\(\int \frac {\sqrt {e x} (a+b x^2)^2}{\sqrt {c+d x^2}} \, dx\) [841]

Optimal result

Integrand size = 28, antiderivative size = 375 \[ \int \frac {\sqrt {e x} \left (a+b x^2\right )^2}{\sqrt {c+d x^2}} \, dx=-\frac {2 b (7 b c-18 a d) (e x)^{3/2} \sqrt {c+d x^2}}{45 d^2 e}+\frac {2 b^2 (e x)^{7/2} \sqrt {c+d x^2}}{9 d e^3}+\frac {2 \left (15 a^2 d^2+b c (7 b c-18 a d)\right ) \sqrt {e x} \sqrt {c+d x^2}}{15 d^{5/2} \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {2 \sqrt [4]{c} \left (15 a^2 d^2+b c (7 b c-18 a d)\right ) \sqrt {e} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{15 d^{11/4} \sqrt {c+d x^2}}+\frac {\sqrt [4]{c} \left (15 a^2 d^2+b c (7 b c-18 a d)\right ) \sqrt {e} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{15 d^{11/4} \sqrt {c+d x^2}} \]

[Out]

-2/45*b*(-18*a*d+7*b*c)*(e*x)^(3/2)*(d*x^2+c)^(1/2)/d^2/e+2/9*b^2*(e*x)^(7/2)*(d*x^2+c)^(1/2)/d/e^3+2/15*(15*a
^2*d^2+b*c*(-18*a*d+7*b*c))*(e*x)^(1/2)*(d*x^2+c)^(1/2)/d^(5/2)/(c^(1/2)+x*d^(1/2))-2/15*c^(1/4)*(15*a^2*d^2+b
*c*(-18*a*d+7*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)))*EllipticE(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))*e^(1/2)*((d*x^2+c)/(c^(1/2)+x*d^(1/2))^2)^(1/2)/d^(11/4)/(d*x^2+c)^(1/2)+1/15*c^(1/4)*(15*a^2*d^2+b*c*(-
18*a*d+7*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))
*e^(1/2)*((d*x^2+c)/(c^(1/2)+x*d^(1/2))^2)^(1/2)/d^(11/4)/(d*x^2+c)^(1/2)

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 375, 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, 335, 311, 226, 1210} \[ \int \frac {\sqrt {e x} \left (a+b x^2\right )^2}{\sqrt {c+d x^2}} \, dx=\frac {\sqrt [4]{c} \sqrt {e} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (15 a^2 d^2+b c (7 b c-18 a d)\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{15 d^{11/4} \sqrt {c+d x^2}}-\frac {2 \sqrt [4]{c} \sqrt {e} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (15 a^2 d^2+b c (7 b c-18 a d)\right ) E\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{15 d^{11/4} \sqrt {c+d x^2}}+\frac {2 \sqrt {e x} \sqrt {c+d x^2} \left (15 a^2 d^2+b c (7 b c-18 a d)\right )}{15 d^{5/2} \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {2 b (e x)^{3/2} \sqrt {c+d x^2} (7 b c-18 a d)}{45 d^2 e}+\frac {2 b^2 (e x)^{7/2} \sqrt {c+d x^2}}{9 d e^3} \]

[In]

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

[Out]

(-2*b*(7*b*c - 18*a*d)*(e*x)^(3/2)*Sqrt[c + d*x^2])/(45*d^2*e) + (2*b^2*(e*x)^(7/2)*Sqrt[c + d*x^2])/(9*d*e^3)
+ (2*(15*a^2*d^2 + b*c*(7*b*c - 18*a*d))*Sqrt[e*x]*Sqrt[c + d*x^2])/(15*d^(5/2)*(Sqrt[c] + Sqrt[d]*x)) - (2*c
^(1/4)*(15*a^2*d^2 + b*c*(7*b*c - 18*a*d))*Sqrt[e]*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x
)^2]*EllipticE[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(15*d^(11/4)*Sqrt[c + d*x^2]) + (c^(1/4)
*(15*a^2*d^2 + b*c*(7*b*c - 18*a*d))*Sqrt[e]*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*E
llipticF[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(15*d^(11/4)*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 311

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, Dist[1/q, Int[1/Sqrt[a + b*x^4], x],
x] - Dist[1/q, Int[(1 - q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

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]

Rule 1210

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a +
c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4
]))*EllipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e}, x] && PosQ[c/a]

