Integrand size = 26, antiderivative size = 252 \[ \int (e x)^{3/2} \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx=\frac {8 a^2 (3 A b-a B) e \sqrt {e x} \sqrt {a+b x^2}}{231 b^2}+\frac {4 a (3 A b-a B) (e x)^{5/2} \sqrt {a+b x^2}}{77 b e}+\frac {2 (3 A b-a B) (e x)^{5/2} \left (a+b x^2\right )^{3/2}}{33 b e}+\frac {2 B (e x)^{5/2} \left (a+b x^2\right )^{5/2}}{15 b e}-\frac {4 a^{11/4} (3 A b-a B) e^{3/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{231 b^{9/4} \sqrt {a+b x^2}} \] Output:
8/231*a^2*(3*A*b-B*a)*e*(e*x)^(1/2)*(b*x^2+a)^(1/2)/b^2+4/77*a*(3*A*b-B*a) *(e*x)^(5/2)*(b*x^2+a)^(1/2)/b/e+2/33*(3*A*b-B*a)*(e*x)^(5/2)*(b*x^2+a)^(3 /2)/b/e+2/15*B*(e*x)^(5/2)*(b*x^2+a)^(5/2)/b/e-4/231*a^(11/4)*(3*A*b-B*a)* e^(3/2)*(a^(1/2)+b^(1/2)*x)*((b*x^2+a)/(a^(1/2)+b^(1/2)*x)^2)^(1/2)*Invers eJacobiAM(2*arctan(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2)),1/2*2^(1/2))/b^(9/ 4)/(b*x^2+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.13 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.45 \[ \int (e x)^{3/2} \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx=\frac {2 e \sqrt {e x} \sqrt {a+b x^2} \left (-\left (a+b x^2\right )^2 \sqrt {1+\frac {b x^2}{a}} \left (-15 A b+5 a B-11 b B x^2\right )+5 a^2 (-3 A b+a B) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4},\frac {5}{4},-\frac {b x^2}{a}\right )\right )}{165 b^2 \sqrt {1+\frac {b x^2}{a}}} \] Input:
Integrate[(e*x)^(3/2)*(a + b*x^2)^(3/2)*(A + B*x^2),x]
Output:
(2*e*Sqrt[e*x]*Sqrt[a + b*x^2]*(-((a + b*x^2)^2*Sqrt[1 + (b*x^2)/a]*(-15*A *b + 5*a*B - 11*b*B*x^2)) + 5*a^2*(-3*A*b + a*B)*Hypergeometric2F1[-3/2, 1 /4, 5/4, -((b*x^2)/a)]))/(165*b^2*Sqrt[1 + (b*x^2)/a])
Time = 0.31 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {363, 248, 248, 262, 266, 761}
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.
| \(\displaystyle \int (e x)^{3/2} \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx\) |
| \(\Big \downarrow \) 363 |
| \(\displaystyle \frac {(3 A b-a B) \int (e x)^{3/2} \left (b x^2+a\right )^{3/2}dx}{3 b}+\frac {2 B (e x)^{5/2} \left (a+b x^2\right )^{5/2}}{15 b e}\) |
| \(\Big \downarrow \) 248 |
| \(\displaystyle \frac {(3 A b-a B) \left (\frac {6}{11} a \int (e x)^{3/2} \sqrt {b x^2+a}dx+\frac {2 (e x)^{5/2} \left (a+b x^2\right )^{3/2}}{11 e}\right )}{3 b}+\frac {2 B (e x)^{5/2} \left (a+b x^2\right )^{5/2}}{15 b e}\) |
| \(\Big \downarrow \) 248 |
