\(\int \frac {A+B x}{(e x)^{3/2} (a+c x^2)^{5/2}} \, dx\) [482]

Optimal result

Integrand size = 24, antiderivative size = 373 \[ \int \frac {A+B x}{(e x)^{3/2} \left (a+c x^2\right )^{5/2}} \, dx=\frac {A+B x}{3 a e \sqrt {e x} \left (a+c x^2\right )^{3/2}}+\frac {7 A+5 B x}{6 a^2 e \sqrt {e x} \sqrt {a+c x^2}}-\frac {7 A \sqrt {a+c x^2}}{2 a^3 e \sqrt {e x}}+\frac {7 A \sqrt {c} x \sqrt {a+c x^2}}{2 a^3 e \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {7 A \sqrt [4]{c} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{11/4} e \sqrt {e x} \sqrt {a+c x^2}}+\frac {\left (5 \sqrt {a} B+21 A \sqrt {c}\right ) \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{12 a^{11/4} \sqrt [4]{c} e \sqrt {e x} \sqrt {a+c x^2}} \]

[Out]

1/3*(B*x+A)/a/e/(c*x^2+a)^(3/2)/(e*x)^(1/2)+1/6*(5*B*x+7*A)/a^2/e/(e*x)^(1/2)/(c*x^2+a)^(1/2)-7/2*A*(c*x^2+a)^
(1/2)/a^3/e/(e*x)^(1/2)+7/2*A*x*c^(1/2)*(c*x^2+a)^(1/2)/a^3/e/(a^(1/2)+x*c^(1/2))/(e*x)^(1/2)-7/2*A*c^(1/4)*(c
os(2*arctan(c^(1/4)*x^(1/2)/a^(1/4)))^2)^(1/2)/cos(2*arctan(c^(1/4)*x^(1/2)/a^(1/4)))*EllipticE(sin(2*arctan(c
^(1/4)*x^(1/2)/a^(1/4))),1/2*2^(1/2))*(a^(1/2)+x*c^(1/2))*x^(1/2)*((c*x^2+a)/(a^(1/2)+x*c^(1/2))^2)^(1/2)/a^(1
1/4)/e/(e*x)^(1/2)/(c*x^2+a)^(1/2)+1/12*(cos(2*arctan(c^(1/4)*x^(1/2)/a^(1/4)))^2)^(1/2)/cos(2*arctan(c^(1/4)*
x^(1/2)/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x^(1/2)/a^(1/4))),1/2*2^(1/2))*(5*B*a^(1/2)+21*A*c^(1/2))*(a^
(1/2)+x*c^(1/2))*x^(1/2)*((c*x^2+a)/(a^(1/2)+x*c^(1/2))^2)^(1/2)/a^(11/4)/c^(1/4)/e/(e*x)^(1/2)/(c*x^2+a)^(1/2
)

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {837, 849, 856, 854, 1212, 226, 1210} \[ \int \frac {A+B x}{(e x)^{3/2} \left (a+c x^2\right )^{5/2}} \, dx=\frac {\sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (5 \sqrt {a} B+21 A \sqrt {c}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{12 a^{11/4} \sqrt [4]{c} e \sqrt {e x} \sqrt {a+c x^2}}-\frac {7 A \sqrt [4]{c} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{11/4} e \sqrt {e x} \sqrt {a+c x^2}}-\frac {7 A \sqrt {a+c x^2}}{2 a^3 e \sqrt {e x}}+\frac {7 A \sqrt {c} x \sqrt {a+c x^2}}{2 a^3 e \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {7 A+5 B x}{6 a^2 e \sqrt {e x} \sqrt {a+c x^2}}+\frac {A+B x}{3 a e \sqrt {e x} \left (a+c x^2\right )^{3/2}} \]

[In]

Int[(A + B*x)/((e*x)^(3/2)*(a + c*x^2)^(5/2)),x]

[Out]

