\(\int \frac {(e x)^{9/2} (A+B x)}{(a+c x^2)^{5/2}} \, dx\) [476]

Optimal result

Integrand size = 24, antiderivative size = 368 \[ \int \frac {(e x)^{9/2} (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx=-\frac {e (e x)^{7/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}-\frac {e^3 (e x)^{3/2} (7 A+9 B x)}{6 c^2 \sqrt {a+c x^2}}+\frac {5 B e^4 \sqrt {e x} \sqrt {a+c x^2}}{2 c^3}+\frac {7 A e^5 x \sqrt {a+c x^2}}{2 c^{5/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {7 \sqrt [4]{a} A e^5 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 c^{11/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {\sqrt [4]{a} \left (5 \sqrt {a} B-7 A \sqrt {c}\right ) e^5 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 c^{13/4} \sqrt {e x} \sqrt {a+c x^2}} \]

[Out]

-1/3*e*(e*x)^(7/2)*(B*x+A)/c/(c*x^2+a)^(3/2)-1/6*e^3*(e*x)^(3/2)*(9*B*x+7*A)/c^2/(c*x^2+a)^(1/2)+7/2*A*e^5*x*(
c*x^2+a)^(1/2)/c^(5/2)/(a^(1/2)+x*c^(1/2))/(e*x)^(1/2)+5/2*B*e^4*(e*x)^(1/2)*(c*x^2+a)^(1/2)/c^3-7/2*a^(1/4)*A
*e^5*(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)))*EllipticE(sin(2*a
rctan(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)/c^(11/4)/(e*x)^(1/2)/(c*x^2+a)^(1/2)-1/4*a^(1/4)*e^5*(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)-7
*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)/c^(13/4)/(e*x)^(1/2)/(c*x^2+a)
^(1/2)

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 368, 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 = {833, 847, 856, 854, 1212, 226, 1210} \[ \int \frac {(e x)^{9/2} (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx=-\frac {\sqrt [4]{a} e^5 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (5 \sqrt {a} B-7 A \sqrt {c}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 c^{13/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {7 \sqrt [4]{a} A e^5 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 c^{11/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {e^3 (e x)^{3/2} (7 A+9 B x)}{6 c^2 \sqrt {a+c x^2}}-\frac {e (e x)^{7/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}+\frac {7 A e^5 x \sqrt {a+c x^2}}{2 c^{5/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {5 B e^4 \sqrt {e x} \sqrt {a+c x^2}}{2 c^3} \]

[In]

Int[((e*x)^(9/2)*(A + B*x))/(a + c*x^2)^(5/2),x]

[Out]

-1/3*(e*(e*x)^(7/2)*(A + B*x))/(c*(a + c*x^2)^(3/2)) - (e^3*(e*x)^(3/2)*(7*A + 9*B*x))/(6*c^2*Sqrt[a + c*x^2])
+ (5*B*e^4*Sqrt[e*x]*Sqrt[a + c*x^2])/(2*c^3) + (7*A*e^5*x*Sqrt[a + c*x^2])/(2*c^(5/2)*Sqrt[e*x]*(Sqrt[a] + S
qrt[c]*x)) - (7*a^(1/4)*A*e^5*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*Elliptic
E[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(2*c^(11/4)*Sqrt[e*x]*Sqrt[a + c*x^2]) - (a^(1/4)*(5*Sqrt[a]*B -
7*A*Sqrt[c])*e^5*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])/(4*c^(13/4)*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 833

