\(\int \frac {(e x)^{5/2} (A+B x)}{(a+c x^2)^{3/2}} \, dx\) [467]

Optimal result

Integrand size = 24, antiderivative size = 326 \[ \int \frac {(e x)^{5/2} (A+B x)}{\left (a+c x^2\right )^{3/2}} \, dx=-\frac {e (e x)^{3/2} (A+B x)}{c \sqrt {a+c x^2}}+\frac {5 B e^2 \sqrt {e x} \sqrt {a+c x^2}}{3 c^2}+\frac {3 A e^3 x \sqrt {a+c x^2}}{c^{3/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {3 \sqrt [4]{a} A e^3 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{c^{7/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {\sqrt [4]{a} \left (5 \sqrt {a} B-9 A \sqrt {c}\right ) e^3 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{6 c^{9/4} \sqrt {e x} \sqrt {a+c x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {833, 847, 856, 854, 1212, 226, 1210} \[ \int \frac {(e x)^{5/2} (A+B x)}{\left (a+c x^2\right )^{3/2}} \, dx=-\frac {\sqrt [4]{a} e^3 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (5 \sqrt {a} B-9 A \sqrt {c}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{6 c^{9/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {3 \sqrt [4]{a} A e^3 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{c^{7/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {e (e x)^{3/2} (A+B x)}{c \sqrt {a+c x^2}}+\frac {3 A e^3 x \sqrt {a+c x^2}}{c^{3/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {5 B e^2 \sqrt {e x} \sqrt {a+c x^2}}{3 c^2} \]

[In]

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

[Out]

-((e*(e*x)^(3/2)*(A + B*x))/(c*Sqrt[a + c*x^2])) + (5*B*e^2*Sqrt[e*x]*Sqrt[a + c*x^2])/(3*c^2) + (3*A*e^3*x*Sq
rt[a + c*x^2])/(c^(3/2)*Sqrt[e*x]*(Sqrt[a] + Sqrt[c]*x)) - (3*a^(1/4)*A*e^3*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])/(c^(7/4)*Sqrt[e*x]*
Sqrt[a + c*x^2]) - (a^(1/4)*(5*Sqrt[a]*B - 9*A*Sqrt[c])*e^3*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sq
rt[a] + Sqrt[c]*x)^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(6*c^(9/4)*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)^{3/2} (A+B x)}{c \sqrt {a+c x^2}}+\frac {\int \frac {\sqrt {e x} \left (\frac {3}{2} a A e^2+\frac {5}{2} a B e^2 x\right )}{\sqrt {a+c x^2}} \, dx}{a c} \\ & = -\frac {e (e x)^{3/2} (A+B x)}{c \sqrt {a+c x^2}}+\frac {5 B e^2 \sqrt {e x} \sqrt {a+c x^2}}{3 c^2}+\frac {2 \int \frac {-\frac {5}{4} a^2 B e^3+\frac {9}{4} a A c e^3 x}{\sqrt {e x} \sqrt {a+c x^2}} \, dx}{3 a c^2} \\ & = -\frac {e (e x)^{3/2} (A+B x)}{c \sqrt {a+c x^2}}+\frac {5 B e^2 \sqrt {e x} \sqrt {a+c x^2}}{3 c^2}+\frac {\left (2 \sqrt {x}\right ) \int \frac {-\frac {5}{4} a^2 B e^3+\frac {9}{4} a A c e^3 x}{\sqrt {x} \sqrt {a+c x^2}} \, dx}{3 a c^2 \sqrt {e x}} \\ & = -\frac {e (e x)^{3/2} (A+B x)}{c \sqrt {a+c x^2}}+\frac {5 B e^2 \sqrt {e x} \sqrt {a+c x^2}}{3 c^2}+\frac {\left (4 \sqrt {x}\right ) \text {Subst}\left (\int \frac {-\frac {5}{4} a^2 B e^3+\frac {9}{4} a A c e^3 x^2}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{3 a c^2 \sqrt {e x}} \\ & = -\frac {e (e x)^{3/2} (A+B x)}{c \sqrt {a+c x^2}}+\frac {5 B e^2 \sqrt {e x} \sqrt {a+c x^2}}{3 c^2}-\frac {\left (\sqrt {a} \left (5 \sqrt {a} B-9 A \sqrt {c}\right ) e^3 \sqrt {x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{3 c^2 \sqrt {e x}}-\frac {\left (3 \sqrt {a} A e^3 \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 )}{c^{3/2} \sqrt {e x}} \\ & = -\frac {e (e x)^{3/2} (A+B x)}{c \sqrt {a+c x^2}}+\frac {5 B e^2 \sqrt {e x} \sqrt {a+c x^2}}{3 c^2}+\frac {3 A e^3 x \sqrt {a+c x^2}}{c^{3/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {3 \sqrt [4]{a} A e^3 \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 )}{c^{7/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {\sqrt [4]{a} \left (5 \sqrt {a} B-9 A \sqrt {c}\right ) e^3 \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 )}{6 c^{9/4} \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 = 10.04 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.37 \[ \int \frac {(e x)^{5/2} (A+B x)}{\left (a+c x^2\right )^{3/2}} \, dx=\frac {e^2 \sqrt {e x} \left (5 a B-3 A c x+2 B c x^2-5 a B \sqrt {1+\frac {c x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {c x^2}{a}\right )+3 A c x \sqrt {1+\frac {c x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-\frac {c x^2}{a}\right )\right )}{3 c^2 \sqrt {a+c x^2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.56 (sec) , antiderivative size = 308, normalized size of antiderivative = 0.94

