3.5.52 \(\int \frac {(A+B x) (a+c x^2)^{5/2}}{\sqrt {e x}} \, dx\) [452]

Optimal. Leaf size=369 \[ \frac {8 a^2 \sqrt {e x} (195 A+77 B x) \sqrt {a+c x^2}}{3003 e}+\frac {16 a^3 B x \sqrt {a+c x^2}}{39 \sqrt {c} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {20 a \sqrt {e x} (117 A+77 B x) \left (a+c x^2\right )^{3/2}}{9009 e}+\frac {2 \sqrt {e x} (13 A+11 B x) \left (a+c x^2\right )^{5/2}}{143 e}-\frac {16 a^{13/4} B \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{39 c^{3/4} \sqrt {e x} \sqrt {a+c x^2}}+\frac {8 a^{11/4} \left (77 \sqrt {a} B+195 A \sqrt {c}\right ) \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{3003 c^{3/4} \sqrt {e x} \sqrt {a+c x^2}} \]

[Out]

20/9009*a*(77*B*x+117*A)*(c*x^2+a)^(3/2)*(e*x)^(1/2)/e+2/143*(11*B*x+13*A)*(c*x^2+a)^(5/2)*(e*x)^(1/2)/e+16/39
*a^3*B*x*(c*x^2+a)^(1/2)/c^(1/2)/(a^(1/2)+x*c^(1/2))/(e*x)^(1/2)+8/3003*a^2*(77*B*x+195*A)*(e*x)^(1/2)*(c*x^2+
a)^(1/2)/e-16/39*a^(13/4)*B*(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^(3/4)/(e*x)^(1/2)/(c*x^2+a)^(1/2)+8/3003*a^(11/4)*(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))*(77*B*a^(1/2)+195*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^(3/
4)/(e*x)^(1/2)/(c*x^2+a)^(1/2)

