3.5.50 \(\int (e x)^{3/2} (A+B x) (a+c x^2)^{5/2} \, dx\) [450]

Optimal. Leaf size=437 \[ -\frac {8 a^3 e \sqrt {e x} (221 A+231 B x) \sqrt {a+c x^2}}{51051 c}-\frac {16 a^4 B e^2 x \sqrt {a+c x^2}}{221 c^{3/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {4 a^2 e \sqrt {e x} (221 A+385 B x) \left (a+c x^2\right )^{3/2}}{51051 c}-\frac {2 a e \sqrt {e x} (221 A+495 B x) \left (a+c x^2\right )^{5/2}}{36465 c}+\frac {2 A e \sqrt {e x} \left (a+c x^2\right )^{7/2}}{15 c}+\frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{7/2}}{17 c}+\frac {16 a^{17/4} B e^2 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{221 c^{7/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {8 a^{15/4} \left (231 \sqrt {a} B+221 A \sqrt {c}\right ) e^2 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{51051 c^{7/4} \sqrt {e x} \sqrt {a+c x^2}} \]

[Out]

2/17*B*(e*x)^(3/2)*(c*x^2+a)^(7/2)/c-4/51051*a^2*e*(385*B*x+221*A)*(c*x^2+a)^(3/2)*(e*x)^(1/2)/c-2/36465*a*e*(
495*B*x+221*A)*(c*x^2+a)^(5/2)*(e*x)^(1/2)/c+2/15*A*e*(c*x^2+a)^(7/2)*(e*x)^(1/2)/c-16/221*a^4*B*e^2*x*(c*x^2+
a)^(1/2)/c^(3/2)/(a^(1/2)+x*c^(1/2))/(e*x)^(1/2)-8/51051*a^3*e*(231*B*x+221*A)*(e*x)^(1/2)*(c*x^2+a)^(1/2)/c+1
6/221*a^(17/4)*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)))*E
llipticE(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)-8/51051*a^(15/4)*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*arctan(c^(1/4)*x^(1/2)/a^(1/4))),1/2*2^(
1/2))*(231*B*a^(1/2)+221*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^(7/4
)/(e*x)^(1/2)/(c*x^2+a)^(1/2)

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Rubi [A]
time = 0.39, antiderivative size = 437, 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 = {847, 829, 856, 854, 1212, 226, 1210} \begin {gather*} -\frac {8 a^{15/4} 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 (231 \sqrt {a} B+221 A \sqrt {c}\right ) F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{51051 c^{7/4} \sqrt {e x} \sqrt {a+c x^2}}+\frac {16 a^{17/4} 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 )}{221 c^{7/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {16 a^4 B e^2 x \sqrt {a+c x^2}}{221 c^{3/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {8 a^3 e \sqrt {e x} \sqrt {a+c x^2} (221 A+231 B x)}{51051 c}-\frac {4 a^2 e \sqrt {e x} \left (a+c x^2\right )^{3/2} (221 A+385 B x)}{51051 c}-\frac {2 a e \sqrt {e x} \left (a+c x^2\right )^{5/2} (221 A+495 B x)}{36465 c}+\frac {2 A e \sqrt {e x} \left (a+c x^2\right )^{7/2}}{15 c}+\frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{7/2}}{17 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*a^3*e*Sqrt[e*x]*(221*A + 231*B*x)*Sqrt[a + c*x^2])/(51051*c) - (16*a^4*B*e^2*x*Sqrt[a + c*x^2])/(221*c^(3/
2)*Sqrt[e*x]*(Sqrt[a] + Sqrt[c]*x)) - (4*a^2*e*Sqrt[e*x]*(221*A + 385*B*x)*(a + c*x^2)^(3/2))/(51051*c) - (2*a
*e*Sqrt[e*x]*(221*A + 495*B*x)*(a + c*x^2)^(5/2))/(36465*c) + (2*A*e*Sqrt[e*x]*(a + c*x^2)^(7/2))/(15*c) + (2*
B*(e*x)^(3/2)*(a + c*x^2)^(7/2))/(17*c) + (16*a^(17/4)*B*e^2*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(S
qrt[a] + Sqrt[c]*x)^2]*EllipticE[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(221*c^(7/4)*Sqrt[e*x]*Sqrt[a + c*
x^2]) - (8*a^(15/4)*(231*Sqrt[a]*B + 221*A*Sqrt[c])*e^2*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])/(51051*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 829