| \(\displaystyle \frac {(3 A b-a B) \left (\frac {6}{11} a \left (\frac {2}{7} a \int \frac {(e x)^{3/2}}{\sqrt {b x^2+a}}dx+\frac {2 (e x)^{5/2} \sqrt {a+b x^2}}{7 e}\right )+\frac {2 (e x)^{5/2} \left (a+b x^2\right )^{3/2}}{11 e}\right )}{3 b}+\frac {2 B (e x)^{5/2} \left (a+b x^2\right )^{5/2}}{15 b e}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {(3 A b-a B) \left (\frac {6}{11} a \left (\frac {2}{7} a \left (\frac {2 e \sqrt {e x} \sqrt {a+b x^2}}{3 b}-\frac {a e^2 \int \frac {1}{\sqrt {e x} \sqrt {b x^2+a}}dx}{3 b}\right )+\frac {2 (e x)^{5/2} \sqrt {a+b x^2}}{7 e}\right )+\frac {2 (e x)^{5/2} \left (a+b x^2\right )^{3/2}}{11 e}\right )}{3 b}+\frac {2 B (e x)^{5/2} \left (a+b x^2\right )^{5/2}}{15 b e}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {(3 A b-a B) \left (\frac {6}{11} a \left (\frac {2}{7} a \left (\frac {2 e \sqrt {e x} \sqrt {a+b x^2}}{3 b}-\frac {2 a e \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {e x}}{3 b}\right )+\frac {2 (e x)^{5/2} \sqrt {a+b x^2}}{7 e}\right )+\frac {2 (e x)^{5/2} \left (a+b x^2\right )^{3/2}}{11 e}\right )}{3 b}+\frac {2 B (e x)^{5/2} \left (a+b x^2\right )^{5/2}}{15 b e}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {(3 A b-a B) \left (\frac {6}{11} a \left (\frac {2}{7} a \left (\frac {2 e \sqrt {e x} \sqrt {a+b x^2}}{3 b}-\frac {a^{3/4} \sqrt {e} \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{3 b^{5/4} \sqrt {a+b x^2}}\right )+\frac {2 (e x)^{5/2} \sqrt {a+b x^2}}{7 e}\right )+\frac {2 (e x)^{5/2} \left (a+b x^2\right )^{3/2}}{11 e}\right )}{3 b}+\frac {2 B (e x)^{5/2} \left (a+b x^2\right )^{5/2}}{15 b e}\) |
Input:
Int[(e*x)^(3/2)*(a + b*x^2)^(3/2)*(A + B*x^2),x]
Output:
(2*B*(e*x)^(5/2)*(a + b*x^2)^(5/2))/(15*b*e) + ((3*A*b - a*B)*((2*(e*x)^(5 /2)*(a + b*x^2)^(3/2))/(11*e) + (6*a*((2*(e*x)^(5/2)*Sqrt[a + b*x^2])/(7*e ) + (2*a*((2*e*Sqrt[e*x]*Sqrt[a + b*x^2])/(3*b) - (a^(3/4)*Sqrt[e]*(Sqrt[a ]*e + Sqrt[b]*e*x)*Sqrt[(a*e^2 + b*e^2*x^2)/(Sqrt[a]*e + Sqrt[b]*e*x)^2]*E llipticF[2*ArcTan[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], 1/2])/(3*b^(5/4) *Sqrt[a + b*x^2])))/7))/11))/(3*b)
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^p/(c*(m + 2*p + 1))), x] + Simp[2*a*(p/(m + 2*p + 1)) Int[(c*x)^m*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x] && GtQ[ p, 0] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3)) Int[(e*x)^ m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d , 0] && NeQ[m + 2*p + 3, 0]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Time = 0.97 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.94
| method | result | size |
| risch | \(\frac {2 \left (77 b^{3} B \,x^{6}+105 A \,x^{4} b^{3}+119 B \,x^{4} a \,b^{2}+195 a A \,b^{2} x^{2}+12 B \,a^{2} b \,x^{2}+60 a^{2} b A -20 a^{3} B \right ) x \sqrt {b \,x^{2}+a}\, e^{2}}{1155 b^{2} \sqrt {e x}}-\frac {4 a^{3} \left (3 A b -B a \right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) e^{2} \sqrt {\left (b \,x^{2}+a \right ) e x}}{231 b^{3} \sqrt {b e \,x^{3}+a e x}\, \sqrt {e x}\, \sqrt {b \,x^{2}+a}}\) | \(238\) |