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Rubi [A]
time = 0.28, antiderivative size = 369, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {829, 856, 854, 1212, 226, 1210} \begin {gather*} \frac {8 a^{11/4} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (77 \sqrt {a} B+195 A \sqrt {c}\right ) F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{3003 c^{3/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {16 a^{13/4} B \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 )}{39 c^{3/4} \sqrt {e x} \sqrt {a+c x^2}}+\frac {16 a^3 B x \sqrt {a+c x^2}}{39 \sqrt {c} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {8 a^2 \sqrt {e x} \sqrt {a+c x^2} (195 A+77 B x)}{3003 e}+\frac {20 a \sqrt {e x} \left (a+c x^2\right )^{3/2} (117 A+77 B x)}{9009 e}+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{5/2} (13 A+11 B x)}{143 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(8*a^2*Sqrt[e*x]*(195*A + 77*B*x)*Sqrt[a + c*x^2])/(3003*e) + (16*a^3*B*x*Sqrt[a + c*x^2])/(39*Sqrt[c]*Sqrt[e*
x]*(Sqrt[a] + Sqrt[c]*x)) + (20*a*Sqrt[e*x]*(117*A + 77*B*x)*(a + c*x^2)^(3/2))/(9009*e) + (2*Sqrt[e*x]*(13*A
+ 11*B*x)*(a + c*x^2)^(5/2))/(143*e) - (16*a^(13/4)*B*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])/(39*c^(3/4)*Sqrt[e*x]*Sqrt[a + c*x^2]) +
(8*a^(11/4)*(77*Sqrt[a]*B + 195*A*Sqrt[c])*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])/(3003*c^(3/4)*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 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 {(A+B x) \left (a+c x^2\right )^{5/2}}{\sqrt {e x}} \, dx &=\frac {2 \sqrt {e x} (13 A+11 B x) \left (a+c x^2\right )^{5/2}}{143 e}+\frac {20 \int \frac {\left (\frac {13}{2} a A c e^2+\frac {11}{2} a B c e^2 x\right ) \left (a+c x^2\right )^{3/2}}{\sqrt {e x}} \, dx}{143 c e^2}\\ &=\frac {20 a \sqrt {e x} (117 A+77 B x) \left (a+c x^2\right )^{3/2}}{9009 e}+\frac {2 \sqrt {e x} (13 A+11 B x) \left (a+c x^2\right )^{5/2}}{143 e}+\frac {80 \int \frac {\left (\frac {117}{4} a^2 A c^2 e^4+\frac {77}{4} a^2 B c^2 e^4 x\right ) \sqrt {a+c x^2}}{\sqrt {e x}} \, dx}{3003 c^2 e^4}\\ &=\frac {8 a^2 \sqrt {e x} (195 A+77 B x) \sqrt {a+c x^2}}{3003 e}+\frac {20 a \sqrt {e x} (117 A+77 B x) \left (a+c x^2\right )^{3/2}}{9009 e}+\frac {2 \sqrt {e x} (13 A+11 B x) \left (a+c x^2\right )^{5/2}}{143 e}+\frac {64 \int \frac {\frac {585}{8} a^3 A c^3 e^6+\frac {231}{8} a^3 B c^3 e^6 x}{\sqrt {e x} \sqrt {a+c x^2}} \, dx}{9009 c^3 e^6}\\ &=\frac {8 a^2 \sqrt {e x} (195 A+77 B x) \sqrt {a+c x^2}}{3003 e}+\frac {20 a \sqrt {e x} (117 A+77 B x) \left (a+c x^2\right )^{3/2}}{9009 e}+\frac {2 \sqrt {e x} (13 A+11 B x) \left (a+c x^2\right )^{5/2}}{143 e}+\frac {\left (64 \sqrt {x}\right ) \int \frac {\frac {585}{8} a^3 A c^3 e^6+\frac {231}{8} a^3 B c^3 e^6 x}{\sqrt {x} \sqrt {a+c x^2}} \, dx}{9009 c^3 e^6 \sqrt {e x}}\\ &=\frac {8 a^2 \sqrt {e x} (195 A+77 B x) \sqrt {a+c x^2}}{3003 e}+\frac {20 a \sqrt {e x} (117 A+77 B x) \left (a+c x^2\right )^{3/2}}{9009 e}+\frac {2 \sqrt {e x} (13 A+11 B x) \left (a+c x^2\right )^{5/2}}{143 e}+\frac {\left (128 \sqrt {x}\right ) \text {Subst}\left (\int \frac {\frac {585}{8} a^3 A c^3 e^6+\frac {231}{8} a^3 B c^3 e^6 x^2}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{9009 c^3 e^6 \sqrt {e x}}\\ &=\frac {8 a^2 \sqrt {e x} (195 A+77 B x) \sqrt {a+c x^2}}{3003 e}+\frac {20 a \sqrt {e x} (117 A+77 B x) \left (a+c x^2\right )^{3/2}}{9009 e}+\frac {2 \sqrt {e x} (13 A+11 B x) \left (a+c x^2\right )^{5/2}}{143 e}-\frac {\left (16 a^{7/2} B \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 )}{39 \sqrt {c} \sqrt {e x}}+\frac {\left (16 a^3 \left (77 \sqrt {a} B+195 A \sqrt {c}\right ) \sqrt {x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{3003 \sqrt {c} \sqrt {e x}}\\ &=\frac {8 a^2 \sqrt {e x} (195 A+77 B x) \sqrt {a+c x^2}}{3003 e}+\frac {16 a^3 B x \sqrt {a+c x^2}}{39 \sqrt {c} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {20 a \sqrt {e x} (117 A+77 B x) \left (a+c x^2\right )^{3/2}}{9009 e}+\frac {2 \sqrt {e x} (13 A+11 B x) \left (a+c x^2\right )^{5/2}}{143 e}-\frac {16 a^{13/4} B \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 )}{39 c^{3/4} \sqrt {e x} \sqrt {a+c x^2}}+\frac {8 a^{11/4} \left (77 \sqrt {a} B+195 A \sqrt {c}\right ) \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 )}{3003 c^{3/4} \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 = 10.03, size = 85, normalized size = 0.23 \begin {gather*} \frac {2 a^2 x \sqrt {a+c x^2} \left (3 A \, _2F_1\left (-\frac {5}{2},\frac {1}{4};\frac {5}{4};-\frac {c x^2}{a}\right )+B x \, _2F_1\left (-\frac {5}{2},\frac {3}{4};\frac {7}{4};-\frac {c x^2}{a}\right )\right )}{3 \sqrt {e x} \sqrt {1+\frac {c x^2}{a}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.59, size = 362, normalized size = 0.98