(A + B*x)/(3*a*e*Sqrt[e*x]*(a + c*x^2)^(3/2)) + (7*A + 5*B*x)/(6*a^2*e*Sqrt[e*x]*Sqrt[a + c*x^2]) - (7*A*Sqrt[
a + c*x^2])/(2*a^3*e*Sqrt[e*x]) + (7*A*Sqrt[c]*x*Sqrt[a + c*x^2])/(2*a^3*e*Sqrt[e*x]*(Sqrt[a] + Sqrt[c]*x)) -
(7*A*c^(1/4)*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticE[2*ArcTan[(c^(1/
4)*Sqrt[x])/a^(1/4)], 1/2])/(2*a^(11/4)*e*Sqrt[e*x]*Sqrt[a + c*x^2]) + ((5*Sqrt[a]*B + 21*A*Sqrt[c])*Sqrt[x]*(
Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)],
1/2])/(12*a^(11/4)*c^(1/4)*e*Sqrt[e*x]*Sqrt[a + c*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 837

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(
m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] +
Dist[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^
2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g},
x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 849

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*f - d*g)*
(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/((m + 1)*(c*d^2 + a*e^2))), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[
(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; Fr
eeQ[{a, c, d, e, f, g, p}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || Integer
sQ[2*m, 2*p])

Rule 854

Int[((f_) + (g_.)*(x_))/(Sqrt[x_]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[2, Subst[Int[(f + g*x^2)/Sqrt[
a + c*x^4], x], x, Sqrt[x]], x] /; FreeQ[{a, c, f, g}, x]

Rule 856

Int[((f_) + (g_.)*(x_))/(Sqrt[(e_)*(x_)]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[x]/Sqrt[e*x], Int[
(f + g*x)/(Sqrt[x]*Sqrt[a + c*x^2]), x], x] /; FreeQ[{a, c, e, f, g}, x]

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]

Rule 1212

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

Rubi steps \begin{align*} \text {integral}& = \frac {A+B x}{3 a e \sqrt {e x} \left (a+c x^2\right )^{3/2}}-\frac {\int \frac {-\frac {7}{2} a A c e^2-\frac {5}{2} a B c e^2 x}{(e x)^{3/2} \left (a+c x^2\right )^{3/2}} \, dx}{3 a^2 c e^2} \\ & = \frac {A+B x}{3 a e \sqrt {e x} \left (a+c x^2\right )^{3/2}}+\frac {7 A+5 B x}{6 a^2 e \sqrt {e x} \sqrt {a+c x^2}}+\frac {\int \frac {\frac {21}{4} a^2 A c^2 e^4+\frac {5}{4} a^2 B c^2 e^4 x}{(e x)^{3/2} \sqrt {a+c x^2}} \, dx}{3 a^4 c^2 e^4} \\ & = \frac {A+B x}{3 a e \sqrt {e x} \left (a+c x^2\right )^{3/2}}+\frac {7 A+5 B x}{6 a^2 e \sqrt {e x} \sqrt {a+c x^2}}-\frac {7 A \sqrt {a+c x^2}}{2 a^3 e \sqrt {e x}}-\frac {2 \int \frac {-\frac {5}{8} a^3 B c^2 e^5-\frac {21}{8} a^2 A c^3 e^5 x}{\sqrt {e x} \sqrt {a+c x^2}} \, dx}{3 a^5 c^2 e^6} \\ & = \frac {A+B x}{3 a e \sqrt {e x} \left (a+c x^2\right )^{3/2}}+\frac {7 A+5 B x}{6 a^2 e \sqrt {e x} \sqrt {a+c x^2}}-\frac {7 A \sqrt {a+c x^2}}{2 a^3 e \sqrt {e x}}-\frac {\left (2 \sqrt {x}\right ) \int \frac {-\frac {5}{8} a^3 B c^2 e^5-\frac {21}{8} a^2 A c^3 e^5 x}{\sqrt {x} \sqrt {a+c x^2}} \, dx}{3 a^5 c^2 e^6 \sqrt {e x}} \\ & = \frac {A+B x}{3 a e \sqrt {e x} \left (a+c x^2\right )^{3/2}}+\frac {7 A+5 B x}{6 a^2 e \sqrt {e x} \sqrt {a+c x^2}}-\frac {7 A \sqrt {a+c x^2}}{2 a^3 e \sqrt {e x}}-\frac {\left (4 \sqrt {x}\right ) \text {Subst}\left (\int \frac {-\frac {5}{8} a^3 B c^2 e^5-\frac {21}{8} a^2 A c^3 e^5 x^2}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{3 a^5 c^2 e^6 \sqrt {e x}} \\ & = \frac {A+B x}{3 a e \sqrt {e x} \left (a+c x^2\right )^{3/2}}+\frac {7 A+5 B x}{6 a^2 e \sqrt {e x} \sqrt {a+c x^2}}-\frac {7 A \sqrt {a+c x^2}}{2 a^3 e \sqrt {e x}}+\frac {\left (\left (5 \sqrt {a} B+21 A \sqrt {c}\right ) \sqrt {x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{6 a^{5/2} e \sqrt {e x}}-\frac {\left (7 A \sqrt {c} \sqrt {x}\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{2 a^{5/2} e \sqrt {e x}} \\ & = \frac {A+B x}{3 a e \sqrt {e x} \left (a+c x^2\right )^{3/2}}+\frac {7 A+5 B x}{6 a^2 e \sqrt {e x} \sqrt {a+c x^2}}-\frac {7 A \sqrt {a+c x^2}}{2 a^3 e \sqrt {e x}}+\frac {7 A \sqrt {c} x \sqrt {a+c x^2}}{2 a^3 e \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {7 A \sqrt [4]{c} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{11/4} e \sqrt {e x} \sqrt {a+c x^2}}+\frac {\left (5 \sqrt {a} B+21 A \sqrt {c}\right ) \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{12 a^{11/4} \sqrt [4]{c} e \sqrt {e x} \sqrt {a+c x^2}} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 10.07 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.37 \[ \int \frac {A+B x}{(e x)^{3/2} \left (a+c x^2\right )^{5/2}} \, dx=\frac {x \left (9 a A+7 a B x+7 A c x^2+5 B c x^3-21 A \left (a+c x^2\right ) \sqrt {1+\frac {c x^2}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},-\frac {c x^2}{a}\right )+5 B x \left (a+c x^2\right ) \sqrt {1+\frac {c x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {c x^2}{a}\right )\right )}{6 a^2 (e x)^{3/2} \left (a+c x^2\right )^{3/2}} \]