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

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 (e x)^{3/2} (A+B x) \left (a+c x^2\right )^{5/2} \, dx &=\frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{7/2}}{17 c}+\frac {2 \int \sqrt {e x} \left (-\frac {3}{2} a B e+\frac {17}{2} A c e x\right ) \left (a+c x^2\right )^{5/2} \, dx}{17 c}\\ &=\frac {2 A e \sqrt {e x} \left (a+c x^2\right )^{7/2}}{15 c}+\frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{7/2}}{17 c}+\frac {4 \int \frac {\left (-\frac {17}{4} a A c e^2-\frac {45}{4} a B c e^2 x\right ) \left (a+c x^2\right )^{5/2}}{\sqrt {e x}} \, dx}{255 c^2}\\ &=-\frac {2 a e \sqrt {e x} (221 A+495 B x) \left (a+c x^2\right )^{5/2}}{36465 c}+\frac {2 A e \sqrt {e x} \left (a+c x^2\right )^{7/2}}{15 c}+\frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{7/2}}{17 c}+\frac {16 \int \frac {\left (-\frac {221}{8} a^2 A c^2 e^4-\frac {495}{8} a^2 B c^2 e^4 x\right ) \left (a+c x^2\right )^{3/2}}{\sqrt {e x}} \, dx}{7293 c^3 e^2}\\ &=-\frac {4 a^2 e \sqrt {e x} (221 A+385 B x) \left (a+c x^2\right )^{3/2}}{51051 c}-\frac {2 a e \sqrt {e x} (221 A+495 B x) \left (a+c x^2\right )^{5/2}}{36465 c}+\frac {2 A e \sqrt {e x} \left (a+c x^2\right )^{7/2}}{15 c}+\frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{7/2}}{17 c}+\frac {64 \int \frac {\left (-\frac {1989}{16} a^3 A c^3 e^6-\frac {3465}{16} a^3 B c^3 e^6 x\right ) \sqrt {a+c x^2}}{\sqrt {e x}} \, dx}{153153 c^4 e^4}\\ &=-\frac {8 a^3 e \sqrt {e x} (221 A+231 B x) \sqrt {a+c x^2}}{51051 c}-\frac {4 a^2 e \sqrt {e x} (221 A+385 B x) \left (a+c x^2\right )^{3/2}}{51051 c}-\frac {2 a e \sqrt {e x} (221 A+495 B x) \left (a+c x^2\right )^{5/2}}{36465 c}+\frac {2 A e \sqrt {e x} \left (a+c x^2\right )^{7/2}}{15 c}+\frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{7/2}}{17 c}+\frac {256 \int \frac {-\frac {9945}{32} a^4 A c^4 e^8-\frac {10395}{32} a^4 B c^4 e^8 x}{\sqrt {e x} \sqrt {a+c x^2}} \, dx}{2297295 c^5 e^6}\\ &=-\frac {8 a^3 e \sqrt {e x} (221 A+231 B x) \sqrt {a+c x^2}}{51051 c}-\frac {4 a^2 e \sqrt {e x} (221 A+385 B x) \left (a+c x^2\right )^{3/2}}{51051 c}-\frac {2 a e \sqrt {e x} (221 A+495 B x) \left (a+c x^2\right )^{5/2}}{36465 c}+\frac {2 A e \sqrt {e x} \left (a+c x^2\right )^{7/2}}{15 c}+\frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{7/2}}{17 c}+\frac {\left (256 \sqrt {x}\right ) \int \frac {-\frac {9945}{32} a^4 A c^4 e^8-\frac {10395}{32} a^4 B c^4 e^8 x}{\sqrt {x} \sqrt {a+c x^2}} \, dx}{2297295 c^5 e^6 \sqrt {e x}}\\ &=-\frac {8 a^3 e \sqrt {e x} (221 A+231 B