\(\int (e x)^{3/2} (a+b x^2)^2 \sqrt {c+d x^2} \, dx\) [822]

Optimal result

Integrand size = 28, antiderivative size = 288 \[ \int (e x)^{3/2} \left (a+b x^2\right )^2 \sqrt {c+d x^2} \, dx=\frac {4 c \left (11 a^2 d^2+b c (3 b c-10 a d)\right ) e \sqrt {e x} \sqrt {c+d x^2}}{231 d^3}+\frac {2 \left (11 a^2 d^2+b c (3 b c-10 a d)\right ) (e x)^{5/2} \sqrt {c+d x^2}}{77 d^2 e}-\frac {2 b (3 b c-10 a d) (e x)^{5/2} \left (c+d x^2\right )^{3/2}}{55 d^2 e}+\frac {2 b^2 (e x)^{9/2} \left (c+d x^2\right )^{3/2}}{15 d e^3}-\frac {2 c^{7/4} \left (11 a^2 d^2+b c (3 b c-10 a d)\right ) e^{3/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{231 d^{13/4} \sqrt {c+d x^2}} \]

[Out]

-2/55*b*(-10*a*d+3*b*c)*(e*x)^(5/2)*(d*x^2+c)^(3/2)/d^2/e+2/15*b^2*(e*x)^(9/2)*(d*x^2+c)^(3/2)/d/e^3+2/77*(11*
a^2*d^2+b*c*(-10*a*d+3*b*c))*(e*x)^(5/2)*(d*x^2+c)^(1/2)/d^2/e+4/231*c*(11*a^2*d^2+b*c*(-10*a*d+3*b*c))*e*(e*x
)^(1/2)*(d*x^2+c)^(1/2)/d^3-2/231*c^(7/4)*(11*a^2*d^2+b*c*(-10*a*d+3*b*c))*e^(3/2)*(cos(2*arctan(d^(1/4)*(e*x)
^(1/2)/c^(1/4)/e^(1/2)))^2)^(1/2)/cos(2*arctan(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2)))*EllipticF(sin(2*arctan(d^
(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2))),1/2*2^(1/2))*(c^(1/2)+x*d^(1/2))*((d*x^2+c)/(c^(1/2)+x*d^(1/2))^2)^(1/2)/d
^(13/4)/(d*x^2+c)^(1/2)

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {475, 470, 285, 327, 335, 226} \[ \int (e x)^{3/2} \left (a+b x^2\right )^2 \sqrt {c+d x^2} \, dx=-\frac {2 c^{7/4} e^{3/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (11 a^2 d^2+b c (3 b c-10 a d)\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{231 d^{13/4} \sqrt {c+d x^2}}+\frac {2 (e x)^{5/2} \sqrt {c+d x^2} \left (11 a^2 d^2+b c (3 b c-10 a d)\right )}{77 d^2 e}+\frac {4 c e \sqrt {e x} \sqrt {c+d x^2} \left (11 a^2 d^2+b c (3 b c-10 a d)\right )}{231 d^3}-\frac {2 b (e x)^{5/2} \left (c+d x^2\right )^{3/2} (3 b c-10 a d)}{55 d^2 e}+\frac {2 b^2 (e x)^{9/2} \left (c+d x^2\right )^{3/2}}{15 d e^3} \]

[In]

Int[(e*x)^(3/2)*(a + b*x^2)^2*Sqrt[c + d*x^2],x]

[Out]

(4*c*(11*a^2*d^2 + b*c*(3*b*c - 10*a*d))*e*Sqrt[e*x]*Sqrt[c + d*x^2])/(231*d^3) + (2*(11*a^2*d^2 + b*c*(3*b*c
- 10*a*d))*(e*x)^(5/2)*Sqrt[c + d*x^2])/(77*d^2*e) - (2*b*(3*b*c - 10*a*d)*(e*x)^(5/2)*(c + d*x^2)^(3/2))/(55*
d^2*e) + (2*b^2*(e*x)^(9/2)*(c + d*x^2)^(3/2))/(15*d*e^3) - (2*c^(7/4)*(11*a^2*d^2 + b*c*(3*b*c - 10*a*d))*e^(
3/2)*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*EllipticF[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c
^(1/4)*Sqrt[e])], 1/2])/(231*d^(13/4)*Sqrt[c + d*x^2])