[In]

Integrate[(A + B*x)/((e*x)^(3/2)*(a + c*x^2)^(5/2)),x]

[Out]

(x*(9*a*A + 7*a*B*x + 7*A*c*x^2 + 5*B*c*x^3 - 21*A*(a + c*x^2)*Sqrt[1 + (c*x^2)/a]*Hypergeometric2F1[-1/4, 1/2
, 3/4, -((c*x^2)/a)] + 5*B*x*(a + c*x^2)*Sqrt[1 + (c*x^2)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^2)/a)]))/
(6*a^2*(e*x)^(3/2)*(a + c*x^2)^(3/2))

Maple [A] (verified)

Time = 2.32 (sec) , antiderivative size = 437, normalized size of antiderivative = 1.17

method	result	size
elliptic	\(\frac {\sqrt {\left (c \,x^{2}+a \right ) e x}\, \left (\frac {\left (-\frac {A x}{3 a^{2} e^{2} c}+\frac {B}{3 a \,e^{2} c^{2}}\right ) \sqrt {c e \,x^{3}+a e x}}{\left (x^{2}+\frac {a}{c}\right )^{2}}-\frac {2 x e c \left (\frac {3 A x}{4 a^{3} e^{2}}-\frac {5 B}{12 a^{2} e^{2} c}\right )}{\sqrt {\left (x^{2}+\frac {a}{c}\right ) x e c}}-\frac {2 \left (c e \,x^{2}+a e \right ) A}{a^{3} e^{2} \sqrt {x \left (c e \,x^{2}+a e \right )}}+\frac {5 B \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{12 a^{2} e c \sqrt {c e \,x^{3}+a e x}}+\frac {7 A \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{4 a^{3} e \sqrt {c e \,x^{3}+a e x}}\right )}{\sqrt {e x}\, \sqrt {c \,x^{2}+a}}\)	\(437\)
default	\(-\frac {21 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a \,c^{2} x^{2}-42 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, E\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a \,c^{2} x^{2}-5 B \sqrt {-a c}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a c \,x^{2}+21 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c -42 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, E\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c -5 B \sqrt {-a c}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{2}+42 A \,c^{3} x^{4}-10 a B \,c^{2} x^{3}+70 a A \,c^{2} x^{2}-14 a^{2} B c x +24 A \,a^{2} c}{12 a^{3} e \sqrt {e x}\, c \left (c \,x^{2}+a \right )^{\frac {3}{2}}}\)	\(605\)
