d*x*(a + b*x^3)^(3/2)*(c + d*x^3))/(17*b) + ((2*d*(23*b*c - 8*a*d)*x*(a + b*x^3)^(3/2))/(11*b) + ((187*b^2*c^2 - 68*a*b*c*d + 16*a^2*d^2)*((2*x*S qrt[a + b*x^3])/5 + (2*3^(3/4)*Sqrt[2 + Sqrt[3]]*a*(a^(1/3) + b^(1/3)*x)*S qrt[(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 + Sqrt[3])*a^(1/3) + b ^(1/3)*x)^2]*EllipticF[ArcSin[((1 - Sqrt[3])*a^(1/3) + b^(1/3)*x)/((1 + Sq rt[3])*a^(1/3) + b^(1/3)*x)], -7 - 4*Sqrt[3]])/(5*b^(1/3)*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*b))/(17*b)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Simp[a*n*(p/(n*p + 1)) Int[(a + b*x^n)^(p - 1), x], x] /; Fre eQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || LtQ[Denominat or[p + 1/n], Denominator[p]])
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[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Si mp[d*x*((a + b*x^n)^(p + 1)/(b*(n*(p + 1) + 1))), x] - Simp[(a*d - b*c*(n*( p + 1) + 1))/(b*(n*(p + 1) + 1)) Int[(a + b*x^n)^p, x], x] /; FreeQ[{a, b , c, d, n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[d*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q - 1)/(b*(n*(p + q) + 1))), x] + Simp[1/(b*(n*(p + q) + 1)) Int[(a + b*x^n)^p*(c + d*x^n)^(q - 2)*Sim p[c*(b*c*(n*(p + q) + 1) - a*d) + d*(b*c*(n*(p + 2*q - 1) + 1) - a*d*(n*(q - 1) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d , 0] && GtQ[q, 1] && NeQ[n*(p + q) + 1, 0] && !IGtQ[p, 1] && IntBinomialQ[ a, b, c, d, n, p, q, x]
Time = 1.53 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.16
| method | result | size |
| risch | \(-\frac {2 x \left (-55 b^{2} d^{2} x^{6}-15 x^{3} a b \,d^{2}-170 x^{3} b^{2} c d +24 a^{2} d^{2}-102 a b c d -187 b^{2} c^{2}\right ) \sqrt {b \,x^{3}+a}}{935 b^{2}}-\frac {2 i a \left (16 a^{2} d^{2}-68 a b c d +187 b^{2} c^{2}\right ) \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{935 b^{3} \sqrt {b \,x^{3}+a}}\) | \(377\) |
| elliptic | \(\frac {2 d^{2} x^{7} \sqrt {b \,x^{3}+a}}{17}+\frac {2 \left (\frac {3}{17} a \,d^{2}+2 b c d \right ) x^{4} \sqrt {b \,x^{3}+a}}{11 b}+\frac {2 \left (2 a c d +b \,c^{2}-\frac {8 a \left (\frac {3}{17} a \,d^{2}+2 b c d \right )}{11 b}\right ) x \sqrt {b \,x^{3}+a}}{5 b}-\frac {2 i \left (a \,c^{2}-\frac {2 a \left (2 a c d +b \,c^{2}-\frac {8 a \left (\frac {3}{17} a \,d^{2}+2 b c d \right )}{11 b}\right )}{5 b}\right ) \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{3 b \sqrt {b \,x^{3}+a}}\) | \(415\) |
| default | \(\text {Expression too large to display}\) | \(962\) |
Input:
int((b*x^3+a)^(1/2)*(d*x^3+c)^2,x,method=_RETURNVERBOSE)
Output:
-2/935*x*(-55*b^2*d^2*x^6-15*a*b*d^2*x^3-170*b^2*c*d*x^3+24*a^2*d^2-102*a* b*c*d-187*b^2*c^2)/b^2*(b*x^3+a)^(1/2)-2/935*I*a*(16*a^2*d^2-68*a*b*c*d+18 7*b^2*c^2)/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 = 0.10 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.36 \[ \int \sqrt {a+b x^3} \left (c+d x^3\right )^2 \, dx=\frac {2 \, {\left (3 \, {\left (187 \, a b^{2} c^{2} - 68 \, a^{2} b c d + 16 \, a^{3} d^{2}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + {\left (55 \, b^{3} d^{2} x^{7} + 5 \, {\left (34 \, b^{3} c d + 3 \, a b^{2} d^{2}\right )} x^{4} + {\left (187 \, b^{3} c^{2} + 102 \, a b^{2} c d - 24 \, a^{2} b d^{2}\right )} x\right )} \sqrt {b x^{3} + a}\right )}}{935 \, b^{3}} \] Input:
integrate((b*x^3+a)^(1/2)*(d*x^3+c)^2,x, algorithm="fricas")
Output:
2/935*(3*(187*a*b^2*c^2 - 68*a^2*b*c*d + 16*a^3*d^2)*sqrt(b)*weierstrassPI nverse(0, -4*a/b, x) + (55*b^3*d^2*x^7 + 5*(34*b^3*c*d + 3*a*b^2*d^2)*x^4 + (187*b^3*c^2 + 102*a*b^2*c*d - 24*a^2*b*d^2)*x)*sqrt(b*x^3 + a))/b^3
Time = 1.53 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.40 \[ \int \sqrt {a+b x^3} \left (c+d x^3\right )^2 \, dx=\frac {\sqrt {a} c^{2} x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} + \frac {2 \sqrt {a} c d x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {7}{3}\right )} + \frac {\sqrt {a} d^{2} x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {10}{3}\right )} \] Input:
integrate((b*x**3+a)**(1/2)*(d*x**3+c)**2,x)
Output:
sqrt(a)*c**2*x*gamma(1/3)*hyper((-1/2, 1/3), (4/3,), b*x**3*exp_polar(I*pi )/a)/(3*gamma(4/3)) + 2*sqrt(a)*c*d*x**4*gamma(4/3)*hyper((-1/2, 4/3), (7/ 3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(7/3)) + sqrt(a)*d**2*x**7*gamma(7/ 3)*hyper((-1/2, 7/3), (10/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(10/3))
\[ \int \sqrt {a+b x^3} \left (c+d x^3\right )^2 \, dx=\int { \sqrt {b x^{3} + a} {\left (d x^{3} + c\right )}^{2} \,d x } \] Input:
integrate((b*x^3+a)^(1/2)*(d*x^3+c)^2,x, algorithm="maxima")
Output:
integrate(sqrt(b*x^3 + a)*(d*x^3 + c)^2, x)
\[ \int \sqrt {a+b x^3} \left (c+d x^3\right )^2 \, dx=\int { \sqrt {b x^{3} + a} {\left (d x^{3} + c\right )}^{2} \,d x } \] Input:
integrate((b*x^3+a)^(1/2)*(d*x^3+c)^2,x, algorithm="giac")
Output:
integrate(sqrt(b*x^3 + a)*(d*x^3 + c)^2, x)
Timed out. \[ \int \sqrt {a+b x^3} \left (c+d x^3\right )^2 \, dx=\int \sqrt {b\,x^3+a}\,{\left (d\,x^3+c\right )}^2 \,d x \] Input:
int((a + b*x^3)^(1/2)*(c + d*x^3)^2,x)
Output:
int((a + b*x^3)^(1/2)*(c + d*x^3)^2, x)
\[ \int \sqrt {a+b x^3} \left (c+d x^3\right )^2 \, dx=\frac {-48 \sqrt {b \,x^{3}+a}\, a^{2} d^{2} x +204 \sqrt {b \,x^{3}+a}\, a b c d x +30 \sqrt {b \,x^{3}+a}\, a b \,d^{2} x^{4}+374 \sqrt {b \,x^{3}+a}\, b^{2} c^{2} x +340 \sqrt {b \,x^{3}+a}\, b^{2} c d \,x^{4}+110 \sqrt {b \,x^{3}+a}\, b^{2} d^{2} x^{7}+48 \left (\int \frac {\sqrt {b \,x^{3}+a}}{b \,x^{3}+a}d x \right ) a^{3} d^{2}-204 \left (\int \frac {\sqrt {b \,x^{3}+a}}{b \,x^{3}+a}d x \right ) a^{2} b c d +561 \left (\int \frac {\sqrt {b \,x^{3}+a}}{b \,x^{3}+a}d x \right ) a \,b^{2} c^{2}}{935 b^{2}} \] Input:
int((b*x^3+a)^(1/2)*(d*x^3+c)^2,x)
Output:
( - 48*sqrt(a + b*x**3)*a**2*d**2*x + 204*sqrt(a + b*x**3)*a*b*c*d*x + 30* sqrt(a + b*x**3)*a*b*d**2*x**4 + 374*sqrt(a + b*x**3)*b**2*c**2*x + 340*sq rt(a + b*x**3)*b**2*c*d*x**4 + 110*sqrt(a + b*x**3)*b**2*d**2*x**7 + 48*in t(sqrt(a + b*x**3)/(a + b*x**3),x)*a**3*d**2 - 204*int(sqrt(a + b*x**3)/(a + b*x**3),x)*a**2*b*c*d + 561*int(sqrt(a + b*x**3)/(a + b*x**3),x)*a*b**2 *c**2)/(935*b**2)