Rule 226

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]

Rule 285

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] + Dist[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]
&& IGtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 327

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] - Dist[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]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[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]

Rule 470

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] - Dist[(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]

Rule 475

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^2, x_Symbol] :> Simp[d^2*(e*x)^(
m + n + 1)*((a + b*x^n)^(p + 1)/(b*e^(n + 1)*(m + n*(p + 2) + 1))), x] + Dist[1/(b*(m + n*(p + 2) + 1)), Int[(
e*x)^m*(a + b*x^n)^p*Simp[b*c^2*(m + n*(p + 2) + 1) + d*((2*b*c - a*d)*(m + n + 1) + 2*b*c*n*(p + 1))*x^n, x],
x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && NeQ[m + n*(p + 2) + 1, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {2 b^2 (e x)^{9/2} \left (c+d x^2\right )^{3/2}}{15 d e^3}+\frac {2 \int (e x)^{3/2} \sqrt {c+d x^2} \left (\frac {15 a^2 d}{2}-\frac {3}{2} b (3 b c-10 a d) x^2\right ) \, dx}{15 d} \\ & = -\frac {2 b (3 b c-10 a d) (e x)^{5/2} \left (c+d x^2\right )^{3/2}}{55 d^2 e}+\frac {2 b^2 (e x)^{9/2} \left (c+d x^2\right )^{3/2}}{15 d e^3}+\frac {1}{11} \left (11 a^2+\frac {b c (3 b c-10 a d)}{d^2}\right ) \int (e x)^{3/2} \sqrt {c+d x^2} \, dx \\ & = \frac {2 \left (11 a^2+\frac {b c (3 b c-10 a d)}{d^2}\right ) (e x)^{5/2} \sqrt {c+d x^2}}{77 e}-\frac {2 b (3 b c-10 a d) (e x)^{5/2} \left (c+d x^2\right )^{3/2}}{55 d^2 e}+\frac {2 b^2 (e x)^{9/2} \left (c+d x^2\right )^{3/2}}{15 d e^3}+\frac {1}{77} \left (2 c \left (11 a^2+\frac {b c (3 b c-10 a d)}{d^2}\right )\right ) \int \frac {(e x)^{3/2}}{\sqrt {c+d x^2}} \, dx \\ & = \frac {4 c \left (11 a^2+\frac {b c (3 b c-10 a d)}{d^2}\right ) e \sqrt {e x} \sqrt {c+d x^2}}{231 d}+\frac {2 \left (11 a^2+\frac {b c (3 b c-10 a d)}{d^2}\right ) (e x)^{5/2} \sqrt {c+d x^2}}{77 e}-\frac {2 b (3 b c-10 a d) (e x)^{5/2} \left (c+d x^2\right )^{3/2}}{55 d^2 e}+\frac {2 b^2 (e x)^{9/2} \left (c+d x^2\right )^{3/2}}{15 d e^3}-\frac {\left (2 c^2 \left (11 a^2+\frac {b c (3 b c-10 a d)}{d^2}\right ) e^2\right ) \int \frac {1}{\sqrt {e x} \sqrt {c+d x^2}} \, dx}{231 d} \\ & = \frac {4 c \left (11 a^2+\frac {b c (3 b c-10 a d)}{d^2}\right ) e \sqrt {e x} \sqrt {c+d x^2}}{231 d}+\frac {2 \left (11 a^2+\frac {b c (3 b c-10 a d)}{d^2}\right ) (e x)^{5/2} \sqrt {c+d x^2}}{77 e}-\frac {2 b (3 b c-10 a d) (e x)^{5/2} \left (c+d x^2\right )^{3/2}}{55 d^2 e}+\frac {2 b^2 (e x)^{9/2} \left (c+d x^2\right )^{3/2}}{15 d e^3}-\frac {\left (4 c^2 \left (11 a^2+\frac {b c (3 b c-10 a d)}{d^2}\right ) e\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{231 d} \\ & = \frac {4 c \left (11 a^2+\frac {b c (3 b c-10 a d)}{d^2}\right ) e \sqrt {e x} \sqrt {c+d x^2}}{231 d}+\frac {2 \left (11 a^2+\frac {b c (3 b c-10 a d)}{d^2}\right ) (e x)^{5/2} \sqrt {c+d x^2}}{77 e}-\frac {2 b (3 b c-10 a d) (e x)^{5/2} \left (c+d x^2\right )^{3/2}}{55 d^2 e}+\frac {2 b^2 (e x)^{9/2} \left (c+d x^2\right )^{3/2}}{15 d e^3}-\frac {2 c^{7/4} \left (11 a^2+\frac {b c (3 b c-10 a d)}{d^2}\right ) e^{3/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{231 d^{5/4} \sqrt {c+d x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 22.35 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.78 \[ \int (e x)^{3/2} \left (a+b x^2\right )^2 \sqrt {c+d x^2} \, dx=\frac {(e x)^{3/2} \left (\frac {2 \sqrt {x} \left (c+d x^2\right ) \left (55 a^2 d^2 \left (2 c+3 d x^2\right )+10 a b d \left (-10 c^2+6 c d x^2+21 d^2 x^4\right )+b^2 \left (30 c^3-18 c^2 d x^2+14 c d^2 x^4+77 d^3 x^6\right )\right )}{5 d^3}-\frac {4 i c^2 \left (3 b^2 c^2-10 a b c d+11 a^2 d^2\right ) \sqrt {1+\frac {c}{d x^2}} x \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}}}{\sqrt {x}}\right ),-1\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}} d^3}\right )}{231 x^{3/2} \sqrt {c+d x^2}} \]