| default | \(-\frac {2 e \sqrt {e x}\, \left (-77 b^{5} B \,x^{9}-105 A \,b^{5} x^{7}-196 B a \,b^{4} x^{7}+30 A \sqrt {2}\, \sqrt {-a b}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{3} b -300 a A \,b^{4} x^{5}-10 B \sqrt {2}\, \sqrt {-a b}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{4}-131 B \,a^{2} b^{3} x^{5}-255 a^{2} A \,b^{3} x^{3}+8 B \,a^{3} b^{2} x^{3}-60 a^{3} b^{2} A x +20 a^{4} b B x \right )}{1155 x \sqrt {b \,x^{2}+a}\, b^{3}}\) | \(300\) |
| elliptic | \(\frac {\sqrt {e x}\, \sqrt {\left (b \,x^{2}+a \right ) e x}\, \left (\frac {2 B b e \,x^{6} \sqrt {b e \,x^{3}+a e x}}{15}+\frac {2 \left (b \left (A b +2 B a \right ) e^{2}-\frac {13 B a b \,e^{2}}{15}\right ) x^{4} \sqrt {b e \,x^{3}+a e x}}{11 b e}+\frac {2 \left (a \left (2 A b +B a \right ) e^{2}-\frac {9 \left (b \left (A b +2 B a \right ) e^{2}-\frac {13 B a b \,e^{2}}{15}\right ) a}{11 b}\right ) x^{2} \sqrt {b e \,x^{3}+a e x}}{7 b e}+\frac {2 \left (a^{2} A \,e^{2}-\frac {5 \left (a \left (2 A b +B a \right ) e^{2}-\frac {9 \left (b \left (A b +2 B a \right ) e^{2}-\frac {13 B a b \,e^{2}}{15}\right ) a}{11 b}\right ) a}{7 b}\right ) \sqrt {b e \,x^{3}+a e x}}{3 b e}-\frac {\left (a^{2} A \,e^{2}-\frac {5 \left (a \left (2 A b +B a \right ) e^{2}-\frac {9 \left (b \left (A b +2 B a \right ) e^{2}-\frac {13 B a b \,e^{2}}{15}\right ) a}{11 b}\right ) a}{7 b}\right ) a \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{3 b^{2} \sqrt {b e \,x^{3}+a e x}}\right )}{e x \sqrt {b \,x^{2}+a}}\) | \(414\) |
Input:
int((e*x)^(3/2)*(b*x^2+a)^(3/2)*(B*x^2+A),x,method=_RETURNVERBOSE)
Output:
2/1155/b^2*(77*B*b^3*x^6+105*A*b^3*x^4+119*B*a*b^2*x^4+195*A*a*b^2*x^2+12* B*a^2*b*x^2+60*A*a^2*b-20*B*a^3)*x*(b*x^2+a)^(1/2)*e^2/(e*x)^(1/2)-4/231*a ^3/b^3*(3*A*b-B*a)*(-a*b)^(1/2)*((x+1/b*(-a*b)^(1/2))*b/(-a*b)^(1/2))^(1/2 )*(-2*(x-1/b*(-a*b)^(1/2))*b/(-a*b)^(1/2))^(1/2)*(-b/(-a*b)^(1/2)*x)^(1/2) /(b*e*x^3+a*e*x)^(1/2)*EllipticF(((x+1/b*(-a*b)^(1/2))*b/(-a*b)^(1/2))^(1/ 2),1/2*2^(1/2))*e^2*((b*x^2+a)*e*x)^(1/2)/(e*x)^(1/2)/(b*x^2+a)^(1/2)
Time = 0.13 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.49 \[ \int (e x)^{3/2} \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx=\frac {2 \, {\left (20 \, {\left (B a^{4} - 3 \, A a^{3} b\right )} \sqrt {b e} e {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) + {\left (77 \, B b^{4} e x^{6} + 7 \, {\left (17 \, B a b^{3} + 15 \, A b^{4}\right )} e x^{4} + 3 \, {\left (4 \, B a^{2} b^{2} + 65 \, A a b^{3}\right )} e x^{2} - 20 \, {\left (B a^{3} b - 3 \, A a^{2} b^{2}\right )} e\right )} \sqrt {b x^{2} + a} \sqrt {e x}\right )}}{1155 \, b^{3}} \] Input:
integrate((e*x)^(3/2)*(b*x^2+a)^(3/2)*(B*x^2+A),x, algorithm="fricas")
Output:
2/1155*(20*(B*a^4 - 3*A*a^3*b)*sqrt(b*e)*e*weierstrassPInverse(-4*a/b, 0, x) + (77*B*b^4*e*x^6 + 7*(17*B*a*b^3 + 15*A*b^4)*e*x^4 + 3*(4*B*a^2*b^2 + 65*A*a*b^3)*e*x^2 - 20*(B*a^3*b - 3*A*a^2*b^2)*e)*sqrt(b*x^2 + a)*sqrt(e*x ))/b^3
Result contains complex when optimal does not.