x) \sqrt {a+c x^2}}{51051 c}-\frac {4 a^2 e \sqrt {e x} (221 A+385 B x) \left (a+c x^2\right )^{3/2}}{51051 c}-\frac {2 a e \sqrt {e x} (221 A+495 B x) \left (a+c x^2\right )^{5/2}}{36465 c}+\frac {2 A e \sqrt {e x} \left (a+c x^2\right )^{7/2}}{15 c}+\frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{7/2}}{17 c}+\frac {\left (512 \sqrt {x}\right ) \text {Subst}\left (\int \frac {-\frac {9945}{32} a^4 A c^4 e^8-\frac {10395}{32} a^4 B c^4 e^8 x^2}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{2297295 c^5 e^6 \sqrt {e x}}\\ &=-\frac {8 a^3 e \sqrt {e x} (221 A+231 B x) \sqrt {a+c x^2}}{51051 c}-\frac {4 a^2 e \sqrt {e x} (221 A+385 B x) \left (a+c x^2\right )^{3/2}}{51051 c}-\frac {2 a e \sqrt {e x} (221 A+495 B x) \left (a+c x^2\right )^{5/2}}{36465 c}+\frac {2 A e \sqrt {e x} \left (a+c x^2\right )^{7/2}}{15 c}+\frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{7/2}}{17 c}+\frac {\left (16 a^{9/2} 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 )}{221 c^{3/2} \sqrt {e x}}-\frac {\left (16 a^4 \left (231 \sqrt {a} B+221 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 )}{51051 c^{3/2} \sqrt {e x}}\\ &=-\frac {8 a^3 e \sqrt {e x} (221 A+231 B x) \sqrt {a+c x^2}}{51051 c}-\frac {16 a^4 B e^2 x \sqrt {a+c x^2}}{221 c^{3/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {4 a^2 e \sqrt {e x} (221 A+385 B x) \left (a+c x^2\right )^{3/2}}{51051 c}-\frac {2 a e \sqrt {e x} (221 A+495 B x) \left (a+c x^2\right )^{5/2}}{36465 c}+\frac {2 A e \sqrt {e x} \left (a+c x^2\right )^{7/2}}{15 c}+\frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{7/2}}{17 c}+\frac {16 a^{17/4} 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 )}{221 c^{7/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {8 a^{15/4} \left (231 \sqrt {a} B+221 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 )}{51051 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 = 124, normalized size = 0.28 \begin {gather*} \frac {2 e \sqrt {e x} \sqrt {a+c x^2} \left ((17 A+15 B x) \left (a+c x^2\right )^3 \sqrt {1+\frac {c x^2}{a}}-17 a^3 A \, _2F_1\left (-\frac {5}{2},\frac {1}{4};\frac {5}{4};-\frac {c x^2}{a}\right )-15 a^3 B x \, _2F_1\left (-\frac {5}{2},\frac {3}{4};\frac {7}{4};-\frac {c x^2}{a}\right )\right )}{255 c \sqrt {1+\frac {c x^2}{a}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*e*Sqrt[e*x]*Sqrt[a + c*x^2]*((17*A + 15*B*x)*(a + c*x^2)^3*Sqrt[1 + (c*x^2)/a] - 17*a^3*A*Hypergeometric2F1
[-5/2, 1/4, 5/4, -((c*x^2)/a)] - 15*a^3*B*x*Hypergeometric2F1[-5/2, 3/4, 7/4, -((c*x^2)/a)]))/(255*c*Sqrt[1 +
(c*x^2)/a])

