\(\int \frac {(a+b x^2)^2}{(e x)^{9/2} \sqrt {c+d x^2}} \, dx\) [846]

Optimal result

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

[Out]

-2/7*a^2*(d*x^2+c)^(1/2)/c/e/(e*x)^(7/2)-2/21*a*(-5*a*d+14*b*c)*(d*x^2+c)^(1/2)/c^2/e^3/(e*x)^(3/2)+1/21*(5*a^
2*d^2-14*a*b*c*d+21*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^(9/4)/d^(1/4)/e^(9/2)/(d*x^2+c)^(1/2)

Rubi [A] (verified)

Time = 0.12 (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 = {473, 464, 335, 226} \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{9/2} \sqrt {c+d x^2}} \, dx=\frac {\left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (5 a^2 d^2-14 a b c d+21 b^2 c^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{21 c^{9/4} \sqrt [4]{d} e^{9/2} \sqrt {c+d x^2}}-\frac {2 a^2 \sqrt {c+d x^2}}{7 c e (e x)^{7/2}}-\frac {2 a \sqrt {c+d x^2} (14 b c-5 a d)}{21 c^2 e^3 (e x)^{3/2}} \]

[In]

Int[(a + b*x^2)^2/((e*x)^(9/2)*Sqrt[c + d*x^2]),x]

[Out]

(-2*a^2*Sqrt[c + d*x^2])/(7*c*e*(e*x)^(7/2)) - (2*a*(14*b*c - 5*a*d)*Sqrt[c + d*x^2])/(21*c^2*e^3*(e*x)^(3/2))
+ ((21*b^2*c^2 - 14*a*b*c*d + 5*a^2*d^2)*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*Elli
pticF[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(21*c^(9/4)*d^(1/4)*e^(9/2)*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 464

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

Rule 473

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

Rubi steps \begin{align*} \text {integral}& = -\frac {2 a^2 \sqrt {c+d x^2}}{7 c e (e x)^{7/2}}+\frac {2 \int \frac {\frac {1}{2} a (14 b c-5 a d)+\frac {7}{2} b^2 c x^2}{(e x)^{5/2} \sqrt {c+d x^2}} \, dx}{7 c e^2} \\ & = -\frac {2 a^2 \sqrt {c+d x^2}}{7 c e (e x)^{7/2}}-\frac {2 a (14 b c-5 a d) \sqrt {c+d x^2}}{21 c^2 e^3 (e x)^{3/2}}-\frac {\left (4 \left (-\frac {21}{4} b^2 c^2+\frac {1}{4} a d (14 b c-5 a d)\right )\right ) \int \frac {1}{\sqrt {e x} \sqrt {c+d x^2}} \, dx}{21 c^2 e^4} \\ & = -\frac {2 a^2 \sqrt {c+d x^2}}{7 c e (e x)^{7/2}}-\frac {2 a (14 b c-5 a d) \sqrt {c+d x^2}}{21 c^2 e^3 (e x)^{3/2}}-\frac {\left (8 \left (-\frac {21}{4} b^2 c^2+\frac {1}{4} a d (14 b c-5 a d)\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{21 c^2 e^5} \\ & = -\frac {2 a^2 \sqrt {c+d x^2}}{7 c e (e x)^{7/2}}-\frac {2 a (14 b c-5 a d) \sqrt {c+d x^2}}{21 c^2 e^3 (e x)^{3/2}}+\frac {\left (21 b^2 c^2-a d (14 b c-5 a d)\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 c^{9/4} \sqrt [4]{d} e^{9/2} \sqrt {c+d x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 11.15 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.82 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{9/2} \sqrt {c+d x^2}} \, dx=\frac {x^{9/2} \left (\frac {2 a \left (c+d x^2\right ) \left (-3 a c-14 b c x^2+5 a d x^2\right )}{c^2 x^{7/2}}+\frac {2 i \left (21 b^2 c^2-14 a b c d+5 a^2 d^2\right ) \sqrt {1+\frac {c}{d x^2}} x \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}}}{\sqrt {x}}\right ),-1\right )}{c^2 \sqrt {\frac {i \sqrt {c}}{\sqrt {d}}}}\right )}{21 (e x)^{9/2} \sqrt {c+d x^2}} \]

[In]

Integrate[(a + b*x^2)^2/((e*x)^(9/2)*Sqrt[c + d*x^2]),x]

[Out]

(x^(9/2)*((2*a*(c + d*x^2)*(-3*a*c - 14*b*c*x^2 + 5*a*d*x^2))/(c^2*x^(7/2)) + ((2*I)*(21*b^2*c^2 - 14*a*b*c*d
+ 5*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1])/(c^2*Sqrt[(I*S
qrt[c])/Sqrt[d]])))/(21*(e*x)^(9/2)*Sqrt[c + d*x^2])

Maple [A] (verified)