Time = 12.24 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.79 \[ \int (e x)^{3/2} \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx=\frac {A a^{\frac {3}{2}} e^{\frac {3}{2}} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {9}{4}\right )} + \frac {A \sqrt {a} b e^{\frac {3}{2}} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {13}{4}\right )} + \frac {B a^{\frac {3}{2}} e^{\frac {3}{2}} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {13}{4}\right )} + \frac {B \sqrt {a} b e^{\frac {3}{2}} x^{\frac {13}{2}} \Gamma \left (\frac {13}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {13}{4} \\ \frac {17}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {17}{4}\right )} \] Input:
integrate((e*x)**(3/2)*(b*x**2+a)**(3/2)*(B*x**2+A),x)
Output:
A*a**(3/2)*e**(3/2)*x**(5/2)*gamma(5/4)*hyper((-1/2, 5/4), (9/4,), b*x**2* exp_polar(I*pi)/a)/(2*gamma(9/4)) + A*sqrt(a)*b*e**(3/2)*x**(9/2)*gamma(9/ 4)*hyper((-1/2, 9/4), (13/4,), b*x**2*exp_polar(I*pi)/a)/(2*gamma(13/4)) + B*a**(3/2)*e**(3/2)*x**(9/2)*gamma(9/4)*hyper((-1/2, 9/4), (13/4,), b*x** 2*exp_polar(I*pi)/a)/(2*gamma(13/4)) + B*sqrt(a)*b*e**(3/2)*x**(13/2)*gamm a(13/4)*hyper((-1/2, 13/4), (17/4,), b*x**2*exp_polar(I*pi)/a)/(2*gamma(17 /4))
\[ \int (e x)^{3/2} \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx=\int { {\left (B x^{2} + A\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {3}{2}} \,d x } \] Input:
integrate((e*x)^(3/2)*(b*x^2+a)^(3/2)*(B*x^2+A),x, algorithm="maxima")
Output:
integrate((B*x^2 + A)*(b*x^2 + a)^(3/2)*(e*x)^(3/2), x)
\[ \int (e x)^{3/2} \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx=\int { {\left (B x^{2} + A\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {3}{2}} \,d x } \] Input:
integrate((e*x)^(3/2)*(b*x^2+a)^(3/2)*(B*x^2+A),x, algorithm="giac")
Output:
integrate((B*x^2 + A)*(b*x^2 + a)^(3/2)*(e*x)^(3/2), x)
Timed out. \[ \int (e x)^{3/2} \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx=\int \left (B\,x^2+A\right )\,{\left (e\,x\right )}^{3/2}\,{\left (b\,x^2+a\right )}^{3/2} \,d x \] Input:
int((A + B*x^2)*(e*x)^(3/2)*(a + b*x^2)^(3/2),x)
Output:
int((A + B*x^2)*(e*x)^(3/2)*(a + b*x^2)^(3/2), x)
\[ \int (e x)^{3/2} \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx=\frac {2 \sqrt {e}\, e \left (40 \sqrt {x}\, \sqrt {b \,x^{2}+a}\, a^{3}+207 \sqrt {x}\, \sqrt {b \,x^{2}+a}\, a^{2} b \,x^{2}+224 \sqrt {x}\, \sqrt {b \,x^{2}+a}\, a \,b^{2} x^{4}+77 \sqrt {x}\, \sqrt {b \,x^{2}+a}\, b^{3} x^{6}-20 \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}}{b \,x^{3}+a x}d x \right ) a^{4}\right )}{1155 b} \] Input:
int((e*x)^(3/2)*(b*x^2+a)^(3/2)*(B*x^2+A),x)
Output:
(2*sqrt(e)*e*(40*sqrt(x)*sqrt(a + b*x**2)*a**3 + 207*sqrt(x)*sqrt(a + b*x* *2)*a**2*b*x**2 + 224*sqrt(x)*sqrt(a + b*x**2)*a*b**2*x**4 + 77*sqrt(x)*sq rt(a + b*x**2)*b**3*x**6 - 20*int((sqrt(x)*sqrt(a + b*x**2))/(a*x + b*x**3 ),x)*a**4))/(1155*b)