Rubi steps \begin{align*} \text {integral}& = \frac {2 b^2 (e x)^{7/2} \sqrt {c+d x^2}}{9 d e^3}+\frac {2 \int \frac {\sqrt {e x} \left (\frac {9 a^2 d}{2}-\frac {1}{2} b (7 b c-18 a d) x^2\right )}{\sqrt {c+d x^2}} \, dx}{9 d} \\ & = -\frac {2 b (7 b c-18 a d) (e x)^{3/2} \sqrt {c+d x^2}}{45 d^2 e}+\frac {2 b^2 (e x)^{7/2} \sqrt {c+d x^2}}{9 d e^3}+\frac {1}{15} \left (15 a^2+\frac {b c (7 b c-18 a d)}{d^2}\right ) \int \frac {\sqrt {e x}}{\sqrt {c+d x^2}} \, dx \\ & = -\frac {2 b (7 b c-18 a d) (e x)^{3/2} \sqrt {c+d x^2}}{45 d^2 e}+\frac {2 b^2 (e x)^{7/2} \sqrt {c+d x^2}}{9 d e^3}+\frac {\left (2 \left (15 a^2+\frac {b c (7 b c-18 a d)}{d^2}\right )\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{15 e} \\ & = -\frac {2 b (7 b c-18 a d) (e x)^{3/2} \sqrt {c+d x^2}}{45 d^2 e}+\frac {2 b^2 (e x)^{7/2} \sqrt {c+d x^2}}{9 d e^3}+\frac {\left (2 \sqrt {c} \left (15 a^2+\frac {b c (7 b c-18 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 )}{15 \sqrt {d}}-\frac {\left (2 \sqrt {c} \left (15 a^2+\frac {b c (7 b c-18 a d)}{d^2}\right )\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt {d} x^2}{\sqrt {c} e}}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{15 \sqrt {d}} \\ & = -\frac {2 b (7 b c-18 a d) (e x)^{3/2} \sqrt {c+d x^2}}{45 d^2 e}+\frac {2 b^2 (e x)^{7/2} \sqrt {c+d x^2}}{9 d e^3}+\frac {2 \left (15 a^2+\frac {b c (7 b c-18 a d)}{d^2}\right ) \sqrt {e x} \sqrt {c+d x^2}}{15 \sqrt {d} \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {2 \sqrt [4]{c} \left (15 a^2+\frac {b c (7 b c-18 a d)}{d^2}\right ) \sqrt {e} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{15 d^{3/4} \sqrt {c+d x^2}}+\frac {\sqrt [4]{c} \left (15 a^2+\frac {b c (7 b c-18 a d)}{d^2}\right ) \sqrt {e} \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 )}{15 d^{3/4} \sqrt {c+d x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 11.10 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.30 \[ \int \frac {\sqrt {e x} \left (a+b x^2\right )^2}{\sqrt {c+d x^2}} \, dx=\frac {2 \sqrt {e x} \left (b x \left (c+d x^2\right ) \left (-7 b c+18 a d+5 b d x^2\right )+3 \left (7 b^2 c^2-18 a b c d+15 a^2 d^2\right ) \sqrt {1+\frac {c}{d x^2}} x \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},-\frac {c}{d x^2}\right )\right )}{45 d^2 \sqrt {c+d x^2}} \]

[In]

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

[Out]

(2*Sqrt[e*x]*(b*x*(c + d*x^2)*(-7*b*c + 18*a*d + 5*b*d*x^2) + 3*(7*b^2*c^2 - 18*a*b*c*d + 15*a^2*d^2)*Sqrt[1 +
c/(d*x^2)]*x*Hypergeometric2F1[-1/4, 1/2, 3/4, -(c/(d*x^2))]))/(45*d^2*Sqrt[c + d*x^2])

Maple [A] (verified)