Time = 3.19 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.10

method	result	size
risch	\(-\frac {2 \sqrt {d \,x^{2}+c}\, a \left (-5 a d \,x^{2}+14 c b \,x^{2}+3 a c \right )}{21 c^{2} x^{3} e^{4} \sqrt {e x}}+\frac {\left (5 a^{2} d^{2}-14 a b c d +21 b^{2} c^{2}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {e x \left (d \,x^{2}+c \right )}}{21 c^{2} d \sqrt {d e \,x^{3}+c e x}\, e^{4} \sqrt {e x}\, \sqrt {d \,x^{2}+c}}\)	\(212\)
elliptic	\(\frac {\sqrt {e x \left (d \,x^{2}+c \right )}\, \left (-\frac {2 a^{2} \sqrt {d e \,x^{3}+c e x}}{7 e^{5} c \,x^{4}}+\frac {2 a \left (5 a d -14 b c \right ) \sqrt {d e \,x^{3}+c e x}}{21 e^{5} c^{2} x^{2}}+\frac {\left (\frac {b^{2}}{e^{4}}+\frac {d a \left (5 a d -14 b c \right )}{21 c^{2} e^{4}}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d \sqrt {d e \,x^{3}+c e x}}\right )}{\sqrt {e x}\, \sqrt {d \,x^{2}+c}}\)	\(226\)
default	\(\frac {5 \sqrt {2}\, \sqrt {-c d}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} d^{2} x^{3}-14 \sqrt {2}\, \sqrt {-c d}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b c d \,x^{3}+21 \sqrt {2}\, \sqrt {-c d}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{2} x^{3}+10 a^{2} d^{3} x^{4}-28 a b c \,d^{2} x^{4}+4 a^{2} c \,d^{2} x^{2}-28 a b \,c^{2} d \,x^{2}-6 a^{2} c^{2} d}{21 \sqrt {d \,x^{2}+c}\, x^{3} d \,c^{2} e^{4} \sqrt {e x}}\)	\(370\)

[In]

int((b*x^2+a)^2/(e*x)^(9/2)/(d*x^2+c)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2/21*(d*x^2+c)^(1/2)*a*(-5*a*d*x^2+14*b*c*x^2+3*a*c)/c^2/x^3/e^4/(e*x)^(1/2)+1/21*(5*a^2*d^2-14*a*b*c*d+21*b^
2*c^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))/e^4*(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 = 0.08 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.51 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{9/2} \sqrt {c+d x^2}} \, dx=\frac {2 \, {\left ({\left (21 \, b^{2} c^{2} - 14 \, a b c d + 5 \, a^{2} d^{2}\right )} \sqrt {d e} x^{4} {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right ) - {\left (3 \, a^{2} c d + {\left (14 \, a b c d - 5 \, a^{2} d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {e x}\right )}}{21 \, c^{2} d e^{5} x^{4}} \]

[In]

integrate((b*x^2+a)^2/(e*x)^(9/2)/(d*x^2+c)^(1/2),x, algorithm="fricas")

[Out]

2/21*((21*b^2*c^2 - 14*a*b*c*d + 5*a^2*d^2)*sqrt(d*e)*x^4*weierstrassPInverse(-4*c/d, 0, x) - (3*a^2*c*d + (14
*a*b*c*d - 5*a^2*d^2)*x^2)*sqrt(d*x^2 + c)*sqrt(e*x))/(c^2*d*e^5*x^4)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 70.80 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{9/2} \sqrt {c+d x^2}} \, dx=\frac {a^{2} \Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{4}, \frac {1}{2} \\ - \frac {3}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt {c} e^{\frac {9}{2}} x^{\frac {7}{2}} \Gamma \left (- \frac {3}{4}\right )} + \frac {a b \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{\sqrt {c} e^{\frac {9}{2}} x^{\frac {3}{2}} \Gamma \left (\frac {1}{4}\right )} + \frac {b^{2} \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt {c} e^{\frac {9}{2}} \Gamma \left (\frac {5}{4}\right )} \]

[In]

integrate((b*x**2+a)**2/(e*x)**(9/2)/(d*x**2+c)**(1/2),x)

[Out]

a**2*gamma(-7/4)*hyper((-7/4, 1/2), (-3/4,), d*x**2*exp_polar(I*pi)/c)/(2*sqrt(c)*e**(9/2)*x**(7/2)*gamma(-3/4
)) + a*b*gamma(-3/4)*hyper((-3/4, 1/2), (1/4,), d*x**2*exp_polar(I*pi)/c)/(sqrt(c)*e**(9/2)*x**(3/2)*gamma(1/4
)) + b**2*sqrt(x)*gamma(1/4)*hyper((1/4, 1/2), (5/4,), d*x**2*exp_polar(I*pi)/c)/(2*sqrt(c)*e**(9/2)*gamma(5/4
))

Maxima [F]

\[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{9/2} \sqrt {c+d x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2}}{\sqrt {d x^{2} + c} \left (e x\right )^{\frac {9}{2}}} \,d x } \]

[In]

integrate((b*x^2+a)^2/(e*x)^(9/2)/(d*x^2+c)^(1/2),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^2/(sqrt(d*x^2 + c)*(e*x)^(9/2)), x)

Giac [F]

\[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{9/2} \sqrt {c+d x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2}}{\sqrt {d x^{2} + c} \left (e x\right )^{\frac {9}{2}}} \,d x } \]

[In]

integrate((b*x^2+a)^2/(e*x)^(9/2)/(d*x^2+c)^(1/2),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^2/(sqrt(d*x^2 + c)*(e*x)^(9/2)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{9/2} \sqrt {c+d x^2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^2}{{\left (e\,x\right )}^{9/2}\,\sqrt {d\,x^2+c}} \,d x \]

[In]

int((a + b*x^2)^2/((e*x)^(9/2)*(c + d*x^2)^(1/2)),x)

[Out]

int((a + b*x^2)^2/((e*x)^(9/2)*(c + d*x^2)^(1/2)), x)