3.5.79 \(\int \frac {(e x)^{3/2} (A+B x)}{(a+c x^2)^{5/2}} \, dx\) [479]

Optimal. Leaf size=341 \[ -\frac {e \sqrt {e x} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}+\frac {e \sqrt {e x} (A+3 B x)}{6 a c \sqrt {a+c x^2}}-\frac {B e^2 x \sqrt {a+c x^2}}{2 a c^{3/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {B e^2 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{3/4} c^{7/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {\left (3 \sqrt {a} B-A \sqrt {c}\right ) e^2 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{12 a^{5/4} c^{7/4} \sqrt {e x} \sqrt {a+c x^2}} \]

[Out]

-1/3*e*(B*x+A)*(e*x)^(1/2)/c/(c*x^2+a)^(3/2)+1/6*e*(3*B*x+A)*(e*x)^(1/2)/a/c/(c*x^2+a)^(1/2)-1/2*B*e^2*x*(c*x^
2+a)^(1/2)/a/c^(3/2)/(a^(1/2)+x*c^(1/2))/(e*x)^(1/2)+1/2*B*e^2*(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)/a^(3/4)/c^(7/4)/(e*x)^(1/2)/(c*x^2+a)^(1/2)-1/12*
e^2*(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*ar
ctan(c^(1/4)*x^(1/2)/a^(1/4))),1/2*2^(1/2))*(3*B*a^(1/2)-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^(5/4)/c^(7/4)/(e*x)^(1/2)/(c*x^2+a)^(1/2)

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Rubi [A]
time = 0.23, antiderivative size = 341, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {833, 837, 856, 854, 1212, 226, 1210} \begin {gather*} -\frac {e^2 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (3 \sqrt {a} B-A \sqrt {c}\right ) F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{12 a^{5/4} c^{7/4} \sqrt {e x} \sqrt {a+c x^2}}+\frac {B e^2 \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 \text {ArcTan}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{3/4} c^{7/4} \sqrt {e x} \sqrt {a+c x^2}}+\frac {e \sqrt {e x} (A+3 B x)}{6 a c \sqrt {a+c x^2}}-\frac {e \sqrt {e x} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}-\frac {B e^2 x \sqrt {a+c x^2}}{2 a c^{3/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(e*Sqrt[e*x]*(A + B*x))/(c*(a + c*x^2)^(3/2)) + (e*Sqrt[e*x]*(A + 3*B*x))/(6*a*c*Sqrt[a + c*x^2]) - (B*e^
2*x*Sqrt[a + c*x^2])/(2*a*c^(3/2)*Sqrt[e*x]*(Sqrt[a] + Sqrt[c]*x)) + (B*e^2*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^(3/4)*c^(7/4)*
Sqrt[e*x]*Sqrt[a + c*x^2]) - ((3*Sqrt[a]*B - A*Sqrt[c])*e^2*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])/(12*a^(5/4)*c^(7/4)*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 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 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*} \int \frac {(e x)^{3/2} (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx &=-\frac {e \sqrt {e x} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}+\frac {\int \frac {\frac {1}{2} a A e^2+\frac {3}{2} a B e^2 x}{\sqrt {e x} \left (a+c x^2\right )^{3/2}} \, dx}{3 a c}\\ &=-\frac {e \sqrt {e x} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}+\frac {e \sqrt {e x} (A+3 B x)}{6 a c \sqrt {a+c x^2}}-\frac {\int \frac {-\frac {1}{4} a^2 A c e^4+\frac {3}{4} a^2 B c e^4 x}{\sqrt {e x} \sqrt {a+c x^2}} \, dx}{3 a^3 c^2 e^2}\\ &=-\frac {e \sqrt {e x} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}+\frac {e \sqrt {e x} (A+3 B x)}{6 a c \sqrt {a+c x^2}}-\frac {\sqrt {x} \int \frac {-\frac {1}{4} a^2 A c e^4+\frac {3}{4} a^2 B c e^4 x}{\sqrt {x} \sqrt {a+c x^2}} \, dx}{3 a^3 c^2 e^2 \sqrt {e x}}\\ &=-\frac {e \sqrt {e x} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}+\frac {e \sqrt {e x} (A+3 B x)}{6 a c \sqrt {a+c x^2}}-\frac {\left (2 \sqrt {x}\right ) \text {Subst}\left (\int \frac {-\frac {1}{4} a^2 A c e^4+\frac {3}{4} a^2 B c e^4 x^2}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{3 a^3 c^2 e^2 \sqrt {e x}}\\ &=-\frac {e \sqrt {e x} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}+\frac {e \sqrt {e x} (A+3 B x)}{6 a c \sqrt {a+c x^2}}+\frac {\left (B e^2 \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 \sqrt {a} c^{3/2} \sqrt {e x}}-\frac {\left (\left (3 \sqrt {a} B-A \sqrt {c}\right ) e^2 \sqrt {x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{6 a c^{3/2} \sqrt {e x}}\\ &=-\frac {e \sqrt {e x} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}+\frac {e \sqrt {e x} (A+3 B x)}{6 a c \sqrt {a+c x^2}}-\frac {B e^2 x \sqrt {a+c x^2}}{2 a c^{3/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {B e^2 \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^{3/4} c^{7/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {\left (3 \sqrt {a} B-A \sqrt {c}\right ) e^2 \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^{5/4} c^{7/4} \sqrt {e x} \sqrt {a+c x^2}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.12, size = 137, normalized size = 0.40 \begin {gather*} \frac {e \sqrt {e x} \left (-a A+a B x+A c x^2+3 B c x^3+A \left (a+c x^2\right ) \sqrt {1+\frac {c x^2}{a}} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\frac {c x^2}{a}\right )-B x \left (a+c x^2\right ) \sqrt {1+\frac {c x^2}{a}} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-\frac {c x^2}{a}\right )\right )}{6 a c \left (a+c x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.61, size = 583, normalized size = 1.71

