\(\int \frac {(a+b x^2)^2}{\sqrt {e x} \sqrt {c+d x^2}} \, dx\) [842]

Optimal result

Integrand size = 28, antiderivative size = 193 \[ \int \frac {\left (a+b x^2\right )^2}{\sqrt {e x} \sqrt {c+d x^2}} \, dx=-\frac {2 b (5 b c-14 a d) \sqrt {e x} \sqrt {c+d x^2}}{21 d^2 e}+\frac {2 b^2 (e x)^{5/2} \sqrt {c+d x^2}}{7 d e^3}+\frac {\left (5 b^2 c^2-14 a b c d+21 a^2 d^2\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{21 \sqrt [4]{c} d^{9/4} \sqrt {e} \sqrt {c+d x^2}} \]

[Out]

2/7*b^2*(e*x)^(5/2)*(d*x^2+c)^(1/2)/d/e^3-2/21*b*(-14*a*d+5*b*c)*(e*x)^(1/2)*(d*x^2+c)^(1/2)/d^2/e+1/21*(21*a^
2*d^2-14*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^(1/4)/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 = {475, 470, 335, 226} \[ \int \frac {\left (a+b x^2\right )^2}{\sqrt {e x} \sqrt {c+d x^2}} \, dx=\frac {\left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (21 a^2 d^2-14 a b c d+5 b^2 c^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{21 \sqrt [4]{c} d^{9/4} \sqrt {e} \sqrt {c+d x^2}}-\frac {2 b \sqrt {e x} \sqrt {c+d x^2} (5 b c-14 a d)}{21 d^2 e}+\frac {2 b^2 (e x)^{5/2} \sqrt {c+d x^2}}{7 d e^3} \]

[In]

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

[Out]

(-2*b*(5*b*c - 14*a*d)*Sqrt[e*x]*Sqrt[c + d*x^2])/(21*d^2*e) + (2*b^2*(e*x)^(5/2)*Sqrt[c + d*x^2])/(7*d*e^3) +
((5*b^2*c^2 - 14*a*b*c*d + 21*a^2*d^2)*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*Ellipt
icF[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(21*c^(1/4)*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 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} \sqrt {c+d x^2}}{7 d e^3}+\frac {2 \int \frac {\frac {7 a^2 d}{2}-\frac {1}{2} b (5 b c-14 a d) x^2}{\sqrt {e x} \sqrt {c+d x^2}} \, dx}{7 d} \\ & = -\frac {2 b (5 b c-14 a d) \sqrt {e x} \sqrt {c+d x^2}}{21 d^2 e}+\frac {2 b^2 (e x)^{5/2} \sqrt {c+d x^2}}{7 d e^3}-\frac {1}{21} \left (-21 a^2-\frac {b c (5 b c-14 a d)}{d^2}\right ) \int \frac {1}{\sqrt {e x} \sqrt {c+d x^2}} \, dx \\ & = -\frac {2 b (5 b c-14 a d) \sqrt {e x} \sqrt {c+d x^2}}{21 d^2 e}+\frac {2 b^2 (e x)^{5/2} \sqrt {c+d x^2}}{7 d e^3}+\frac {\left (2 \left (21 a^2+\frac {b c (5 b c-14 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 )}{21 e} \\ & = -\frac {2 b (5 b c-14 a d) \sqrt {e x} \sqrt {c+d x^2}}{21 d^2 e}+\frac {2 b^2 (e x)^{5/2} \sqrt {c+d x^2}}{7 d e^3}+\frac {\left (21 a^2+\frac {b c (5 b c-14 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 )}{21 \sqrt [4]{c} \sqrt [4]{d} \sqrt {e} \sqrt {c+d x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 11.19 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.77 \[ \int \frac {\left (a+b x^2\right )^2}{\sqrt {e x} \sqrt {c+d x^2}} \, dx=\frac {2 x \left (-b \left (c+d x^2\right ) \left (5 b c-14 a d-3 b d x^2\right )+\frac {i \left (5 b^2 c^2-14 a b c d+21 a^2 d^2\right ) \sqrt {1+\frac {c}{d x^2}} \sqrt {x} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}}}{\sqrt {x}}\right ),-1\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}}}\right )}{21 d^2 \sqrt {e x} \sqrt {c+d x^2}} \]

[In]

