3.5.74 \(\int \frac {(e x)^{13/2} (A+B x)}{(a+c x^2)^{5/2}} \, dx\) [474]

Optimal. Leaf size=428 \[ -\frac {e (e x)^{11/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}-\frac {e^3 (e x)^{7/2} (11 A+13 B x)}{6 c^2 \sqrt {a+c x^2}}-\frac {65 a B e^6 \sqrt {e x} \sqrt {a+c x^2}}{14 c^4}+\frac {77 A e^5 (e x)^{3/2} \sqrt {a+c x^2}}{30 c^3}+\frac {39 B e^4 (e x)^{5/2} \sqrt {a+c x^2}}{14 c^3}-\frac {77 a A e^7 x \sqrt {a+c x^2}}{10 c^{7/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {77 a^{5/4} A e^7 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{10 c^{15/4} \sqrt {e x} \sqrt {a+c x^2}}+\frac {a^{5/4} \left (325 \sqrt {a} B-539 A \sqrt {c}\right ) e^7 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{140 c^{17/4} \sqrt {e x} \sqrt {a+c x^2}} \]

[Out]

-1/3*e*(e*x)^(11/2)*(B*x+A)/c/(c*x^2+a)^(3/2)-1/6*e^3*(e*x)^(7/2)*(13*B*x+11*A)/c^2/(c*x^2+a)^(1/2)+77/30*A*e^
5*(e*x)^(3/2)*(c*x^2+a)^(1/2)/c^3+39/14*B*e^4*(e*x)^(5/2)*(c*x^2+a)^(1/2)/c^3-77/10*a*A*e^7*x*(c*x^2+a)^(1/2)/
c^(7/2)/(a^(1/2)+x*c^(1/2))/(e*x)^(1/2)-65/14*a*B*e^6*(e*x)^(1/2)*(c*x^2+a)^(1/2)/c^4+77/10*a^(5/4)*A*e^7*(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^(15/
4)/(e*x)^(1/2)/(c*x^2+a)^(1/2)+1/140*a^(5/4)*e^7*(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))*(325*B*a^(1/2)-539*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^(17/4)/(e*x)^(1/2)/(c*x^2+a)^(1
/2)

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Rubi [A]
time = 0.38, antiderivative size = 428, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {833, 847, 856, 854, 1212, 226, 1210} \begin {gather*} \frac {a^{5/4} e^7 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (325 \sqrt {a} B-539 A \sqrt {c}\right ) F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{140 c^{17/4} \sqrt {e x} \sqrt {a+c x^2}}+\frac {77 a^{5/4} A e^7 \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 )}{10 c^{15/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {e^3 (e x)^{7/2} (11 A+13 B x)}{6 c^2 \sqrt {a+c x^2}}-\frac {e (e x)^{11/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}-\frac {77 a A e^7 x \sqrt {a+c x^2}}{10 c^{7/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {77 A e^5 (e x)^{3/2} \sqrt {a+c x^2}}{30 c^3}-\frac {65 a B e^6 \sqrt {e x} \sqrt {a+c x^2}}{14 c^4}+\frac {39 B e^4 (e x)^{5/2} \sqrt {a+c x^2}}{14 c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((e*x)^(13/2)*(A + B*x))/(a + c*x^2)^(5/2),x]

[Out]