risch	\(-\frac {2 A \sqrt {c \,x^{2}+a}}{a^{3} e \sqrt {e x}}+\frac {\left (\frac {A \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{\sqrt {c e \,x^{3}+a e x}}-A a c \left (\frac {x^{2}}{a \sqrt {\left (x^{2}+\frac {a}{c}\right ) x e c}}-\frac {\sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{2 a c \sqrt {c e \,x^{3}+a e x}}\right )-a^{2} \left (\frac {\left (\frac {A x}{3 a e c}-\frac {B}{3 e \,c^{2}}\right ) \sqrt {c e \,x^{3}+a e x}}{\left (x^{2}+\frac {a}{c}\right )^{2}}-\frac {2 c e x \left (-\frac {A x}{4 a^{2} e}+\frac {5 B}{12 a e c}\right )}{\sqrt {\left (x^{2}+\frac {a}{c}\right ) x e c}}-\frac {5 B \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{12 a c \sqrt {c e \,x^{3}+a e x}}-\frac {A \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{4 a^{2} \sqrt {c e \,x^{3}+a e x}}\right )\right ) \sqrt {\left (c \,x^{2}+a \right ) e x}}{a^{3} e \sqrt {e x}\, \sqrt {c \,x^{2}+a}}\)	\(779\)

[In]

int((B*x+A)/(e*x)^(3/2)/(c*x^2+a)^(5/2),x,method=_RETURNVERBOSE)

[Out]

((c*x^2+a)*e*x)^(1/2)/(e*x)^(1/2)/(c*x^2+a)^(1/2)*((-1/3/a^2/e^2/c*A*x+1/3/a/e^2/c^2*B)*(c*e*x^3+a*e*x)^(1/2)/
(x^2+a/c)^2-2*x*e*c*(3/4/a^3/e^2*A*x-5/12/a^2/e^2*B/c)/((x^2+a/c)*x*e*c)^(1/2)-2*(c*e*x^2+a*e)/a^3/e^2*A/(x*(c
*e*x^2+a*e))^(1/2)+5/12/a^2*B/e*(-a*c)^(1/2)/c*((x+(-a*c)^(1/2)/c)/(-a*c)^(1/2)*c)^(1/2)*(-2*(x-(-a*c)^(1/2)/c
)/(-a*c)^(1/2)*c)^(1/2)*(-x/(-a*c)^(1/2)*c)^(1/2)/(c*e*x^3+a*e*x)^(1/2)*EllipticF(((x+(-a*c)^(1/2)/c)/(-a*c)^(
1/2)*c)^(1/2),1/2*2^(1/2))+7/4/a^3*A/e*(-a*c)^(1/2)*((x+(-a*c)^(1/2)/c)/(-a*c)^(1/2)*c)^(1/2)*(-2*(x-(-a*c)^(1
/2)/c)/(-a*c)^(1/2)*c)^(1/2)*(-x/(-a*c)^(1/2)*c)^(1/2)/(c*e*x^3+a*e*x)^(1/2)*(-2*(-a*c)^(1/2)/c*EllipticE(((x+
(-a*c)^(1/2)/c)/(-a*c)^(1/2)*c)^(1/2),1/2*2^(1/2))+(-a*c)^(1/2)/c*EllipticF(((x+(-a*c)^(1/2)/c)/(-a*c)^(1/2)*c
)^(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 = 195, normalized size of antiderivative = 0.52 \[ \int \frac {A+B x}{(e x)^{3/2} \left (a+c x^2\right )^{5/2}} \, dx=\frac {5 \, {\left (B a c^{2} x^{5} + 2 \, B a^{2} c x^{3} + B a^{3} x\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) - 21 \, {\left (A c^{3} x^{5} + 2 \, A a c^{2} x^{3} + A a^{2} c x\right )} \sqrt {c e} {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) - {\left (21 \, A c^{3} x^{4} - 5 \, B a c^{2} x^{3} + 35 \, A a c^{2} x^{2} - 7 \, B a^{2} c x + 12 \, A a^{2} c\right )} \sqrt {c x^{2} + a} \sqrt {e x}}{6 \, {\left (a^{3} c^{3} e^{2} x^{5} + 2 \, a^{4} c^{2} e^{2} x^{3} + a^{5} c e^{2} x\right )}} \]