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

Rule 847

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

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 {e (e x)^{7/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}+\frac {\int \frac {(e x)^{5/2} \left (\frac {7}{2} a A e^2+\frac {9}{2} a B e^2 x\right )}{\left (a+c x^2\right )^{3/2}} \, dx}{3 a c} \\ & = -\frac {e (e x)^{7/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}-\frac {e^3 (e x)^{3/2} (7 A+9 B x)}{6 c^2 \sqrt {a+c x^2}}+\frac {\int \frac {\sqrt {e x} \left (\frac {21}{4} a^2 A e^4+\frac {45}{4} a^2 B e^4 x\right )}{\sqrt {a+c x^2}} \, dx}{3 a^2 c^2} \\ & = -\frac {e (e x)^{7/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}-\frac {e^3 (e x)^{3/2} (7 A+9 B x)}{6 c^2 \sqrt {a+c x^2}}+\frac {5 B e^4 \sqrt {e x} \sqrt {a+c x^2}}{2 c^3}+\frac {2 \int \frac {-\frac {45}{8} a^3 B e^5+\frac {63}{8} a^2 A c e^5 x}{\sqrt {e x} \sqrt {a+c x^2}} \, dx}{9 a^2 c^3} \\ & = -\frac {e (e x)^{7/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}-\frac {e^3 (e x)^{3/2} (7 A+9 B x)}{6 c^2 \sqrt {a+c x^2}}+\frac {5 B e^4 \sqrt {e x} \sqrt {a+c x^2}}{2 c^3}+\frac {\left (2 \sqrt {x}\right ) \int \frac {-\frac {45}{8} a^3 B e^5+\frac {63}{8} a^2 A c e^5 x}{\sqrt {x} \sqrt {a+c x^2}} \, dx}{9 a^2 c^3 \sqrt {e x}} \\ & = -\frac {e (e x)^{7/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}-\frac {e^3 (e x)^{3/2} (7 A+9 B x)}{6 c^2 \sqrt {a+c x^2}}+\frac {5 B e^4 \sqrt {e x} \sqrt {a+c x^2}}{2 c^3}+\frac {\left (4 \sqrt {x}\right ) \text {Subst}\left (\int \frac {-\frac {45}{8} a^3 B e^5+\frac {63}{8} a^2 A c e^5 x^2}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{9 a^2 c^3 \sqrt {e x}} \\ & = -\frac {e (e x)^{7/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}-\frac {e^3 (e x)^{3/2} (7 A+9 B x)}{6 c^2 \sqrt {a+c x^2}}+\frac {5 B e^4 \sqrt {e x} \sqrt {a+c x^2}}{2 c^3}-\frac {\left (\sqrt {a} \left (5 \sqrt {a} B-7 A \sqrt {c}\right ) e^5 \sqrt {x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{2 c^3 \sqrt {e x}}-\frac {\left (7 \sqrt {a} A e^5 \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 c^{5/2} \sqrt {e x}} \\ & = -\frac {e (e x)^{7/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}-\frac {e^3 (e x)^{3/2} (7 A+9 B x)}{6 c^2 \sqrt {a+c x^2}}+\frac {5 B e^4 \sqrt {e x} \sqrt {a+c x^2}}{2 c^3}+\frac {7 A e^5 x \sqrt {a+c x^2}}{2 c^{5/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {7 \sqrt [4]{a} A e^5 \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 c^{11/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {\sqrt [4]{a} \left (5 \sqrt {a} B-7 A \sqrt {c}\right ) e^5 \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 )}{4 c^{13/4} \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 = 10.10 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.42 \[ \int \frac {(e x)^{9/2} (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx=\frac {e^4 \sqrt {e x} \left (15 a^2 B-7 a A c x+21 a B c x^2-9 A c^2 x^3+4 B c^2 x^4-15 a B \left (a+c x^2\right ) \sqrt {1+\frac {c x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {c x^2}{a}\right )+7 A c x \left (a+c x^2\right ) \sqrt {1+\frac {c x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-\frac {c x^2}{a}\right )\right )}{6 c^3 \left (a+c x^2\right )^{3/2}} \]

[In]

Integrate[((e*x)^(9/2)*(A + B*x))/(a + c*x^2)^(5/2),x]

[Out]

(e^4*Sqrt[e*x]*(15*a^2*B - 7*a*A*c*x + 21*a*B*c*x^2 - 9*A*c^2*x^3 + 4*B*c^2*x^4 - 15*a*B*(a + c*x^2)*Sqrt[1 +
(c*x^2)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^2)/a)] + 7*A*c*x*(a + c*x^2)*Sqrt[1 + (c*x^2)/a]*Hypergeome
tric2F1[1/2, 3/4, 7/4, -((c*x^2)/a)]))/(6*c^3*(a + c*x^2)^(3/2))

Maple [A] (verified)

Time = 2.06 (sec) , antiderivative size = 426, normalized size of antiderivative = 1.16

method	result	size
elliptic	\(\frac {\sqrt {e x}\, \sqrt {\left (c \,x^{2}+a \right ) e x}\, \left (\frac {\left (\frac {a \,e^{4} A x}{3 c^{4}}-\frac {a^{2} e^{4} B}{3 c^{5}}\right ) \sqrt {c e \,x^{3}+a e x}}{\left (x^{2}+\frac {a}{c}\right )^{2}}-\frac {2 x e c \left (\frac {3 e^{4} A x}{4 c^{3}}-\frac {13 e^{4} B a}{12 c^{4}}\right )}{\sqrt {\left (x^{2}+\frac {a}{c}\right ) x e c}}+\frac {2 B \,e^{4} \sqrt {c e \,x^{3}+a e x}}{3 c^{3}}-\frac {5 B a \,e^{5} \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{4 c^{4} \sqrt {c e \,x^{3}+a e x}}+\frac {7 A \,e^{5} \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{4 c^{3} \sqrt {c e \,x^{3}+a e x}}\right )}{e x \sqrt {c \,x^{2}+a}}\)	\(426\)
default	\(-\frac {\left (21 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a \,c^{2} x^{2}-42 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, E\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a \,c^{2} x^{2}+15 B \sqrt {-a c}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a c \,x^{2}+21 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c -42 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, E\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c +15 B \sqrt {-a c}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{2}-8 B \,c^{3} x^{5}+18 A \,c^{3} x^{4}-42 a B \,c^{2} x^{3}+14 a A \,c^{2} x^{2}-30 a^{2} B c x \right ) e^{4} \sqrt {e x}}{12 x \,c^{4} \left (c \,x^{2}+a \right )^{\frac {3}{2}}}\)	\(607\)
risch	\(\text {Expression too large to display}\)	\(1019\)