-1/3*(e*(e*x)^(11/2)*(A + B*x))/(c*(a + c*x^2)^(3/2)) - (e^3*(e*x)^(7/2)*(11*A + 13*B*x))/(6*c^2*Sqrt[a + c*x^
2]) - (65*a*B*e^6*Sqrt[e*x]*Sqrt[a + c*x^2])/(14*c^4) + (77*A*e^5*(e*x)^(3/2)*Sqrt[a + c*x^2])/(30*c^3) + (39*
B*e^4*(e*x)^(5/2)*Sqrt[a + c*x^2])/(14*c^3) - (77*a*A*e^7*x*Sqrt[a + c*x^2])/(10*c^(7/2)*Sqrt[e*x]*(Sqrt[a] +
Sqrt[c]*x)) + (77*a^(5/4)*A*e^7*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*Ellipt
icE[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(10*c^(15/4)*Sqrt[e*x]*Sqrt[a + c*x^2]) + (a^(5/4)*(325*Sqrt[a]
*B - 539*A*Sqrt[c])*e^7*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticF[2*Ar
cTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(140*c^(17/4)*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*} \int \frac {(e x)^{13/2} (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx &=-\frac {e (e x)^{11/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}+\frac {\int \frac {(e x)^{9/2} \left (\frac {11}{2} a A e^2+\frac {13}{2} a B e^2 x\right )}{\left (a+c x^2\right )^{3/2}} \, dx}{3 a c}\\ &=-\frac {e (e x)^{11/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}-\frac {e^3 (e x)^{7/2} (11 A+13 B x)}{6 c^2 \sqrt {a+c x^2}}+\frac {\int \frac {(e x)^{5/2} \left (\frac {77}{4} a^2 A e^4+\frac {117}{4} a^2 B e^4 x\right )}{\sqrt {a+c x^2}} \, dx}{3 a^2 c^2}\\ &=-\frac {e (e x)^{11/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}-\frac {e^3 (e x)^{7/2} (11 A+13 B x)}{6 c^2 \sqrt {a+c x^2}}+\frac {39 B e^4 (e x)^{5/2} \sqrt {a+c x^2}}{14 c^3}+\frac {2 \int \frac {(e x)^{3/2} \left (-\frac {585}{8} a^3 B e^5+\frac {539}{8} a^2 A c e^5 x\right )}{\sqrt {a+c x^2}} \, dx}{21 a^2 c^3}\\ &=-\frac {e (e x)^{11/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}-\frac {e^3 (e x)^{7/2} (11 A+13 B x)}{6 c^2 \sqrt {a+c x^2}}+\frac {77 A e^5 (e x)^{3/2} \sqrt {a+c x^2}}{30 c^3}+\frac {39 B e^4 (e x)^{5/2} \sqrt {a+c x^2}}{14 c^3}+\frac {4 \int \frac {\sqrt {e x} \left (-\frac {1617}{16} a^3 A c e^6-\frac {2925}{16} a^3 B c e^6 x\right )}{\sqrt {a+c x^2}} \, dx}{105 a^2 c^4}\\ &=-\frac {e (e x)^{11/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}-\frac {e^3 (e x)^{7/2} (11 A+13 B x)}{6 c^2 \sqrt {a+c x^2}}-\frac {65 a B e^6 \sqrt {e x} \sqrt {a+c x^2}}{14 c^4}+\frac {77 A e^5 (e x)^{3/2} \sqrt {a+c x^2}}{30 c^3}+\frac {39 B e^4 (e x)^{5/2} \sqrt {a+c x^2}}{14 c^3}+\frac {8 \int \frac {\frac {2925}{32} a^4 B c e^7-\frac {4851}{32} a^3 A c^2 e^7 x}{\sqrt {e x} \sqrt {a+c x^2}} \, dx}{315 a^2 c^5}\\ &=-\frac {e (e x)^{11/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}-\frac {e^3 (e x)^{7/2} (11 A+13 B x)}{6 c^2 \sqrt {a+c x^2}}-\frac {65 a B e^6 \sqrt {e x} \sqrt {a+c x^2}}{14 c^4}+\frac {77 A e^5 (e x)^{3/2} \sqrt {a+c x^2}}{30 c^3}+\frac {39 B e^4 (e x)^{5/2} \sqrt {a+c x^2}}{14 c^3}+\frac {\left (8 \sqrt {x}\right ) \int \frac {\frac {2925}{32} a^4 B c e^7-\frac {4851}{32} a^3 A c^2 e^7 x}{\sqrt {x} \sqrt {a+c x^2}} \, dx}{315 a^2 c^5 \sqrt {e x}}\\ &=-\frac {e (e x)^{11/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}-\frac {e^3 (e x)^{7/2} (11 A+13 B x)}{6 c^2 \sqrt {a+c x^2}}-\frac {65 a B e^6 \sqrt {e x} \sqrt {a+c x^2}}{14 c^4}+\frac {77 A e^5 (e x)^{3/2} \sqrt {a+c x^2}}{30 c^3}+\frac {39 B e^4 (e x)^{5/2} \sqrt {a+c x^2}}{14 c^3}+\frac {\left (16 \sqrt {x}\right ) \text {Subst}\left (\int \frac {\frac {2925}{32} a^4 B c e^7-\frac {4851}{32} a^3 A c^2 e^7 x^2}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{315 a^2 c^5 \sqrt {e x}}\\ &=-\frac {e (e x)^{11/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}-\frac {e^3 (e x)^{7/2} (11 A+13 B x)}{6 c^2 \sqrt {a+c x^2}}-\frac {65 a B e^6 \sqrt {e x} \sqrt {a+c x^2}}{14 c^4}+\frac {77 A e^5 (e x)^{3/2} \sqrt {a+c x^2}}{30 c^3}+\frac {39 B e^4 (e x)^{5/2} \sqrt {a+c x^2}}{14 c^3}+\frac {\left (a^{3/2} \left (325 \sqrt {a} B-539 A \sqrt {c}\right ) e^7 \sqrt {x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{70 c^4 \sqrt {e x}}+\frac {\left (77 a^{3/2} A e^7 \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 )}{10 c^{7/2} \sqrt {e x}}\\ &=-\frac {e (e x)^{11/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}-\frac {e^3 (e x)^{7/2} (11 A+13 B x)}{6 c^2 \sqrt {a+c x^2}}-\frac {65 a B e^6 \sqrt {e x} \sqrt {a+c x^2}}{14 c^4}+\frac {77 A e^5 (e x)^{3/2} \sqrt {a+c x^2}}{30 c^3}+\frac {39 B e^4 (e x)^{5/2} \sqrt {a+c x^2}}{14 c^3}-\frac {77 a A e^7 x \sqrt {a+c x^2}}{10 c^{7/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {77 a^{5/4} A e^7 \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 )}{10 c^{15/4} \sqrt {e x} \sqrt {a+c x^2}}+\frac {a^{5/4} \left (325 \sqrt {a} B-539 A \sqrt {c}\right ) e^7 \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 )}{140 c^{17/4} \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 = 183, normalized size = 0.43 \begin {gather*} \frac {e^6 \sqrt {e x} \left (-975 a^3 B+539 a^2 A c x-1365 a^2 B c x^2+693 a A c^2 x^3-260 a B c^2 x^4+84 A c^3 x^5+60 B c^3 x^6+975 a^2 B \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 )-539 a A c 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 )}{210 c^4 \left (a+c x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((e*x)^(13/2)*(A + B*x))/(a + c*x^2)^(5/2),x]

