\(\int \frac {(a+b x^2)^2}{\sqrt {e x} (c+d x^2)^{3/2}} \, dx\) [853]

Optimal result

Integrand size = 28, antiderivative size = 193 \[ \int \frac {\left (a+b x^2\right )^2}{\sqrt {e x} \left (c+d x^2\right )^{3/2}} \, dx=\frac {(b c-a d)^2 \sqrt {e x}}{c d^2 e \sqrt {c+d x^2}}+\frac {2 b^2 \sqrt {e x} \sqrt {c+d x^2}}{3 d^2 e}-\frac {\left (5 b^2 c^2-6 a b c d-3 a^2 d^2\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{6 c^{5/4} d^{9/4} \sqrt {e} \sqrt {c+d x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {474, 470, 335, 226} \[ \int \frac {\left (a+b x^2\right )^2}{\sqrt {e x} \left (c+d x^2\right )^{3/2}} \, dx=-\frac {\left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (-3 a^2 d^2-6 a b c d+5 b^2 c^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{6 c^{5/4} d^{9/4} \sqrt {e} \sqrt {c+d x^2}}+\frac {\sqrt {e x} (b c-a d)^2}{c d^2 e \sqrt {c+d x^2}}+\frac {2 b^2 \sqrt {e x} \sqrt {c+d x^2}}{3 d^2 e} \]

[In]

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

[Out]

((b*c - a*d)^2*Sqrt[e*x])/(c*d^2*e*Sqrt[c + d*x^2]) + (2*b^2*Sqrt[e*x]*Sqrt[c + d*x^2])/(3*d^2*e) - ((5*b^2*c^
2 - 6*a*b*c*d - 3*a^2*d^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])/(6*c^(5/4)*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 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 474

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

Rubi steps \begin{align*} \text {integral}& = \frac {(b c-a d)^2 \sqrt {e x}}{c d^2 e \sqrt {c+d x^2}}-\frac {\int \frac {\frac {1}{2} \left (-2 a^2 d^2+(b c-a d)^2\right )-b^2 c d x^2}{\sqrt {e x} \sqrt {c+d x^2}} \, dx}{c d^2} \\ & = \frac {(b c-a d)^2 \sqrt {e x}}{c d^2 e \sqrt {c+d x^2}}+\frac {2 b^2 \sqrt {e x} \sqrt {c+d x^2}}{3 d^2 e}-\frac {\left (5 b^2 c^2-6 a b c d-3 a^2 d^2\right ) \int \frac {1}{\sqrt {e x} \sqrt {c+d x^2}} \, dx}{6 c d^2} \\ & = \frac {(b c-a d)^2 \sqrt {e x}}{c d^2 e \sqrt {c+d x^2}}+\frac {2 b^2 \sqrt {e x} \sqrt {c+d x^2}}{3 d^2 e}-\frac {\left (5 b^2 c^2-6 a b c d-3 a^2 d^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{3 c d^2 e} \\ & = \frac {(b c-a d)^2 \sqrt {e x}}{c d^2 e \sqrt {c+d x^2}}+\frac {2 b^2 \sqrt {e x} \sqrt {c+d x^2}}{3 d^2 e}-\frac {\left (5 b^2 c^2-6 a b c d-3 a^2 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 )}{6 c^{5/4} d^{9/4} \sqrt {e} \sqrt {c+d x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 11.12 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b x^2\right )^2}{\sqrt {e x} \left (c+d x^2\right )^{3/2}} \, dx=\frac {\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}} x \left (-6 a b c d+3 a^2 d^2+b^2 c \left (5 c+2 d x^2\right )\right )+i \left (-5 b^2 c^2+6 a b c d+3 a^2 d^2\right ) \sqrt {1+\frac {c}{d x^2}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}}}{\sqrt {x}}\right ),-1\right )}{3 c \sqrt {\frac {i \sqrt {c}}{\sqrt {d}}} d^2 \sqrt {e x} \sqrt {c+d x^2}} \]

[In]

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

[Out]