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Maple [A]
time = 0.56, size = 390, normalized size = 0.89

method	result	size
default	\(-\frac {2 e \sqrt {e x}\, \left (-15015 B \,c^{5} x^{10}-17017 A \,c^{5} x^{9}-56595 a B \,c^{4} x^{8}-66521 A a \,c^{4} x^{7}-75845 B \,a^{2} c^{3} x^{6}+4420 A \sqrt {2}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a c}\, a^{4}-95251 A \,a^{2} c^{3} x^{5}+9240 B \sqrt {2}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \EllipticE \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{5}-4620 B \sqrt {2}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{5}-37345 B \,a^{3} c^{2} x^{4}-54587 A \,a^{3} c^{2} x^{3}-3080 B \,a^{4} c \,x^{2}-8840 A \,a^{4} c x \right )}{255255 x \sqrt {c \,x^{2}+a}\, c^{2}}\)	\(390\)
risch	\(\frac {2 \left (15015 B \,c^{3} x^{7}+17017 A \,c^{3} x^{6}+41580 a B \,c^{2} x^{5}+49504 a A \,c^{2} x^{4}+34265 a^{2} B c \,x^{3}+45747 a^{2} A c \,x^{2}+3080 B \,a^{3} x +8840 A \,a^{3}\right ) x \sqrt {c \,x^{2}+a}\, e^{2}}{255255 c \sqrt {e x}}-\frac {8 a^{4} \left (\frac {231 B \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{c \sqrt {c e \,x^{3}+a e x}}+\frac {221 A \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c \sqrt {c e \,x^{3}+a e x}}\right ) e^{2} \sqrt {\left (c \,x^{2}+a \right ) e x}}{51051 c \sqrt {e x}\, \sqrt {c \,x^{2}+a}}\)	\(407\)
elliptic	\(\frac {\sqrt {e x}\, \sqrt {\left (c \,x^{2}+a \right ) e x}\, \left (\frac {2 B \,c^{2} e \,x^{7} \sqrt {c e \,x^{3}+a e x}}{17}+\frac {2 A \,c^{2} e \,x^{6} \sqrt {c e \,x^{3}+a e x}}{15}+\frac {72 B c a e \,x^{5} \sqrt {c e \,x^{3}+a e x}}{221}+\frac {64 a A c e \,x^{4} \sqrt {c e \,x^{3}+a e x}}{165}+\frac {178 B \,a^{2} e \,x^{3} \sqrt {c e \,x^{3}+a e x}}{663}+\frac {138 a^{2} A e \,x^{2} \sqrt {c e \,x^{3}+a e x}}{385}+\frac {16 B \,a^{3} e x \sqrt {c e \,x^{3}+a e x}}{663 c}+\frac {16 A \,a^{3} e \sqrt {c e \,x^{3}+a e x}}{231 c}-\frac {8 A \,a^{4} e^{2} \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{231 c^{2} \sqrt {c e \,x^{3}+a e x}}-\frac {8 B \,a^{4} e^{2} \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{221 c^{2} \sqrt {c e \,x^{3}+a e x}}\right )}{e x \sqrt {c \,x^{2}+a}}\)	\(502\)

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]

-2/255255*e/x*(e*x)^(1/2)/(c*x^2+a)^(1/2)/c^2*(-15015*B*c^5*x^10-17017*A*c^5*x^9-56595*a*B*c^4*x^8-66521*A*a*c
^4*x^7-75845*B*a^2*c^3*x^6+4420*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*c)^(
1/2)*a^4-95251*A*a^2*c^3*x^5+9240*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)*EllipticE(((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2),1/2*2^(1/2))*a^5-46
20*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^5-37345*B*a^3*c^2*x^4-54587*A*a^3*c
^2*x^3-3080*B*a^4*c*x^2-8840*A*a^4*c*x)