method	result	size
default	\(\frac {\frac {2 B \,c^{4} x^{8}}{13}+\frac {2 A \,c^{4} x^{7}}{11}+\frac {74 a B \,c^{3} x^{6}}{117}+\frac {40 A \sqrt {2}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a c}\, a^{3}}{77}+\frac {62 a A \,c^{3} x^{5}}{77}+\frac {16 B \sqrt {2}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \EllipticE \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{4}}{39}-\frac {8 B \sqrt {2}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{4}}{39}+\frac {118 a^{2} B \,c^{2} x^{4}}{117}+\frac {122 a^{2} A \,c^{2} x^{3}}{77}+\frac {62 a^{3} B c \,x^{2}}{117}+\frac {74 a^{3} A c x}{77}}{\sqrt {c \,x^{2}+a}\, c \sqrt {e x}}\)	\(362\)
risch	\(\frac {2 \left (693 B \,c^{2} x^{5}+819 A \,c^{2} x^{4}+2156 a B c \,x^{3}+2808 a A c \,x^{2}+2387 a^{2} B x +4329 a^{2} A \right ) x \sqrt {c \,x^{2}+a}}{9009 \sqrt {e x}}+\frac {8 a^{3} \left (\frac {77 B \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{c \sqrt {c e \,x^{3}+a e x}}+\frac {195 A \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c \sqrt {c e \,x^{3}+a e x}}\right ) \sqrt {\left (c \,x^{2}+a \right ) e x}}{3003 \sqrt {e x}\, \sqrt {c \,x^{2}+a}}\)	\(371\)
elliptic	\(\frac {\sqrt {\left (c \,x^{2}+a \right ) e x}\, \left (\frac {2 B \,c^{2} x^{5} \sqrt {c e \,x^{3}+a e x}}{13 e}+\frac {2 A \,c^{2} x^{4} \sqrt {c e \,x^{3}+a e x}}{11 e}+\frac {56 B c a \,x^{3} \sqrt {c e \,x^{3}+a e x}}{117 e}+\frac {48 A a c \,x^{2} \sqrt {c e \,x^{3}+a e x}}{77 e}+\frac {62 a^{2} B x \sqrt {c e \,x^{3}+a e x}}{117 e}+\frac {74 a^{2} A \sqrt {c e \,x^{3}+a e x}}{77 e}+\frac {40 A \,a^{3} \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{77 c \sqrt {c e \,x^{3}+a e x}}+\frac {8 B \,a^{3} \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{39 c \sqrt {c e \,x^{3}+a e x}}\right )}{\sqrt {e x}\, \sqrt {c \,x^{2}+a}}\)	\(450\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+a)^(5/2)/(e*x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/9009/(c*x^2+a)^(1/2)/c*(693*B*c^4*x^8+819*A*c^4*x^7+2849*a*B*c^3*x^6+2340*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^3+3627*a*A*c^3*x^5+1848*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^4-924*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^4+4543*a^2*B*c^2*x^4+7137*a^2*A*c^2*x^3+2387*a^3*B*c*x^2+4329*a^3*A*c*x)/(e*x)^(1/2)

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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((B*x+A)*(c*x^2+a)^(5/2)/(e*x)^(1/2),x, algorithm="maxima")