[In]

Integrate[(e*x)^(3/2)*(a + b*x^2)^2*Sqrt[c + d*x^2],x]

[Out]

((e*x)^(3/2)*((2*Sqrt[x]*(c + d*x^2)*(55*a^2*d^2*(2*c + 3*d*x^2) + 10*a*b*d*(-10*c^2 + 6*c*d*x^2 + 21*d^2*x^4)
+ b^2*(30*c^3 - 18*c^2*d*x^2 + 14*c*d^2*x^4 + 77*d^3*x^6)))/(5*d^3) - ((4*I)*c^2*(3*b^2*c^2 - 10*a*b*c*d + 11
*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1])/(Sqrt[(I*Sqrt[c])
/Sqrt[d]]*d^3)))/(231*x^(3/2)*Sqrt[c + d*x^2])

Maple [A] (verified)

Time = 3.15 (sec) , antiderivative size = 283, normalized size of antiderivative = 0.98

method	result	size
risch	\(\frac {2 \left (77 b^{2} d^{3} x^{6}+210 a b \,d^{3} x^{4}+14 b^{2} c \,d^{2} x^{4}+165 a^{2} d^{3} x^{2}+60 a b c \,d^{2} x^{2}-18 b^{2} c^{2} d \,x^{2}+110 c \,a^{2} d^{2}-100 a b \,c^{2} d +30 b^{2} c^{3}\right ) x \sqrt {d \,x^{2}+c}\, e^{2}}{1155 d^{3} \sqrt {e x}}-\frac {2 c^{2} \left (11 a^{2} d^{2}-10 a b c d +3 b^{2} c^{2}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) e^{2} \sqrt {e x \left (d \,x^{2}+c \right )}}{231 d^{4} \sqrt {d e \,x^{3}+c e x}\, \sqrt {e x}\, \sqrt {d \,x^{2}+c}}\)	\(283\)
elliptic	\(\frac {\sqrt {e x \left (d \,x^{2}+c \right )}\, \sqrt {e x}\, \left (\frac {2 b^{2} e \,x^{6} \sqrt {d e \,x^{3}+c e x}}{15}+\frac {2 \left (b \left (2 a d +b c \right ) e^{2}-\frac {13 b^{2} e^{2} c}{15}\right ) x^{4} \sqrt {d e \,x^{3}+c e x}}{11 d e}+\frac {2 \left (a \left (a d +2 b c \right ) e^{2}-\frac {9 \left (b \left (2 a d +b c \right ) e^{2}-\frac {13 b^{2} e^{2} c}{15}\right ) c}{11 d}\right ) x^{2} \sqrt {d e \,x^{3}+c e x}}{7 d e}+\frac {2 \left (a^{2} c \,e^{2}-\frac {5 \left (a \left (a d +2 b c \right ) e^{2}-\frac {9 \left (b \left (2 a d +b c \right ) e^{2}-\frac {13 b^{2} e^{2} c}{15}\right ) c}{11 d}\right ) c}{7 d}\right ) \sqrt {d e \,x^{3}+c e x}}{3 d e}-\frac {\left (a^{2} c \,e^{2}-\frac {5 \left (a \left (a d +2 b c \right ) e^{2}-\frac {9 \left (b \left (2 a d +b c \right ) e^{2}-\frac {13 b^{2} e^{2} c}{15}\right ) c}{11 d}\right ) c}{7 d}\right ) c \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{3 d^{2} \sqrt {d e \,x^{3}+c e x}}\right )}{e x \sqrt {d \,x^{2}+c}}\)	\(419\)
