3.5.58 \(\int \frac {(e x)^{7/2} (A+B x)}{\sqrt {a+c x^2}} \, dx\) [458]

Optimal. Leaf size=388 \[ -\frac {10 a A e^3 \sqrt {e x} \sqrt {a+c x^2}}{21 c^2}-\frac {14 a B e^2 (e x)^{3/2} \sqrt {a+c x^2}}{45 c^2}+\frac {2 A e (e x)^{5/2} \sqrt {a+c x^2}}{7 c}+\frac {2 B (e x)^{7/2} \sqrt {a+c x^2}}{9 c}+\frac {14 a^2 B e^4 x \sqrt {a+c x^2}}{15 c^{5/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {14 a^{9/4} B e^4 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 c^{11/4} \sqrt {e x} \sqrt {a+c x^2}}+\frac {a^{7/4} \left (49 \sqrt {a} B+25 A \sqrt {c}\right ) e^4 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{105 c^{11/4} \sqrt {e x} \sqrt {a+c x^2}} \]

[Out]

-14/45*a*B*e^2*(e*x)^(3/2)*(c*x^2+a)^(1/2)/c^2+2/7*A*e*(e*x)^(5/2)*(c*x^2+a)^(1/2)/c+2/9*B*(e*x)^(7/2)*(c*x^2+
a)^(1/2)/c+14/15*a^2*B*e^4*x*(c*x^2+a)^(1/2)/c^(5/2)/(a^(1/2)+x*c^(1/2))/(e*x)^(1/2)-10/21*a*A*e^3*(e*x)^(1/2)
*(c*x^2+a)^(1/2)/c^2-14/15*a^(9/4)*B*e^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)))*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^(11/4)/(e*x)^(1/2)/(c*x^2+a)^(1/2)+1/105*a^(7/4)*e^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))*(49*B*a^(1/2)+25*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^(11/4)/(e*x)^(1/2)/(c*x^2+a)^(1/2)

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Rubi [A]
time = 0.33, antiderivative size = 388, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {847, 856, 854, 1212, 226, 1210} \begin {gather*} \frac {a^{7/4} e^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 (49 \sqrt {a} B+25 A \sqrt {c}\right ) F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{105 c^{11/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {14 a^{9/4} B e^4 \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 )}{15 c^{11/4} \sqrt {e x} \sqrt {a+c x^2}}+\frac {14 a^2 B e^4 x \sqrt {a+c x^2}}{15 c^{5/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {10 a A e^3 \sqrt {e x} \sqrt {a+c x^2}}{21 c^2}+\frac {2 A e (e x)^{5/2} \sqrt {a+c x^2}}{7 c}-\frac {14 a B e^2 (e x)^{3/2} \sqrt {a+c x^2}}{45 c^2}+\frac {2 B (e x)^{7/2} \sqrt {a+c x^2}}{9 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((e*x)^(7/2)*(A + B*x))/Sqrt[a + c*x^2],x]

[Out]