method	result	size
default	\(-\frac {e^{2} \sqrt {e x}\, \left (9 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a c -18 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, E\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a c +5 B \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a c}\, a -4 B \,c^{2} x^{3}+6 A \,c^{2} x^{2}-10 a B c x \right )}{6 x \sqrt {c \,x^{2}+a}\, c^{3}}\)	\(308\)
elliptic	\(\frac {\sqrt {e x}\, \sqrt {\left (c \,x^{2}+a \right ) e x}\, \left (-\frac {2 x e c \left (\frac {e^{2} A x}{2 c^{2}}-\frac {e^{2} B a}{2 c^{3}}\right )}{\sqrt {\left (x^{2}+\frac {a}{c}\right ) x e c}}+\frac {2 B \,e^{2} \sqrt {c e \,x^{3}+a e x}}{3 c^{2}}-\frac {5 B a \,e^{3} \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{6 c^{3} \sqrt {c e \,x^{3}+a e x}}+\frac {3 A \,e^{3} \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{2 c^{2} \sqrt {c e \,x^{3}+a e x}}\right )}{e x \sqrt {c \,x^{2}+a}}\)	\(377\)
risch	\(\frac {2 B x \sqrt {c \,x^{2}+a}\, e^{3}}{3 c^{2} \sqrt {e x}}+\frac {\left (-\frac {4 B 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}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c \sqrt {c e \,x^{3}+a e x}}+\frac {3 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}}-3 a \left (-\frac {2 c e x \left (-\frac {A x}{2 a e}+\frac {B}{2 e c}\right )}{\sqrt {\left (x^{2}+\frac {a}{c}\right ) x e c}}-\frac {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 )}{2 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 )}{2 a \sqrt {c e \,x^{3}+a e x}}\right )\right ) e^{3} \sqrt {\left (c \,x^{2}+a \right ) e x}}{3 c^{2} \sqrt {e x}\, \sqrt {c \,x^{2}+a}}\)	\(647\)

[In]

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

[Out]

-1/6*e^2/x*(e*x)^(1/2)*(9*A*((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2)*2^(1/2)*((-c*x+(-a*c)^(1/2))/(-a*c)^(1/2))
^(1/2)*(-x/(-a*c)^(1/2)*c)^(1/2)*EllipticF(((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2),1/2*2^(1/2))*a*c-18*A*((c*x
+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2)*2^(1/2)*((-c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2)*(-x/(-a*c)^(1/2)*c)^(1/2)*
EllipticE(((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2),1/2*2^(1/2))*a*c+5*B*((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2)
*2^(1/2)*((-c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2)*(-x/(-a*c)^(1/2)*c)^(1/2)*EllipticF(((c*x+(-a*c)^(1/2))/(-a*
c)^(1/2))^(1/2),1/2*2^(1/2))*(-a*c)^(1/2)*a-4*B*c^2*x^3+6*A*c^2*x^2-10*a*B*c*x)/(c*x^2+a)^(1/2)/c^3

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.10 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.44 \[ \int \frac {(e x)^{5/2} (A+B x)}{\left (a+c x^2\right )^{3/2}} \, dx=-\frac {5 \, {\left (B a c e^{2} x^{2} + B a^{2} e^{2}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) + 9 \, {\left (A c^{2} e^{2} x^{2} + A a c e^{2}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) - {\left (2 \, B c^{2} e^{2} x^{2} - 3 \, A c^{2} e^{2} x + 5 \, B a c e^{2}\right )} \sqrt {c x^{2} + a} \sqrt {e x}}{3 \, {\left (c^{4} x^{2} + a c^{3}\right )}} \]

[In]

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

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 46.87 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.29 \[ \int \frac {(e x)^{5/2} (A+B x)}{\left (a+c x^2\right )^{3/2}} \, dx=\frac {A e^{\frac {5}{2}} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} \Gamma \left (\frac {11}{4}\right )} + \frac {B e^{\frac {5}{2}} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} \Gamma \left (\frac {13}{4}\right )} \]

[In]

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

[Out]

A*e**(5/2)*x**(7/2)*gamma(7/4)*hyper((3/2, 7/4), (11/4,), c*x**2*exp_polar(I*pi)/a)/(2*a**(3/2)*gamma(11/4)) +
B*e**(5/2)*x**(9/2)*gamma(9/4)*hyper((3/2, 9/4), (13/4,), c*x**2*exp_polar(I*pi)/a)/(2*a**(3/2)*gamma(13/4))

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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