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

[Out]

(2*x*(-(b*(c + d*x^2)*(5*b*c - 14*a*d - 3*b*d*x^2)) + (I*(5*b^2*c^2 - 14*a*b*c*d + 21*a^2*d^2)*Sqrt[1 + c/(d*x
^2)]*Sqrt[x]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1])/Sqrt[(I*Sqrt[c])/Sqrt[d]]))/(21*d^2*
Sqrt[e*x]*Sqrt[c + d*x^2])

Maple [A] (verified)

Time = 3.16 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.03

method	result	size
risch	\(\frac {2 b \left (3 b d \,x^{2}+14 a d -5 b c \right ) x \sqrt {d \,x^{2}+c}}{21 d^{2} \sqrt {e x}}+\frac {\left (21 a^{2} d^{2}-14 a b c d +5 b^{2} c^{2}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {e x \left (d \,x^{2}+c \right )}}{21 d^{3} \sqrt {d e \,x^{3}+c e x}\, \sqrt {e x}\, \sqrt {d \,x^{2}+c}}\)	\(198\)
elliptic	\(\frac {\sqrt {e x \left (d \,x^{2}+c \right )}\, \left (\frac {2 b^{2} x^{2} \sqrt {d e \,x^{3}+c e x}}{7 d e}+\frac {2 \left (2 a b -\frac {5 b^{2} c}{7 d}\right ) \sqrt {d e \,x^{3}+c e x}}{3 d e}+\frac {\left (a^{2}-\frac {c \left (2 a b -\frac {5 b^{2} c}{7 d}\right )}{3 d}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d \sqrt {d e \,x^{3}+c e x}}\right )}{\sqrt {e x}\, \sqrt {d \,x^{2}+c}}\)	\(224\)
default	\(\frac {21 \sqrt {2}\, \sqrt {-c d}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} d^{2}-14 \sqrt {2}\, \sqrt {-c d}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b c d +5 \sqrt {2}\, \sqrt {-c d}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{2}+6 b^{2} d^{3} x^{5}+28 a b \,d^{3} x^{3}-4 b^{2} c \,d^{2} x^{3}+28 x a b c \,d^{2}-10 x \,b^{2} c^{2} d}{21 \sqrt {d \,x^{2}+c}\, \sqrt {e x}\, d^{3}}\)	\(350\)

[In]

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

[Out]

2/21*b*(3*b*d*x^2+14*a*d-5*b*c)/d^2*x*(d*x^2+c)^(1/2)/(e*x)^(1/2)+1/21*(21*a^2*d^2-14*a*b*c*d+5*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)*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 = 88, normalized size of antiderivative = 0.46 \[ \int \frac {\left (a+b x^2\right )^2}{\sqrt {e x} \sqrt {c+d x^2}} \, dx=\frac {2 \, {\left ({\left (5 \, b^{2} c^{2} - 14 \, a b c d + 21 \, a^{2} d^{2}\right )} \sqrt {d e} {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right ) + {\left (3 \, b^{2} d^{2} x^{2} - 5 \, b^{2} c d + 14 \, a b d^{2}\right )} \sqrt {d x^{2} + c} \sqrt {e x}\right )}}{21 \, d^{3} e} \]

[In]

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

[Out]

2/21*((5*b^2*c^2 - 14*a*b*c*d + 21*a^2*d^2)*sqrt(d*e)*weierstrassPInverse(-4*c/d, 0, x) + (3*b^2*d^2*x^2 - 5*b
^2*c*d + 14*a*b*d^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 = 3.92 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.75 \[ \int \frac {\left (a+b x^2\right )^2}{\sqrt {e x} \sqrt {c+d x^2}} \, dx=\frac {a^{2} \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt {c} \sqrt {e} \Gamma \left (\frac {5}{4}\right )} + \frac {a b x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{\sqrt {c} \sqrt {e} \Gamma \left (\frac {9}{4}\right )} + \frac {b^{2} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt {c} \sqrt {e} \Gamma \left (\frac {13}{4}\right )} \]

[In]

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

[Out]

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

Maxima [F]

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

[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(d*x^2 + c)*sqrt(e*x)), x)

Giac [F]

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

[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(d*x^2 + c)*sqrt(e*x)), x)

Mupad [F(-1)]

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

[In]

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

[Out]

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