default	\(-\frac {2 e \sqrt {e x}\, \left (-77 b^{2} d^{5} x^{9}-210 a b \,d^{5} x^{7}-91 b^{2} c \,d^{4} x^{7}+55 \sqrt {2}\, \sqrt {-c d}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c^{2} d^{2}-50 \sqrt {2}\, \sqrt {-c d}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{3} d +15 \sqrt {2}\, \sqrt {-c d}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{4}-165 a^{2} d^{5} x^{5}-270 a b c \,d^{4} x^{5}+4 b^{2} c^{2} d^{3} x^{5}-275 a^{2} c \,d^{4} x^{3}+40 a b \,c^{2} d^{3} x^{3}-12 b^{2} c^{3} d^{2} x^{3}-110 a^{2} c^{2} d^{3} x +100 a b \,c^{3} d^{2} x -30 b^{2} c^{4} d x \right )}{1155 x \sqrt {d \,x^{2}+c}\, d^{4}}\)	\(448\)

[In]

int((e*x)^(3/2)*(b*x^2+a)^2*(d*x^2+c)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/1155*(77*b^2*d^3*x^6+210*a*b*d^3*x^4+14*b^2*c*d^2*x^4+165*a^2*d^3*x^2+60*a*b*c*d^2*x^2-18*b^2*c^2*d*x^2+110*
a^2*c*d^2-100*a*b*c^2*d+30*b^2*c^3)*x*(d*x^2+c)^(1/2)/d^3*e^2/(e*x)^(1/2)-2/231*c^2*(11*a^2*d^2-10*a*b*c*d+3*b
^2*c^2)/d^4*(-c*d)^(1/2)*((x+(-c*d)^(1/2)/d)/(-c*d)^(1/2)*d)^(1/2)*(-2*(x-(-c*d)^(1/2)/d)/(-c*d)^(1/2)*d)^(1/2
)*(-x/(-c*d)^(1/2)*d)^(1/2)/(d*e*x^3+c*e*x)^(1/2)*EllipticF(((x+(-c*d)^(1/2)/d)/(-c*d)^(1/2)*d)^(1/2),1/2*2^(1
/2))*e^2*(e*x*(d*x^2+c))^(1/2)/(e*x)^(1/2)/(d*x^2+c)^(1/2)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.09 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.58 \[ \int (e x)^{3/2} \left (a+b x^2\right )^2 \sqrt {c+d x^2} \, dx=-\frac {2 \, {\left (10 \, {\left (3 \, b^{2} c^{4} - 10 \, a b c^{3} d + 11 \, a^{2} c^{2} d^{2}\right )} \sqrt {d e} e {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right ) - {\left (77 \, b^{2} d^{4} e x^{6} + 14 \, {\left (b^{2} c d^{3} + 15 \, a b d^{4}\right )} e x^{4} - 3 \, {\left (6 \, b^{2} c^{2} d^{2} - 20 \, a b c d^{3} - 55 \, a^{2} d^{4}\right )} e x^{2} + 10 \, {\left (3 \, b^{2} c^{3} d - 10 \, a b c^{2} d^{2} + 11 \, a^{2} c d^{3}\right )} e\right )} \sqrt {d x^{2} + c} \sqrt {e x}\right )}}{1155 \, d^{4}} \]