[In]

int((e*x)^(9/2)*(B*x+A)/(c*x^2+a)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/e/x*(e*x)^(1/2)/(c*x^2+a)^(1/2)*((c*x^2+a)*e*x)^(1/2)*((1/3*a*e^4/c^4*A*x-1/3*a^2*e^4/c^5*B)*(c*e*x^3+a*e*x)
^(1/2)/(x^2+a/c)^2-2*x*e*c*(3/4*e^4*A/c^3*x-13/12*e^4*B*a/c^4)/((x^2+a/c)*x*e*c)^(1/2)+2/3*B/c^3*e^4*(c*e*x^3+
a*e*x)^(1/2)-5/4*B*a*e^5/c^4*(-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)*EllipticF(((x+(-a*c)^(1/2)/c)/(-a*c)^(1/2)*
c)^(1/2),1/2*2^(1/2))+7/4*A/c^3*e^5*(-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 = 0.11 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.58 \[ \int \frac {(e x)^{9/2} (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx=-\frac {15 \, {\left (B a c^{2} e^{4} x^{4} + 2 \, B a^{2} c e^{4} x^{2} + B a^{3} e^{4}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) + 21 \, {\left (A c^{3} e^{4} x^{4} + 2 \, A a c^{2} e^{4} x^{2} + A a^{2} c e^{4}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) - {\left (4 \, B c^{3} e^{4} x^{4} - 9 \, A c^{3} e^{4} x^{3} + 21 \, B a c^{2} e^{4} x^{2} - 7 \, A a c^{2} e^{4} x + 15 \, B a^{2} c e^{4}\right )} \sqrt {c x^{2} + a} \sqrt {e x}}{6 \, {\left (c^{6} x^{4} + 2 \, a c^{5} x^{2} + a^{2} c^{4}\right )}} \]

[In]

integrate((e*x)^(9/2)*(B*x+A)/(c*x^2+a)^(5/2),x, algorithm="fricas")

[Out]

-1/6*(15*(B*a*c^2*e^4*x^4 + 2*B*a^2*c*e^4*x^2 + B*a^3*e^4)*sqrt(c*e)*weierstrassPInverse(-4*a/c, 0, x) + 21*(A
*c^3*e^4*x^4 + 2*A*a*c^2*e^4*x^2 + A*a^2*c*e^4)*sqrt(c*e)*weierstrassZeta(-4*a/c, 0, weierstrassPInverse(-4*a/
c, 0, x)) - (4*B*c^3*e^4*x^4 - 9*A*c^3*e^4*x^3 + 21*B*a*c^2*e^4*x^2 - 7*A*a*c^2*e^4*x + 15*B*a^2*c*e^4)*sqrt(c
*x^2 + a)*sqrt(e*x))/(c^6*x^4 + 2*a*c^5*x^2 + a^2*c^4)

Sympy [F(-1)]

Timed out. \[ \int \frac {(e x)^{9/2} (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate((e*x)**(9/2)*(B*x+A)/(c*x**2+a)**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(e x)^{9/2} (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (B x + A\right )} \left (e x\right )^{\frac {9}{2}}}{{\left (c x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((e*x)^(9/2)*(B*x+A)/(c*x^2+a)^(5/2),x, algorithm="maxima")

[Out]

integrate((B*x + A)*(e*x)^(9/2)/(c*x^2 + a)^(5/2), x)

Giac [F]

\[ \int \frac {(e x)^{9/2} (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (B x + A\right )} \left (e x\right )^{\frac {9}{2}}}{{\left (c x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((e*x)^(9/2)*(B*x+A)/(c*x^2+a)^(5/2),x, algorithm="giac")

[Out]

integrate((B*x + A)*(e*x)^(9/2)/(c*x^2 + a)^(5/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^{9/2} (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx=\int \frac {{\left (e\,x\right )}^{9/2}\,\left (A+B\,x\right )}{{\left (c\,x^2+a\right )}^{5/2}} \,d x \]

[In]

int(((e*x)^(9/2)*(A + B*x))/(a + c*x^2)^(5/2),x)

[Out]

int(((e*x)^(9/2)*(A + B*x))/(a + c*x^2)^(5/2), x)