[Out]

e^(-1/2)*integrate((c*x^2 + a)^(5/2)*(B*x + A)/sqrt(x), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.98, size = 119, normalized size = 0.32 \begin {gather*} \frac {2 \, {\left (4680 \, A a^{3} \sqrt {c} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) - 1848 \, B a^{3} \sqrt {c} {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) + {\left (693 \, B c^{3} x^{5} + 819 \, A c^{3} x^{4} + 2156 \, B a c^{2} x^{3} + 2808 \, A a c^{2} x^{2} + 2387 \, B a^{2} c x + 4329 \, A a^{2} c\right )} \sqrt {c x^{2} + a} \sqrt {x}\right )} e^{\left (-\frac {1}{2}\right )}}{9009 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^(5/2)/(e*x)^(1/2),x, algorithm="fricas")

[Out]

2/9009*(4680*A*a^3*sqrt(c)*weierstrassPInverse(-4*a/c, 0, x) - 1848*B*a^3*sqrt(c)*weierstrassZeta(-4*a/c, 0, w
eierstrassPInverse(-4*a/c, 0, x)) + (693*B*c^3*x^5 + 819*A*c^3*x^4 + 2156*B*a*c^2*x^3 + 2808*A*a*c^2*x^2 + 238
7*B*a^2*c*x + 4329*A*a^2*c)*sqrt(c*x^2 + a)*sqrt(x))*e^(-1/2)/c

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Sympy [C] Result contains complex when optimal does not.
time = 10.56, size = 301, normalized size = 0.82 \begin {gather*} \frac {A a^{\frac {5}{2}} \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {e} \Gamma \left (\frac {5}{4}\right )} + \frac {A a^{\frac {3}{2}} c 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 )}}{\sqrt {e} \Gamma \left (\frac {9}{4}\right )} + \frac {A \sqrt {a} c^{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 )}}{2 \sqrt {e} \Gamma \left (\frac {13}{4}\right )} + \frac {B a^{\frac {5}{2}} x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {e} \Gamma \left (\frac {7}{4}\right )} + \frac {B a^{\frac {3}{2}} c 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 )}}{\sqrt {e} \Gamma \left (\frac {11}{4}\right )} + \frac {B \sqrt {a} c^{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 )}}{2 \sqrt {e} \Gamma \left (\frac {15}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+a)**(5/2)/(e*x)**(1/2),x)

[Out]

A*a**(5/2)*sqrt(x)*gamma(1/4)*hyper((-1/2, 1/4), (5/4,), c*x**2*exp_polar(I*pi)/a)/(2*sqrt(e)*gamma(5/4)) + A*
a**(3/2)*c*x**(5/2)*gamma(5/4)*hyper((-1/2, 5/4), (9/4,), c*x**2*exp_polar(I*pi)/a)/(sqrt(e)*gamma(9/4)) + A*s
qrt(a)*c**2*x**(9/2)*gamma(9/4)*hyper((-1/2, 9/4), (13/4,), c*x**2*exp_polar(I*pi)/a)/(2*sqrt(e)*gamma(13/4))
+ B*a**(5/2)*x**(3/2)*gamma(3/4)*hyper((-1/2, 3/4), (7/4,), c*x**2*exp_polar(I*pi)/a)/(2*sqrt(e)*gamma(7/4)) +
B*a**(3/2)*c*x**(7/2)*gamma(7/4)*hyper((-1/2, 7/4), (11/4,), c*x**2*exp_polar(I*pi)/a)/(sqrt(e)*gamma(11/4))
+ B*sqrt(a)*c**2*x**(11/2)*gamma(11/4)*hyper((-1/2, 11/4), (15/4,), c*x**2*exp_polar(I*pi)/a)/(2*sqrt(e)*gamma
(15/4))

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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((B*x+A)*(c*x^2+a)^(5/2)/(e*x)^(1/2),x, algorithm="giac")

[Out]

integrate((c*x^2 + a)^(5/2)*(B*x + A)*e^(-1/2)/sqrt(x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + c*x^2)^(5/2)*(A + B*x))/(e*x)^(1/2),x)

[Out]

int(((a + c*x^2)^(5/2)*(A + B*x))/(e*x)^(1/2), x)

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