\(\int \frac {(a+b x^2)^2 (c+d x^2)^{3/2}}{\sqrt {e x}} \, dx\) [835]

Optimal result

Integrand size = 28, antiderivative size = 286 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{\sqrt {e x}} \, dx=\frac {4 c \left (33 a^2 d^2+b c (b c-6 a d)\right ) \sqrt {e x} \sqrt {c+d x^2}}{231 d^2 e}+\frac {2 \left (33 a^2 d^2+b c (b c-6 a d)\right ) \sqrt {e x} \left (c+d x^2\right )^{3/2}}{231 d^2 e}-\frac {2 b (b c-6 a d) \sqrt {e x} \left (c+d x^2\right )^{5/2}}{33 d^2 e}+\frac {2 b^2 (e x)^{5/2} \left (c+d x^2\right )^{5/2}}{15 d e^3}+\frac {4 c^{7/4} \left (33 a^2 d^2+b c (b c-6 a d)\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{231 d^{9/4} \sqrt {e} \sqrt {c+d x^2}} \]

[Out]

2/15*b^2*(e*x)^(5/2)*(d*x^2+c)^(5/2)/d/e^3+2/231*(33*a^2*d^2+b*c*(-6*a*d+b*c))*(d*x^2+c)^(3/2)*(e*x)^(1/2)/d^2
/e-2/33*b*(-6*a*d+b*c)*(d*x^2+c)^(5/2)*(e*x)^(1/2)/d^2/e+4/231*c*(33*a^2*d^2+b*c*(-6*a*d+b*c))*(e*x)^(1/2)*(d*
x^2+c)^(1/2)/d^2/e+4/231*c^(7/4)*(33*a^2*d^2+b*c*(-6*a*d+b*c))*(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^(9/4)/e^(1/2)/(d*x^
2+c)^(1/2)

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {475, 470, 285, 335, 226} \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{\sqrt {e x}} \, dx=\frac {4 c^{7/4} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (33 a^2 d^2+b c (b c-6 a d)\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{231 d^{9/4} \sqrt {e} \sqrt {c+d x^2}}+\frac {2 \sqrt {e x} \left (c+d x^2\right )^{3/2} \left (33 a^2 d^2+b c (b c-6 a d)\right )}{231 d^2 e}+\frac {4 c \sqrt {e x} \sqrt {c+d x^2} \left (33 a^2 d^2+b c (b c-6 a d)\right )}{231 d^2 e}-\frac {2 b \sqrt {e x} \left (c+d x^2\right )^{5/2} (b c-6 a d)}{33 d^2 e}+\frac {2 b^2 (e x)^{5/2} \left (c+d x^2\right )^{5/2}}{15 d e^3} \]

[In]

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

[Out]