[Out]

(e^6*Sqrt[e*x]*(-975*a^3*B + 539*a^2*A*c*x - 1365*a^2*B*c*x^2 + 693*a*A*c^2*x^3 - 260*a*B*c^2*x^4 + 84*A*c^3*x
^5 + 60*B*c^3*x^6 + 975*a^2*B*(a + c*x^2)*Sqrt[1 + (c*x^2)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^2)/a)] -
539*a*A*c*x*(a + c*x^2)*Sqrt[1 + (c*x^2)/a]*Hypergeometric2F1[1/2, 3/4, 7/4, -((c*x^2)/a)]))/(210*c^4*(a + c*
x^2)^(3/2))

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Maple [A]
time = 0.64, size = 637, normalized size = 1.49 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^(13/2)*(B*x+A)/(c*x^2+a)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/420*(1617*A*2^(1/2)*((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))*a^2*c^2*x^2-3234*A*2^(1/2)*(
(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)*Elli
pticE(((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2),1/2*2^(1/2))*a^2*c^2*x^2+975*B*2^(1/2)*((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))*(-a*c)^(1/2)*a^2*c*x^2+120*B*c^4*x^7+168*A*c^4*x^6+1617*A*2^(1/2)*((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))*a^3*c-3234*A*2^(1/2)*((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))*a^3*c+975*B*(-a*c)^(1/2)*2^(1/2)*((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))*a^3-520*a*B*c^3*x^5+1386*a*A*c^3*x^4-2730*a^2*B*c^2*x^3+1078*a^2*A*c^2*x^2-1950*a^3*B*c*x)*e^6/x*(e*x)^(1
/2)/c^5/(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)^(13/2)*(B*x+A)/(c*x^2+a)^(5/2),x, algorithm="maxima")

[Out]

e^(13/2)*integrate((B*x + A)*x^(13/2)/(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 = 1.38, size = 207, normalized size = 0.48 \begin {gather*} \frac {975 \, {\left (B a^{2} c^{2} x^{4} + 2 \, B a^{3} c x^{2} + B a^{4}\right )} \sqrt {c} e^{\frac {13}{2}} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) + 1617 \, {\left (A a c^{3} x^{4} + 2 \, A a^{2} c^{2} x^{2} + A a^{3} c\right )} \sqrt {c} e^{\frac {13}{2}} {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) + {\left (60 \, B c^{4} x^{6} + 84 \, A c^{4} x^{5} - 260 \, B a c^{3} x^{4} + 693 \, A a c^{3} x^{3} - 1365 \, B a^{2} c^{2} x^{2} + 539 \, A a^{2} c^{2} x - 975 \, B a^{3} c\right )} \sqrt {c x^{2} + a} \sqrt {x} e^{\frac {13}{2}}}{210 \, {\left (c^{7} x^{4} + 2 \, a c^{6} x^{2} + a^{2} c^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^(13/2)*(B*x+A)/(c*x^2+a)^(5/2),x, algorithm="fricas")

[Out]

1/210*(975*(B*a^2*c^2*x^4 + 2*B*a^3*c*x^2 + B*a^4)*sqrt(c)*e^(13/2)*weierstrassPInverse(-4*a/c, 0, x) + 1617*(
A*a*c^3*x^4 + 2*A*a^2*c^2*x^2 + A*a^3*c)*sqrt(c)*e^(13/2)*weierstrassZeta(-4*a/c, 0, weierstrassPInverse(-4*a/
c, 0, x)) + (60*B*c^4*x^6 + 84*A*c^4*x^5 - 260*B*a*c^3*x^4 + 693*A*a*c^3*x^3 - 1365*B*a^2*c^2*x^2 + 539*A*a^2*
c^2*x - 975*B*a^3*c)*sqrt(c*x^2 + a)*sqrt(x)*e^(13/2))/(c^7*x^4 + 2*a*c^6*x^2 + a^2*c^5)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**(13/2)*(B*x+A)/(c*x**2+a)**(5/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 4497 deep

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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)^(13/2)*(B*x+A)/(c*x^2+a)^(5/2),x, algorithm="giac")

[Out]

integrate((B*x + A)*x^(13/2)*e^(13/2)/(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 )}^{13/2}\,\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)^(13/2)*(A + B*x))/(a + c*x^2)^(5/2),x)

[Out]

int(((e*x)^(13/2)*(A + B*x))/(a + c*x^2)^(5/2), x)

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