method	result	size
elliptic	\(\frac {\sqrt {e x}\, \sqrt {\left (c \,x^{2}+a \right ) e x}\, \left (\frac {\left (-\frac {e B x}{3 c^{3}}-\frac {e A}{3 c^{3}}\right ) \sqrt {c e \,x^{3}+a e x}}{\left (x^{2}+\frac {a}{c}\right )^{2}}-\frac {2 c e x \left (-\frac {e B x}{4 a \,c^{2}}-\frac {A e}{12 a \,c^{2}}\right )}{\sqrt {\left (x^{2}+\frac {a}{c}\right ) c e x}}+\frac {e^{2} A \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{12 a \,c^{2} \sqrt {c e \,x^{3}+a e x}}-\frac {e^{2} B \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{4 a \,c^{2} \sqrt {c e \,x^{3}+a e x}}\right )}{e x \sqrt {c \,x^{2}+a}}\)	\(402\)
default	\(\frac {\left (A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {-a c}\, c \,x^{2}-6 B \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \EllipticE \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a c \,x^{2}+3 B \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a c \,x^{2}+A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {-a c}\, a -6 B \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \EllipticE \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a^{2}+3 B \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a^{2}+6 B \,c^{2} x^{4}+2 A \,c^{2} x^{3}+2 a B c \,x^{2}-2 a A c x \right ) e \sqrt {e x}}{12 x \,c^{2} a \left (c \,x^{2}+a \right )^{\frac {3}{2}}}\)	\(583\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(A*((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(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))*2^(1/2)*(-a*c)^(1/2)*c*x^2-6*B*((c*x+(-a*c
)^(1/2))/(-a*c)^(1/2))^(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))*2^(1/2)*a*c*x^2+3*B*((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(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))*2^(1/2)*a*c*x^2+A*((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(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))*2^(1/2)*(-a*c
)^(1/2)*a-6*B*((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(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))*2^(1/2)*a^2+3*B*((c*x+(-a*c)^(1/2))/
(-a*c)^(1/2))^(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))*2^(1/2)*a^2+6*B*c^2*x^4+2*A*c^2*x^3+2*a*B*c*x^2-2*a*A*c*x)*e/x*(e*x)^(
1/2)/c^2/a/(c*x^2+a)^(3/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.53, size = 158, normalized size = 0.46 \begin {gather*} \frac {{\left (A c^{2} x^{4} + 2 \, A a c x^{2} + A a^{2}\right )} \sqrt {c} e^{\frac {3}{2}} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) + 3 \, {\left (B c^{2} x^{4} + 2 \, B a c x^{2} + B a^{2}\right )} \sqrt {c} e^{\frac {3}{2}} {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) + {\left (3 \, B c^{2} x^{3} + A c^{2} x^{2} + B a c x - A a c\right )} \sqrt {c x^{2} + a} \sqrt {x} e^{\frac {3}{2}}}{6 \, {\left (a c^{4} x^{4} + 2 \, a^{2} c^{3} x^{2} + a^{3} c^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C] Result contains complex when optimal does not.
time = 74.57, size = 94, normalized size = 0.28 \begin {gather*} \frac {A e^{\frac {3}{2}} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, \frac {5}{2} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {5}{2}} \Gamma \left (\frac {9}{4}\right )} + \frac {B e^{\frac {3}{2}} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {7}{4}, \frac {5}{2} \\ \frac {11}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {5}{2}} \Gamma \left (\frac {11}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (e\,x\right )}^{3/2}\,\left (A+B\,x\right )}{{\left (c\,x^2+a\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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