[In]

integrate((e*x)^(3/2)*(b*x^2+a)^2*(d*x^2+c)^(1/2),x, algorithm="fricas")

[Out]

-2/1155*(10*(3*b^2*c^4 - 10*a*b*c^3*d + 11*a^2*c^2*d^2)*sqrt(d*e)*e*weierstrassPInverse(-4*c/d, 0, x) - (77*b^
2*d^4*e*x^6 + 14*(b^2*c*d^3 + 15*a*b*d^4)*e*x^4 - 3*(6*b^2*c^2*d^2 - 20*a*b*c*d^3 - 55*a^2*d^4)*e*x^2 + 10*(3*
b^2*c^3*d - 10*a*b*c^2*d^2 + 11*a^2*c*d^3)*e)*sqrt(d*x^2 + c)*sqrt(e*x))/d^4

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 11.27 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.52 \[ \int (e x)^{3/2} \left (a+b x^2\right )^2 \sqrt {c+d x^2} \, dx=\frac {a^{2} \sqrt {c} e^{\frac {3}{2}} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \Gamma \left (\frac {9}{4}\right )} + \frac {a b \sqrt {c} e^{\frac {3}{2}} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{\Gamma \left (\frac {13}{4}\right )} + \frac {b^{2} \sqrt {c} e^{\frac {3}{2}} x^{\frac {13}{2}} \Gamma \left (\frac {13}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {13}{4} \\ \frac {17}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \Gamma \left (\frac {17}{4}\right )} \]

[In]

integrate((e*x)**(3/2)*(b*x**2+a)**2*(d*x**2+c)**(1/2),x)

[Out]

a**2*sqrt(c)*e**(3/2)*x**(5/2)*gamma(5/4)*hyper((-1/2, 5/4), (9/4,), d*x**2*exp_polar(I*pi)/c)/(2*gamma(9/4))
+ a*b*sqrt(c)*e**(3/2)*x**(9/2)*gamma(9/4)*hyper((-1/2, 9/4), (13/4,), d*x**2*exp_polar(I*pi)/c)/gamma(13/4) +
b**2*sqrt(c)*e**(3/2)*x**(13/2)*gamma(13/4)*hyper((-1/2, 13/4), (17/4,), d*x**2*exp_polar(I*pi)/c)/(2*gamma(1
7/4))

Maxima [F]

\[ \int (e x)^{3/2} \left (a+b x^2\right )^2 \sqrt {c+d x^2} \, dx=\int { {\left (b x^{2} + a\right )}^{2} \sqrt {d x^{2} + c} \left (e x\right )^{\frac {3}{2}} \,d x } \]

[In]

integrate((e*x)^(3/2)*(b*x^2+a)^2*(d*x^2+c)^(1/2),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)*(e*x)^(3/2), x)

Giac [F]

\[ \int (e x)^{3/2} \left (a+b x^2\right )^2 \sqrt {c+d x^2} \, dx=\int { {\left (b x^{2} + a\right )}^{2} \sqrt {d x^{2} + c} \left (e x\right )^{\frac {3}{2}} \,d x } \]

[In]

integrate((e*x)^(3/2)*(b*x^2+a)^2*(d*x^2+c)^(1/2),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)*(e*x)^(3/2), x)

Mupad [F(-1)]

Timed out. \[ \int (e x)^{3/2} \left (a+b x^2\right )^2 \sqrt {c+d x^2} \, dx=\int {\left (e\,x\right )}^{3/2}\,{\left (b\,x^2+a\right )}^2\,\sqrt {d\,x^2+c} \,d x \]

[In]

int((e*x)^(3/2)*(a + b*x^2)^2*(c + d*x^2)^(1/2),x)

[Out]

int((e*x)^(3/2)*(a + b*x^2)^2*(c + d*x^2)^(1/2), x)