[In]

integrate((B*x+A)/(e*x)^(3/2)/(c*x^2+a)^(5/2),x, algorithm="fricas")

[Out]

1/6*(5*(B*a*c^2*x^5 + 2*B*a^2*c*x^3 + B*a^3*x)*sqrt(c*e)*weierstrassPInverse(-4*a/c, 0, x) - 21*(A*c^3*x^5 + 2
*A*a*c^2*x^3 + A*a^2*c*x)*sqrt(c*e)*weierstrassZeta(-4*a/c, 0, weierstrassPInverse(-4*a/c, 0, x)) - (21*A*c^3*
x^4 - 5*B*a*c^2*x^3 + 35*A*a*c^2*x^2 - 7*B*a^2*c*x + 12*A*a^2*c)*sqrt(c*x^2 + a)*sqrt(e*x))/(a^3*c^3*e^2*x^5 +
2*a^4*c^2*e^2*x^3 + a^5*c*e^2*x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 105.88 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.26 \[ \int \frac {A+B x}{(e x)^{3/2} \left (a+c x^2\right )^{5/2}} \, dx=\frac {A \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {5}{2} \\ \frac {3}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {5}{2}} e^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} + \frac {B \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {5}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {5}{2}} e^{\frac {3}{2}} \Gamma \left (\frac {5}{4}\right )} \]

[In]

integrate((B*x+A)/(e*x)**(3/2)/(c*x**2+a)**(5/2),x)

[Out]

A*gamma(-1/4)*hyper((-1/4, 5/2), (3/4,), c*x**2*exp_polar(I*pi)/a)/(2*a**(5/2)*e**(3/2)*sqrt(x)*gamma(3/4)) +
B*sqrt(x)*gamma(1/4)*hyper((1/4, 5/2), (5/4,), c*x**2*exp_polar(I*pi)/a)/(2*a**(5/2)*e**(3/2)*gamma(5/4))

Maxima [F]

\[ \int \frac {A+B x}{(e x)^{3/2} \left (a+c x^2\right )^{5/2}} \, dx=\int { \frac {B x + A}{{\left (c x^{2} + a\right )}^{\frac {5}{2}} \left (e x\right )^{\frac {3}{2}}} \,d x } \]

[In]

integrate((B*x+A)/(e*x)^(3/2)/(c*x^2+a)^(5/2),x, algorithm="maxima")

[Out]

integrate((B*x + A)/((c*x^2 + a)^(5/2)*(e*x)^(3/2)), x)

Giac [F]

\[ \int \frac {A+B x}{(e x)^{3/2} \left (a+c x^2\right )^{5/2}} \, dx=\int { \frac {B x + A}{{\left (c x^{2} + a\right )}^{\frac {5}{2}} \left (e x\right )^{\frac {3}{2}}} \,d x } \]

[In]

integrate((B*x+A)/(e*x)^(3/2)/(c*x^2+a)^(5/2),x, algorithm="giac")

[Out]

integrate((B*x + A)/((c*x^2 + a)^(5/2)*(e*x)^(3/2)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x}{(e x)^{3/2} \left (a+c x^2\right )^{5/2}} \, dx=\int \frac {A+B\,x}{{\left (e\,x\right )}^{3/2}\,{\left (c\,x^2+a\right )}^{5/2}} \,d x \]

[In]

int((A + B*x)/((e*x)^(3/2)*(a + c*x^2)^(5/2)),x)

[Out]

int((A + B*x)/((e*x)^(3/2)*(a + c*x^2)^(5/2)), x)