(-10*a*A*e^3*Sqrt[e*x]*Sqrt[a + c*x^2])/(21*c^2) - (14*a*B*e^2*(e*x)^(3/2)*Sqrt[a + c*x^2])/(45*c^2) + (2*A*e*
(e*x)^(5/2)*Sqrt[a + c*x^2])/(7*c) + (2*B*(e*x)^(7/2)*Sqrt[a + c*x^2])/(9*c) + (14*a^2*B*e^4*x*Sqrt[a + c*x^2]
)/(15*c^(5/2)*Sqrt[e*x]*(Sqrt[a] + Sqrt[c]*x)) - (14*a^(9/4)*B*e^4*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])/(15*c^(11/4)*Sqrt[e*x]*Sqrt[
a + c*x^2]) + (a^(7/4)*(49*Sqrt[a]*B + 25*A*Sqrt[c])*e^4*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])/(105*c^(11/4)*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 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)^{7/2} (A+B x)}{\sqrt {a+c x^2}} \, dx &=\frac {2 B (e x)^{7/2} \sqrt {a+c x^2}}{9 c}+\frac {2 \int \frac {(e x)^{5/2} \left (-\frac {7}{2} a B e+\frac {9}{2} A c e x\right )}{\sqrt {a+c x^2}} \, dx}{9 c}\\ &=\frac {2 A e (e x)^{5/2} \sqrt {a+c x^2}}{7 c}+\frac {2 B (e x)^{7/2} \sqrt {a+c x^2}}{9 c}+\frac {4 \int \frac {(e x)^{3/2} \left (-\frac {45}{4} a A c e^2-\frac {49}{4} a B c e^2 x\right )}{\sqrt {a+c x^2}} \, dx}{63 c^2}\\ &=-\frac {14 a B e^2 (e x)^{3/2} \sqrt {a+c x^2}}{45 c^2}+\frac {2 A e (e x)^{5/2} \sqrt {a+c x^2}}{7 c}+\frac {2 B (e x)^{7/2} \sqrt {a+c x^2}}{9 c}+\frac {8 \int \frac {\sqrt {e x} \left (\frac {147}{8} a^2 B c e^3-\frac {225}{8} a A c^2 e^3 x\right )}{\sqrt {a+c x^2}} \, dx}{315 c^3}\\ &=-\frac {10 a A e^3 \sqrt {e x} \sqrt {a+c x^2}}{21 c^2}-\frac {14 a B e^2 (e x)^{3/2} \sqrt {a+c x^2}}{45 c^2}+\frac {2 A e (e x)^{5/2} \sqrt {a+c x^2}}{7 c}+\frac {2 B (e x)^{7/2} \sqrt {a+c x^2}}{9 c}+\frac {16 \int \frac {\frac {225}{16} a^2 A c^2 e^4+\frac {441}{16} a^2 B c^2 e^4 x}{\sqrt {e x} \sqrt {a+c x^2}} \, dx}{945 c^4}\\ &=-\frac {10 a A e^3 \sqrt {e x} \sqrt {a+c x^2}}{21 c^2}-\frac {14 a B e^2 (e x)^{3/2} \sqrt {a+c x^2}}{45 c^2}+\frac {2 A e (e x)^{5/2} \sqrt {a+c x^2}}{7 c}+\frac {2 B (e x)^{7/2} \sqrt {a+c x^2}}{9 c}+\frac {\left (16 \sqrt {x}\right ) \int \frac {\frac {225}{16} a^2 A c^2 e^4+\frac {441}{16} a^2 B c^2 e^4 x}{\sqrt {x} \sqrt {a+c x^2}} \, dx}{945 c^4 \sqrt {e x}}\\ &=-\frac {10 a A e^3 \sqrt {e x} \sqrt {a+c x^2}}{21 c^2}-\frac {14 a B e^2 (e x)^{3/2} \sqrt {a+c x^2}}{45 c^2}+\frac {2 A e (e x)^{5/2} \sqrt {a+c x^2}}{7 c}+\frac {2 B (e x)^{7/2} \sqrt {a+c x^2}}{9 c}+\frac {\left (32 \sqrt {x}\right ) \text {Subst}\left (\int \frac {\frac {225}{16} a^2 A c^2 e^4+\frac {441}{16} a^2 B c^2 e^4 x^2}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{945 c^4 \sqrt {e x}}\\ &=-\frac {10 a A e^3 \sqrt {e x} \sqrt {a+c x^2}}{21 c^2}-\frac {14 a B e^2 (e x)^{3/2} \sqrt {a+c x^2}}{45 c^2}+\frac {2 A e (e x)^{5/2} \sqrt {a+c x^2}}{7 c}+\frac {2 B (e x)^{7/2} \sqrt {a+c x^2}}{9 c}-\frac {\left (14 a^{5/2} B e^4 \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 )}{15 c^{5/2} \sqrt {e x}}+\frac {\left (2 a^2 \left (49 \sqrt {a} B+25 A \sqrt {c}\right ) e^4 \sqrt {x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{105 c^{5/2} \sqrt {e x}}\\ &=-\frac {10 a A e^3 \sqrt {e x} \sqrt {a+c x^2}}{21 c^2}-\frac {14 a B e^2 (e x)^{3/2} \sqrt {a+c x^2}}{45 c^2}+\frac {2 A e (e x)^{5/2} \sqrt {a+c x^2}}{7 c}+\frac {2 B (e x)^{7/2} \sqrt {a+c x^2}}{9 c}+\frac {14 a^2 B e^4 x \sqrt {a+c x^2}}{15 c^{5/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {14 a^{9/4} B e^4 \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 )}{15 c^{11/4} \sqrt {e x} \sqrt {a+c x^2}}+\frac {a^{7/4} \left (49 \sqrt {a} B+25 A \sqrt {c}\right ) e^4 \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 )}{105 c^{11/4} \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 = 10.07, size = 142, normalized size = 0.37 \begin {gather*} \frac {2 e^3 \sqrt {e x} \left (-\left (\left (a+c x^2\right ) \left (-5 c x^2 (9 A+7 B x)+a (75 A+49 B x)\right )\right )+75 a^2 A \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 )+49 a^2 B x \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 )}{315 c^2 \sqrt {a+c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((e*x)^(7/2)*(A + B*x))/Sqrt[a + c*x^2],x]

[Out]

(2*e^3*Sqrt[e*x]*(-((a + c*x^2)*(-5*c*x^2*(9*A + 7*B*x) + a*(75*A + 49*B*x))) + 75*a^2*A*Sqrt[1 + (c*x^2)/a]*H
ypergeometric2F1[1/4, 1/2, 5/4, -((c*x^2)/a)] + 49*a^2*B*x*Sqrt[1 + (c*x^2)/a]*Hypergeometric2F1[1/2, 3/4, 7/4
, -((c*x^2)/a)]))/(315*c^2*Sqrt[a + c*x^2])

