\(\int \frac {(a+b x^2)^2}{\sqrt {e x} (c+d x^2)^{5/2}} \, dx\) [861]

Optimal result

Integrand size = 28, antiderivative size = 213 \[ \int \frac {\left (a+b x^2\right )^2}{\sqrt {e x} \left (c+d x^2\right )^{5/2}} \, dx=\frac {(b c-a d)^2 \sqrt {e x}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}-\frac {(b c-a d) (7 b c+5 a d) \sqrt {e x}}{6 c^2 d^2 e \sqrt {c+d x^2}}+\frac {\left (5 b^2 c^2+2 a b c d+5 a^2 d^2\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{12 c^{9/4} d^{9/4} \sqrt {e} \sqrt {c+d x^2}} \]

[Out]

1/3*(-a*d+b*c)^2*(e*x)^(1/2)/c/d^2/e/(d*x^2+c)^(3/2)-1/6*(-a*d+b*c)*(5*a*d+7*b*c)*(e*x)^(1/2)/c^2/d^2/e/(d*x^2
+c)^(1/2)+1/12*(5*a^2*d^2+2*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)/co
s(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^(9/4)/d^(9/4)/e^(1/2)/(d*x^2+c)^(1/
2)

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 213, 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, 468, 335, 226} \[ \int \frac {\left (a+b x^2\right )^2}{\sqrt {e x} \left (c+d x^2\right )^{5/2}} \, dx=\frac {\left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (5 a^2 d^2+2 a b c d+5 b^2 c^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{12 c^{9/4} d^{9/4} \sqrt {e} \sqrt {c+d x^2}}-\frac {\sqrt {e x} (5 a d+7 b c) (b c-a d)}{6 c^2 d^2 e \sqrt {c+d x^2}}+\frac {\sqrt {e x} (b c-a d)^2}{3 c d^2 e \left (c+d x^2\right )^{3/2}} \]

[In]

Int[(a + b*x^2)^2/(Sqrt[e*x]*(c + d*x^2)^(5/2)),x]

[Out]

((b*c - a*d)^2*Sqrt[e*x])/(3*c*d^2*e*(c + d*x^2)^(3/2)) - ((b*c - a*d)*(7*b*c + 5*a*d)*Sqrt[e*x])/(6*c^2*d^2*e
*Sqrt[c + d*x^2]) + ((5*b^2*c^2 + 2*a*b*c*d + 5*a^2*d^2)*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqr
t[d]*x)^2]*EllipticF[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(12*c^(9/4)*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 468

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*c - a*d
))*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*b*e*n*(p + 1))), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a
*b*n*(p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0]
&& LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) ||  !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0]
&& LeQ[-1, m, (-n)*(p + 1)]))

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}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}-\frac {\int \frac {\frac {1}{2} \left (-6 a^2 d^2+(b c-a d)^2\right )-3 b^2 c d x^2}{\sqrt {e x} \left (c+d x^2\right )^{3/2}} \, dx}{3 c d^2} \\ & = \frac {(b c-a d)^2 \sqrt {e x}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}-\frac {(b c-a d) (7 b c+5 a d) \sqrt {e x}}{6 c^2 d^2 e \sqrt {c+d x^2}}+\frac {\left (5 b^2 c^2+2 a b c d+5 a^2 d^2\right ) \int \frac {1}{\sqrt {e x} \sqrt {c+d x^2}} \, dx}{12 c^2 d^2} \\ & = \frac {(b c-a d)^2 \sqrt {e x}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}-\frac {(b c-a d) (7 b c+5 a d) \sqrt {e x}}{6 c^2 d^2 e \sqrt {c+d x^2}}+\frac {\left (5 b^2 c^2+2 a b c d+5 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 )}{6 c^2 d^2 e} \\ & = \frac {(b c-a d)^2 \sqrt {e x}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}-\frac {(b c-a d) (7 b c+5 a d) \sqrt {e x}}{6 c^2 d^2 e \sqrt {c+d x^2}}+\frac {\left (5 b^2 c^2+2 a b c d+5 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 )}{12 c^{9/4} d^{9/4} \sqrt {e} \sqrt {c+d x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 11.21 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.79 \[ \int \frac {\left (a+b x^2\right )^2}{\sqrt {e x} \left (c+d x^2\right )^{5/2}} \, dx=\frac {x \left (-7 b^2 c^2+2 a b c d+5 a^2 d^2+\frac {2 c (b c-a d)^2}{c+d x^2}+\frac {i \left (5 b^2 c^2+2 a b c d+5 a^2 d^2\right ) \sqrt {1+\frac {c}{d x^2}} \sqrt {x} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}}}{\sqrt {x}}\right ),-1\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}}}\right )}{6 c^2 d^2 \sqrt {e x} \sqrt {c+d x^2}} \]