(4*c*(33*a^2*d^2 + b*c*(b*c - 6*a*d))*Sqrt[e*x]*Sqrt[c + d*x^2])/(231*d^2*e) + (2*(33*a^2*d^2 + b*c*(b*c - 6*a
*d))*Sqrt[e*x]*(c + d*x^2)^(3/2))/(231*d^2*e) - (2*b*(b*c - 6*a*d)*Sqrt[e*x]*(c + d*x^2)^(5/2))/(33*d^2*e) + (
2*b^2*(e*x)^(5/2)*(c + d*x^2)^(5/2))/(15*d*e^3) + (4*c^(7/4)*(33*a^2*d^2 + b*c*(b*c - 6*a*d))*(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^(9/4)*Sqrt[e]*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 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)^{5/2} \left (c+d x^2\right )^{5/2}}{15 d e^3}+\frac {2 \int \frac {\left (c+d x^2\right )^{3/2} \left (\frac {15 a^2 d}{2}-\frac {5}{2} b (b c-6 a d) x^2\right )}{\sqrt {e x}} \, dx}{15 d} \\ & = -\frac {2 b (b c-6 a d) \sqrt {e x} \left (c+d x^2\right )^{5/2}}{33 d^2 e}+\frac {2 b^2 (e x)^{5/2} \left (c+d x^2\right )^{5/2}}{15 d e^3}+\frac {1}{33} \left (33 a^2+\frac {b c (b c-6 a d)}{d^2}\right ) \int \frac {\left (c+d x^2\right )^{3/2}}{\sqrt {e x}} \, dx \\ & = \frac {2 \left (33 a^2+\frac {b c (b c-6 a d)}{d^2}\right ) \sqrt {e x} \left (c+d x^2\right )^{3/2}}{231 e}-\frac {2 b (b c-6 a d) \sqrt {e x} \left (c+d x^2\right )^{5/2}}{33 d^2 e}+\frac {2 b^2 (e x)^{5/2} \left (c+d x^2\right )^{5/2}}{15 d e^3}+\frac {1}{77} \left (2 c \left (33 a^2+\frac {b c (b c-6 a d)}{d^2}\right )\right ) \int \frac {\sqrt {c+d x^2}}{\sqrt {e x}} \, dx \\ & = \frac {4 c \left (33 a^2+\frac {b c (b c-6 a d)}{d^2}\right ) \sqrt {e x} \sqrt {c+d x^2}}{231 e}+\frac {2 \left (33 a^2+\frac {b c (b c-6 a d)}{d^2}\right ) \sqrt {e x} \left (c+d x^2\right )^{3/2}}{231 e}-\frac {2 b (b c-6 a d) \sqrt {e x} \left (c+d x^2\right )^{5/2}}{33 d^2 e}+\frac {2 b^2 (e x)^{5/2} \left (c+d x^2\right )^{5/2}}{15 d e^3}+\frac {1}{231} \left (4 c^2 \left (33 a^2+\frac {b c (b c-6 a d)}{d^2}\right )\right ) \int \frac {1}{\sqrt {e x} \sqrt {c+d x^2}} \, dx \\ & = \frac {4 c \left (33 a^2+\frac {b c (b c-6 a d)}{d^2}\right ) \sqrt {e x} \sqrt {c+d x^2}}{231 e}+\frac {2 \left (33 a^2+\frac {b c (b c-6 a d)}{d^2}\right ) \sqrt {e x} \left (c+d x^2\right )^{3/2}}{231 e}-\frac {2 b (b c-6 a d) \sqrt {e x} \left (c+d x^2\right )^{5/2}}{33 d^2 e}+\frac {2 b^2 (e x)^{5/2} \left (c+d x^2\right )^{5/2}}{15 d e^3}+\frac {\left (8 c^2 \left (33 a^2+\frac {b c (b c-6 a d)}{d^2}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{231 e} \\ & = \frac {4 c \left (33 a^2+\frac {b c (b c-6 a d)}{d^2}\right ) \sqrt {e x} \sqrt {c+d x^2}}{231 e}+\frac {2 \left (33 a^2+\frac {b c (b c-6 a d)}{d^2}\right ) \sqrt {e x} \left (c+d x^2\right )^{3/2}}{231 e}-\frac {2 b (b c-6 a d) \sqrt {e x} \left (c+d x^2\right )^{5/2}}{33 d^2 e}+\frac {2 b^2 (e x)^{5/2} \left (c+d x^2\right )^{5/2}}{15 d e^3}+\frac {4 c^{7/4} \left (33 a^2+\frac {b c (b c-6 a d)}{d^2}\right ) \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 \sqrt [4]{d} \sqrt {e} \sqrt {c+d x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 11.23 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{\sqrt {e x}} \, dx=\frac {\sqrt {x} \left (\frac {2 \sqrt {x} \left (c+d x^2\right ) \left (165 a^2 d^2 \left (3 c+d x^2\right )+30 a b d \left (4 c^2+13 c d x^2+7 d^2 x^4\right )+b^2 \left (-20 c^3+12 c^2 d x^2+119 c d^2 x^4+77 d^3 x^6\right )\right )}{5 d^2}+\frac {8 i c^2 \left (b^2 c^2-6 a b c d+33 a^2 d^2\right ) \sqrt {1+\frac {c}{d x^2}} x \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}}}{\sqrt {x}}\right ),-1\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}} d^2}\right )}{231 \sqrt {e x} \sqrt {c+d x^2}} \]