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.47, size = 148, normalized size = 0.34 \begin {gather*} -\frac {2 \, {\left (8840 \, A a^{4} \sqrt {c} e^{\frac {3}{2}} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) - 9240 \, B a^{4} \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 (15015 \, B c^{4} x^{7} + 17017 \, A c^{4} x^{6} + 41580 \, B a c^{3} x^{5} + 49504 \, A a c^{3} x^{4} + 34265 \, B a^{2} c^{2} x^{3} + 45747 \, A a^{2} c^{2} x^{2} + 3080 \, B a^{3} c x + 8840 \, A a^{3} c\right )} \sqrt {c x^{2} + a} \sqrt {x} e^{\frac {3}{2}}\right )}}{255255 \, c^{2}} \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]

-2/255255*(8840*A*a^4*sqrt(c)*e^(3/2)*weierstrassPInverse(-4*a/c, 0, x) - 9240*B*a^4*sqrt(c)*e^(3/2)*weierstra
ssZeta(-4*a/c, 0, weierstrassPInverse(-4*a/c, 0, x)) - (15015*B*c^4*x^7 + 17017*A*c^4*x^6 + 41580*B*a*c^3*x^5
+ 49504*A*a*c^3*x^4 + 34265*B*a^2*c^2*x^3 + 45747*A*a^2*c^2*x^2 + 3080*B*a^3*c*x + 8840*A*a^3*c)*sqrt(c*x^2 +
a)*sqrt(x)*e^(3/2))/c^2

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Sympy [C] Result contains complex when optimal does not.
time = 35.51, size = 301, normalized size = 0.69 \begin {gather*} \frac {A a^{\frac {5}{2}} e^{\frac {3}{2}} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {9}{4}\right )} + \frac {A a^{\frac {3}{2}} c e^{\frac {3}{2}} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{\Gamma \left (\frac {13}{4}\right )} + \frac {A \sqrt {a} c^{2} e^{\frac {3}{2}} x^{\frac {13}{2}} \Gamma \left (\frac {13}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {13}{4} \\ \frac {17}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {17}{4}\right )} + \frac {B a^{\frac {5}{2}} e^{\frac {3}{2}} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {11}{4}\right )} + \frac {B a^{\frac {3}{2}} c e^{\frac {3}{2}} x^{\frac {11}{2}} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{\Gamma \left (\frac {15}{4}\right )} + \frac {B \sqrt {a} c^{2} e^{\frac {3}{2}} x^{\frac {15}{2}} \Gamma \left (\frac {15}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {15}{4} \\ \frac {19}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {19}{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*a**(5/2)*e**(3/2)*x**(5/2)*gamma(5/4)*hyper((-1/2, 5/4), (9/4,), c*x**2*exp_polar(I*pi)/a)/(2*gamma(9/4)) +
A*a**(3/2)*c*e**(3/2)*x**(9/2)*gamma(9/4)*hyper((-1/2, 9/4), (13/4,), c*x**2*exp_polar(I*pi)/a)/gamma(13/4) +
A*sqrt(a)*c**2*e**(3/2)*x**(13/2)*gamma(13/4)*hyper((-1/2, 13/4), (17/4,), c*x**2*exp_polar(I*pi)/a)/(2*gamma(
17/4)) + B*a**(5/2)*e**(3/2)*x**(7/2)*gamma(7/4)*hyper((-1/2, 7/4), (11/4,), c*x**2*exp_polar(I*pi)/a)/(2*gamm
a(11/4)) + B*a**(3/2)*c*e**(3/2)*x**(11/2)*gamma(11/4)*hyper((-1/2, 11/4), (15/4,), c*x**2*exp_polar(I*pi)/a)/
gamma(15/4) + B*sqrt(a)*c**2*e**(3/2)*x**(15/2)*gamma(15/4)*hyper((-1/2, 15/4), (19/4,), c*x**2*exp_polar(I*pi
)/a)/(2*gamma(19/4))

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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