[In]

Integrate[(a + b*x^2)^2/(Sqrt[e*x]*(c + d*x^2)^(5/2)),x]

[Out]

(x*(-7*b^2*c^2 + 2*a*b*c*d + 5*a^2*d^2 + (2*c*(b*c - a*d)^2)/(c + d*x^2) + (I*(5*b^2*c^2 + 2*a*b*c*d + 5*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]]))/(6*c^2*d^2*Sqrt[e*x]*Sqrt[c + d*x^2])

Maple [A] (verified)

Time = 3.09 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.31

method	result	size
elliptic	\(\frac {\sqrt {e x \left (d \,x^{2}+c \right )}\, \left (\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {d e \,x^{3}+c e x}}{3 c e \,d^{4} \left (x^{2}+\frac {c}{d}\right )^{2}}+\frac {x \left (5 a^{2} d^{2}+2 a b c d -7 b^{2} c^{2}\right )}{6 d^{2} c^{2} \sqrt {\left (x^{2}+\frac {c}{d}\right ) d e x}}+\frac {\left (\frac {b^{2}}{d^{2}}+\frac {5 a^{2} d^{2}+2 a b c d -7 b^{2} c^{2}}{12 d^{2} c^{2}}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d \sqrt {d e \,x^{3}+c e x}}\right )}{\sqrt {e x}\, \sqrt {d \,x^{2}+c}}\)	\(280\)
default	\(\frac {5 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-c d}\, a^{2} d^{3} x^{2}+2 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-c d}\, a b c \,d^{2} x^{2}+5 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-c d}\, b^{2} c^{2} d \,x^{2}+5 \sqrt {-c d}\, \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c \,d^{2}+2 \sqrt {-c d}\, \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{2} d +5 \sqrt {-c d}\, \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{3}+10 a^{2} d^{4} x^{3}+4 x^{3} d^{3} b a c -14 b^{2} c^{2} d^{2} x^{3}+14 a^{2} c \,d^{3} x -4 a b \,c^{2} d^{2} x -10 b^{2} d x \,c^{3}}{12 \sqrt {e x}\, c^{2} d^{3} \left (d \,x^{2}+c \right )^{\frac {3}{2}}}\)	\(660\)

[In]