[In]

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

[Out]

(Sqrt[x]*((2*Sqrt[x]*(c + d*x^2)*(165*a^2*d^2*(3*c + d*x^2) + 30*a*b*d*(4*c^2 + 13*c*d*x^2 + 7*d^2*x^4) + b^2*
(-20*c^3 + 12*c^2*d*x^2 + 119*c*d^2*x^4 + 77*d^3*x^6)))/(5*d^2) + ((8*I)*c^2*(b^2*c^2 - 6*a*b*c*d + 33*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^2)))/(231*Sqrt[e*x]*Sqrt[c + d*x^2])

Maple [A] (verified)

Time = 3.04 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.97

method	result	size
risch	\(\frac {2 \left (77 b^{2} d^{3} x^{6}+210 a b \,d^{3} x^{4}+119 b^{2} c \,d^{2} x^{4}+165 a^{2} d^{3} x^{2}+390 a b c \,d^{2} x^{2}+12 b^{2} c^{2} d \,x^{2}+495 c \,a^{2} d^{2}+120 a b \,c^{2} d -20 b^{2} c^{3}\right ) x \sqrt {d \,x^{2}+c}}{1155 d^{2} \sqrt {e x}}+\frac {4 c^{2} \left (33 a^{2} d^{2}-6 a b c d +b^{2} c^{2}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {e x \left (d \,x^{2}+c \right )}}{231 d^{3} \sqrt {d e \,x^{3}+c e x}\, \sqrt {e x}\, \sqrt {d \,x^{2}+c}}\)	\(276\)
elliptic	\(\frac {\sqrt {e x \left (d \,x^{2}+c \right )}\, \left (\frac {2 b^{2} d \,x^{6} \sqrt {d e \,x^{3}+c e x}}{15 e}+\frac {2 \left (2 a b \,d^{2}+\frac {17}{15} b^{2} c d \right ) x^{4} \sqrt {d e \,x^{3}+c e x}}{11 d e}+\frac {2 \left (a^{2} d^{2}+4 a b c d +b^{2} c^{2}-\frac {9 c \left (2 a b \,d^{2}+\frac {17}{15} b^{2} c d \right )}{11 d}\right ) x^{2} \sqrt {d e \,x^{3}+c e x}}{7 d e}+\frac {2 \left (2 a^{2} c d +2 b \,c^{2} a -\frac {5 c \left (a^{2} d^{2}+4 a b c d +b^{2} c^{2}-\frac {9 c \left (2 a b \,d^{2}+\frac {17}{15} b^{2} c d \right )}{11 d}\right )}{7 d}\right ) \sqrt {d e \,x^{3}+c e x}}{3 d e}+\frac {\left (a^{2} c^{2}-\frac {c \left (2 a^{2} c d +2 b \,c^{2} a -\frac {5 c \left (a^{2} d^{2}+4 a b c d +b^{2} c^{2}-\frac {9 c \left (2 a b \,d^{2}+\frac {17}{15} b^{2} c d \right )}{11 d}\right )}{7 d}\right )}{3 d}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d \sqrt {d e \,x^{3}+c e x}}\right )}{\sqrt {e x}\, \sqrt {d \,x^{2}+c}}\)	\(429\)
default	\(\frac {\frac {2 b^{2} d^{5} x^{9}}{15}+\frac {4 a b \,d^{5} x^{7}}{11}+\frac {56 b^{2} c \,d^{4} x^{7}}{165}+\frac {4 \sqrt {2}\, \sqrt {-c d}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c^{2} d^{2}}{7}-\frac {8 \sqrt {2}\, \sqrt {-c d}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{3} d}{77}+\frac {4 \sqrt {2}\, \sqrt {-c d}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{4}}{231}+\frac {2 a^{2} d^{5} x^{5}}{7}+\frac {80 a b c \,d^{4} x^{5}}{77}+\frac {262 b^{2} c^{2} d^{3} x^{5}}{1155}+\frac {8 a^{2} c \,d^{4} x^{3}}{7}+\frac {68 a b \,c^{2} d^{3} x^{3}}{77}-\frac {16 b^{2} c^{3} d^{2} x^{3}}{1155}+\frac {6 a^{2} c^{2} d^{3} x}{7}+\frac {16 a b \,c^{3} d^{2} x}{77}-\frac {8 b^{2} c^{4} d x}{231}}{\sqrt {d \,x^{2}+c}\, d^{3} \sqrt {e x}}\)	\(444\)