(Sqrt[(I*Sqrt[c])/Sqrt[d]]*x*(-6*a*b*c*d + 3*a^2*d^2 + b^2*c*(5*c + 2*d*x^2)) + I*(-5*b^2*c^2 + 6*a*b*c*d + 3*
a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1])/(3*c*Sqrt[(I
*Sqrt[c])/Sqrt[d]]*d^2*Sqrt[e*x]*Sqrt[c + d*x^2])

Maple [A] (verified)

Time = 3.71 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.34

method	result	size
elliptic	\(\frac {\sqrt {e x \left (d \,x^{2}+c \right )}\, \left (\frac {x \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{d^{2} c \sqrt {\left (x^{2}+\frac {c}{d}\right ) d e x}}+\frac {2 b^{2} \sqrt {d e \,x^{3}+c e x}}{3 d^{2} e}+\frac {\left (\frac {b \left (2 a d -b c \right )}{d^{2}}+\frac {a^{2} d^{2}-2 a b c d +b^{2} c^{2}}{2 d^{2} c}-\frac {b^{2} c}{3 d^{2}}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d \sqrt {d e \,x^{3}+c e x}}\right )}{\sqrt {e x}\, \sqrt {d \,x^{2}+c}}\)	\(259\)
default	\(\frac {3 \sqrt {2}\, \sqrt {-c d}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} d^{2}+6 \sqrt {2}\, \sqrt {-c d}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b c d -5 \sqrt {2}\, \sqrt {-c d}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{2}+4 b^{2} c \,d^{2} x^{3}+6 x \,a^{2} d^{3}-12 x a b c \,d^{2}+10 x \,b^{2} c^{2} d}{6 \sqrt {d \,x^{2}+c}\, c \sqrt {e x}\, d^{3}}\)	\(341\)
risch	\(\frac {2 b^{2} x \sqrt {d \,x^{2}+c}}{3 d^{2} \sqrt {e x}}+\frac {\left (-\frac {4 b^{2} c \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d \sqrt {d e \,x^{3}+c e x}}+\frac {6 a b \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{\sqrt {d e \,x^{3}+c e x}}+\left (3 a^{2} d^{2}-6 a b c d +3 b^{2} c^{2}\right ) \left (\frac {x}{c \sqrt {\left (x^{2}+\frac {c}{d}\right ) d e x}}+\frac {\sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{2 c d \sqrt {d e \,x^{3}+c e x}}\right )\right ) \sqrt {e x \left (d \,x^{2}+c \right )}}{3 d^{2} \sqrt {e x}\, \sqrt {d \,x^{2}+c}}\)	\(438\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

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

Time = 0.10 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.77 \[ \int \frac {\left (a+b x^2\right )^2}{\sqrt {e x} \left (c+d x^2\right )^{3/2}} \, dx=-\frac {{\left (5 \, b^{2} c^{3} - 6 \, a b c^{2} d - 3 \, a^{2} c d^{2} + {\left (5 \, b^{2} c^{2} d - 6 \, a b c d^{2} - 3 \, a^{2} d^{3}\right )} x^{2}\right )} \sqrt {d e} {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right ) - {\left (2 \, b^{2} c d^{2} x^{2} + 5 \, b^{2} c^{2} d - 6 \, a b c d^{2} + 3 \, a^{2} d^{3}\right )} \sqrt {d x^{2} + c} \sqrt {e x}}{3 \, {\left (c d^{4} e x^{2} + c^{2} d^{3} e\right )}} \]

[In]

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

[Out]

-1/3*((5*b^2*c^3 - 6*a*b*c^2*d - 3*a^2*c*d^2 + (5*b^2*c^2*d - 6*a*b*c*d^2 - 3*a^2*d^3)*x^2)*sqrt(d*e)*weierstr
assPInverse(-4*c/d, 0, x) - (2*b^2*c*d^2*x^2 + 5*b^2*c^2*d - 6*a*b*c*d^2 + 3*a^2*d^3)*sqrt(d*x^2 + c)*sqrt(e*x
))/(c*d^4*e*x^2 + c^2*d^3*e)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[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]

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

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

[In]

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

[Out]

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