int((b*x^2+a)^2/(d*x^2+c)^(5/2)/(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)*(1/3/c/e/d^4*(a^2*d^2-2*a*b*c*d+b^2*c^2)*(d*e*x^3+c*e*x)^(1/
2)/(x^2+c/d)^2+1/6/d^2*x/c^2*(5*a^2*d^2+2*a*b*c*d-7*b^2*c^2)/((x^2+c/d)*d*e*x)^(1/2)+(b^2/d^2+1/12/d^2/c^2*(5*
a^2*d^2+2*a*b*c*d-7*b^2*c^2))*(-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 = 0.09 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.05 \[ \int \frac {\left (a+b x^2\right )^2}{\sqrt {e x} \left (c+d x^2\right )^{5/2}} \, dx=\frac {{\left (5 \, b^{2} c^{4} + 2 \, a b c^{3} d + 5 \, a^{2} c^{2} d^{2} + {\left (5 \, b^{2} c^{2} d^{2} + 2 \, a b c d^{3} + 5 \, a^{2} d^{4}\right )} x^{4} + 2 \, {\left (5 \, b^{2} c^{3} d + 2 \, a b c^{2} d^{2} + 5 \, a^{2} c d^{3}\right )} x^{2}\right )} \sqrt {d e} {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right ) - {\left (5 \, b^{2} c^{3} d + 2 \, a b c^{2} d^{2} - 7 \, a^{2} c d^{3} + {\left (7 \, b^{2} c^{2} d^{2} - 2 \, a b c d^{3} - 5 \, a^{2} d^{4}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {e x}}{6 \, {\left (c^{2} d^{5} e x^{4} + 2 \, c^{3} d^{4} e x^{2} + c^{4} d^{3} e\right )}} \]

[In]

integrate((b*x^2+a)^2/(d*x^2+c)^(5/2)/(e*x)^(1/2),x, algorithm="fricas")

[Out]

1/6*((5*b^2*c^4 + 2*a*b*c^3*d + 5*a^2*c^2*d^2 + (5*b^2*c^2*d^2 + 2*a*b*c*d^3 + 5*a^2*d^4)*x^4 + 2*(5*b^2*c^3*d
+ 2*a*b*c^2*d^2 + 5*a^2*c*d^3)*x^2)*sqrt(d*e)*weierstrassPInverse(-4*c/d, 0, x) - (5*b^2*c^3*d + 2*a*b*c^2*d^
2 - 7*a^2*c*d^3 + (7*b^2*c^2*d^2 - 2*a*b*c*d^3 - 5*a^2*d^4)*x^2)*sqrt(d*x^2 + c)*sqrt(e*x))/(c^2*d^5*e*x^4 + 2
*c^3*d^4*e*x^2 + c^4*d^3*e)

Sympy [F]

\[ \int \frac {\left (a+b x^2\right )^2}{\sqrt {e x} \left (c+d x^2\right )^{5/2}} \, dx=\int \frac {\left (a + b x^{2}\right )^{2}}{\sqrt {e x} \left (c + d x^{2}\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate((b*x**2+a)**2/(d*x**2+c)**(5/2)/(e*x)**(1/2),x)

[Out]

Integral((a + b*x**2)**2/(sqrt(e*x)*(c + d*x**2)**(5/2)), x)

Maxima [F]

\[ \int \frac {\left (a+b x^2\right )^2}{\sqrt {e x} \left (c+d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2}}{{\left (d x^{2} + c\right )}^{\frac {5}{2}} \sqrt {e x}} \,d x } \]

[In]

integrate((b*x^2+a)^2/(d*x^2+c)^(5/2)/(e*x)^(1/2),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^2/((d*x^2 + c)^(5/2)*sqrt(e*x)), x)

Giac [F]

\[ \int \frac {\left (a+b x^2\right )^2}{\sqrt {e x} \left (c+d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2}}{{\left (d x^{2} + c\right )}^{\frac {5}{2}} \sqrt {e x}} \,d x } \]

[In]

integrate((b*x^2+a)^2/(d*x^2+c)^(5/2)/(e*x)^(1/2),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^2/((d*x^2 + c)^(5/2)*sqrt(e*x)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^2}{\sqrt {e x} \left (c+d x^2\right )^{5/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^2}{\sqrt {e\,x}\,{\left (d\,x^2+c\right )}^{5/2}} \,d x \]

[In]

int((a + b*x^2)^2/((e*x)^(1/2)*(c + d*x^2)^(5/2)),x)

[Out]

int((a + b*x^2)^2/((e*x)^(1/2)*(c + d*x^2)^(5/2)), x)