[In]

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

[Out]

2/1155/d^2*(77*b^2*d^3*x^6+210*a*b*d^3*x^4+119*b^2*c*d^2*x^4+165*a^2*d^3*x^2+390*a*b*c*d^2*x^2+12*b^2*c^2*d*x^
2+495*a^2*c*d^2+120*a*b*c^2*d-20*b^2*c^3)*x*(d*x^2+c)^(1/2)/(e*x)^(1/2)+4/231*c^2/d^3*(33*a^2*d^2-6*a*b*c*d+b^
2*c^2)*(-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*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 = 162, normalized size of antiderivative = 0.57 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{\sqrt {e x}} \, dx=\frac {2 \, {\left (20 \, {\left (b^{2} c^{4} - 6 \, a b c^{3} d + 33 \, a^{2} c^{2} d^{2}\right )} \sqrt {d e} {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right ) + {\left (77 \, b^{2} d^{4} x^{6} - 20 \, b^{2} c^{3} d + 120 \, a b c^{2} d^{2} + 495 \, a^{2} c d^{3} + 7 \, {\left (17 \, b^{2} c d^{3} + 30 \, a b d^{4}\right )} x^{4} + 3 \, {\left (4 \, b^{2} c^{2} d^{2} + 130 \, a b c d^{3} + 55 \, a^{2} d^{4}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {e x}\right )}}{1155 \, d^{3} e} \]

[In]

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

[Out]

2/1155*(20*(b^2*c^4 - 6*a*b*c^3*d + 33*a^2*c^2*d^2)*sqrt(d*e)*weierstrassPInverse(-4*c/d, 0, x) + (77*b^2*d^4*
x^6 - 20*b^2*c^3*d + 120*a*b*c^2*d^2 + 495*a^2*c*d^3 + 7*(17*b^2*c*d^3 + 30*a*b*d^4)*x^4 + 3*(4*b^2*c^2*d^2 +
130*a*b*c*d^3 + 55*a^2*d^4)*x^2)*sqrt(d*x^2 + c)*sqrt(e*x))/(d^3*e)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 9.63 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.07 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{\sqrt {e x}} \, dx=\frac {a^{2} c^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt {e} \Gamma \left (\frac {5}{4}\right )} + \frac {a^{2} \sqrt {c} d x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt {e} \Gamma \left (\frac {9}{4}\right )} + \frac {a b c^{\frac {3}{2}} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{\sqrt {e} \Gamma \left (\frac {9}{4}\right )} + \frac {a b \sqrt {c} d x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{\sqrt {e} \Gamma \left (\frac {13}{4}\right )} + \frac {b^{2} c^{\frac {3}{2}} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt {e} \Gamma \left (\frac {13}{4}\right )} + \frac {b^{2} \sqrt {c} d x^{\frac {13}{2}} \Gamma \left (\frac {13}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {13}{4} \\ \frac {17}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt {e} \Gamma \left (\frac {17}{4}\right )} \]

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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