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Maple [A]
time = 0.56, size = 344, normalized size = 0.89

method	result	size
default	\(\frac {e^{3} \sqrt {e x}\, \left (70 B \,c^{3} x^{6}+75 A \sqrt {2}\, \sqrt {-a c}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{2}+90 A \,c^{3} x^{5}+294 B \sqrt {2}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \EllipticE \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{3}-147 B \sqrt {2}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{3}-28 a B \,c^{2} x^{4}-60 a A \,c^{2} x^{3}-98 a^{2} B c \,x^{2}-150 a^{2} A c x \right )}{315 x \sqrt {c \,x^{2}+a}\, c^{3}}\)	\(344\)
risch	\(-\frac {2 \left (-35 B c \,x^{3}-45 A c \,x^{2}+49 B a x +75 A a \right ) x \sqrt {c \,x^{2}+a}\, e^{4}}{315 c^{2} \sqrt {e x}}+\frac {a^{2} \left (\frac {49 B \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{c \sqrt {c e \,x^{3}+a e x}}+\frac {25 A \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c \sqrt {c e \,x^{3}+a e x}}\right ) e^{4} \sqrt {\left (c \,x^{2}+a \right ) e x}}{105 c^{2} \sqrt {e x}\, \sqrt {c \,x^{2}+a}}\)	\(359\)
elliptic	\(\frac {\sqrt {e x}\, \sqrt {\left (c \,x^{2}+a \right ) e x}\, \left (\frac {2 B \,e^{3} x^{3} \sqrt {c e \,x^{3}+a e x}}{9 c}+\frac {2 A \,e^{3} x^{2} \sqrt {c e \,x^{3}+a e x}}{7 c}-\frac {14 B \,e^{3} a x \sqrt {c e \,x^{3}+a e x}}{45 c^{2}}-\frac {10 A \,e^{3} a \sqrt {c e \,x^{3}+a e x}}{21 c^{2}}+\frac {5 A \,e^{4} a^{2} \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{21 c^{3} \sqrt {c e \,x^{3}+a e x}}+\frac {7 B \,e^{4} a^{2} \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{15 c^{3} \sqrt {c e \,x^{3}+a e x}}\right )}{e x \sqrt {c \,x^{2}+a}}\)	\(416\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^(7/2)*(B*x+A)/(c*x^2+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/315*e^3/x*(e*x)^(1/2)/(c*x^2+a)^(1/2)/c^3*(70*B*c^3*x^6+75*A*2^(1/2)*(-a*c)^(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+90*A*c^3*x^5+294*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^3-147*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^3-28*a*B*c^2*x^
4-60*a*A*c^2*x^3-98*a^2*B*c*x^2-150*a^2*A*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)^(7/2)*(B*x+A)/(c*x^2+a)^(1/2),x, algorithm="maxima")

[Out]

e^(7/2)*integrate((B*x + A)*x^(7/2)/sqrt(c*x^2 + a), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.66, size = 99, normalized size = 0.26 \begin {gather*} \frac {2 \, {\left (75 \, A a^{2} \sqrt {c} e^{\frac {7}{2}} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) - 147 \, B a^{2} \sqrt {c} e^{\frac {7}{2}} {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) + {\left (35 \, B c^{2} x^{3} + 45 \, A c^{2} x^{2} - 49 \, B a c x - 75 \, A a c\right )} \sqrt {c x^{2} + a} \sqrt {x} e^{\frac {7}{2}}\right )}}{315 \, c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^(7/2)*(B*x+A)/(c*x^2+a)^(1/2),x, algorithm="fricas")

[Out]

2/315*(75*A*a^2*sqrt(c)*e^(7/2)*weierstrassPInverse(-4*a/c, 0, x) - 147*B*a^2*sqrt(c)*e^(7/2)*weierstrassZeta(
-4*a/c, 0, weierstrassPInverse(-4*a/c, 0, x)) + (35*B*c^2*x^3 + 45*A*c^2*x^2 - 49*B*a*c*x - 75*A*a*c)*sqrt(c*x
^2 + a)*sqrt(x)*e^(7/2))/c^3

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Sympy [C] Result contains complex when optimal does not.
time = 44.81, size = 94, normalized size = 0.24 \begin {gather*} \frac {A e^{\frac {7}{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 {a} \Gamma \left (\frac {13}{4}\right )} + \frac {B e^{\frac {7}{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 {a} \Gamma \left (\frac {15}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**(7/2)*(B*x+A)/(c*x**2+a)**(1/2),x)

[Out]

A*e**(7/2)*x**(9/2)*gamma(9/4)*hyper((1/2, 9/4), (13/4,), c*x**2*exp_polar(I*pi)/a)/(2*sqrt(a)*gamma(13/4)) +
B*e**(7/2)*x**(11/2)*gamma(11/4)*hyper((1/2, 11/4), (15/4,), c*x**2*exp_polar(I*pi)/a)/(2*sqrt(a)*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((e*x)^(7/2)*(B*x+A)/(c*x^2+a)^(1/2),x, algorithm="giac")

[Out]

integrate((B*x + A)*x^(7/2)*e^(7/2)/sqrt(c*x^2 + a), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (e\,x\right )}^{7/2}\,\left (A+B\,x\right )}{\sqrt {c\,x^2+a}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((e*x)^(7/2)*(A + B*x))/(a + c*x^2)^(1/2),x)

[Out]

int(((e*x)^(7/2)*(A + B*x))/(a + c*x^2)^(1/2), x)

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