Time = 3.12 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.67

method	result	size
risch	\(\frac {2 b \,x^{2} \left (5 b d \,x^{2}+18 a d -7 b c \right ) \sqrt {d \,x^{2}+c}\, e}{45 d^{2} \sqrt {e x}}+\frac {\left (15 a^{2} d^{2}-18 a b c d +7 b^{2} c^{2}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right ) e \sqrt {e x \left (d \,x^{2}+c \right )}}{15 d^{3} \sqrt {d e \,x^{3}+c e x}\, \sqrt {e x}\, \sqrt {d \,x^{2}+c}}\)	\(252\)
elliptic	\(\frac {\sqrt {e x \left (d \,x^{2}+c \right )}\, \sqrt {e x}\, \left (\frac {2 b^{2} x^{3} \sqrt {d e \,x^{3}+c e x}}{9 d}+\frac {2 \left (2 a e b -\frac {7 b^{2} c e}{9 d}\right ) x \sqrt {d e \,x^{3}+c e x}}{5 d e}+\frac {\left (a^{2} e -\frac {3 \left (2 a e b -\frac {7 b^{2} c e}{9 d}\right ) c}{5 d}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right )}{d \sqrt {d e \,x^{3}+c e x}}\right )}{e x \sqrt {d \,x^{2}+c}}\)	\(284\)
default	\(\frac {\sqrt {e x}\, \left (10 b^{2} d^{3} x^{6}+90 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, E\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c \,d^{2}-108 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, E\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{2} d +42 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, E\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{3}-45 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c \,d^{2}+54 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{2} d -21 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{3}+36 a b \,d^{3} x^{4}-4 b^{2} c \,d^{2} x^{4}+36 a b c \,d^{2} x^{2}-14 b^{2} c^{2} d \,x^{2}\right )}{45 \sqrt {d \,x^{2}+c}\, d^{3} x}\)	\(604\)

[In]

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

[Out]

2/45*b*x^2*(5*b*d*x^2+18*a*d-7*b*c)*(d*x^2+c)^(1/2)/d^2*e/(e*x)^(1/2)+1/15*(15*a^2*d^2-18*a*b*c*d+7*b^2*c^2)/d
^3*(-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)*(-2*(-c*d)^(1/2)/d*EllipticE(((x+(-c*d)^(1/2)/d)/(-c*d)^(1/2)*d)^(1/2
),1/2*2^(1/2))+(-c*d)^(1/2)/d*EllipticF(((x+(-c*d)^(1/2)/d)/(-c*d)^(1/2)*d)^(1/2),1/2*2^(1/2)))*e*(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 = 99, normalized size of antiderivative = 0.26 \[ \int \frac {\sqrt {e x} \left (a+b x^2\right )^2}{\sqrt {c+d x^2}} \, dx=-\frac {2 \, {\left (3 \, {\left (7 \, b^{2} c^{2} - 18 \, a b c d + 15 \, a^{2} d^{2}\right )} \sqrt {d e} {\rm weierstrassZeta}\left (-\frac {4 \, c}{d}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right )\right ) - {\left (5 \, b^{2} d^{2} x^{3} - {\left (7 \, b^{2} c d - 18 \, a b d^{2}\right )} x\right )} \sqrt {d x^{2} + c} \sqrt {e x}\right )}}{45 \, d^{3}} \]

[In]

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

[Out]

-2/45*(3*(7*b^2*c^2 - 18*a*b*c*d + 15*a^2*d^2)*sqrt(d*e)*weierstrassZeta(-4*c/d, 0, weierstrassPInverse(-4*c/d
, 0, x)) - (5*b^2*d^2*x^3 - (7*b^2*c*d - 18*a*b*d^2)*x)*sqrt(d*x^2 + c)*sqrt(e*x))/d^3

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 4.53 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.38 \[ \int \frac {\sqrt {e x} \left (a+b x^2\right )^2}{\sqrt {c+d x^2}} \, dx=\frac {a^{2} \sqrt {e} x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt {c} \Gamma \left (\frac {7}{4}\right )} + \frac {a b \sqrt {e} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{\sqrt {c} \Gamma \left (\frac {11}{4}\right )} + \frac {b^{2} \sqrt {e} x^{\frac {11}{2}} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt {c} \Gamma \left (\frac {15}{4}\right )} \]

[In]

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

[Out]

a**2*sqrt(e)*x**(3/2)*gamma(3/4)*hyper((1/2, 3/4), (7/4,), d*x**2*exp_polar(I*pi)/c)/(2*sqrt(c)*gamma(7/4)) +
a*b*sqrt(e)*x**(7/2)*gamma(7/4)*hyper((1/2, 7/4), (11/4,), d*x**2*exp_polar(I*pi)/c)/(sqrt(c)*gamma(11/4)) + b
**2*sqrt(e)*x**(11/2)*gamma(11/4)*hyper((1/2, 11/4), (15/4,), d*x**2*exp_polar(I*pi)/c)/(2*sqrt(c)*gamma(15/4)
)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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