3.5.49 \(\int \frac {(A+B x) (a+c x^2)^{3/2}}{(e x)^{9/2}} \, dx\) [449]

Optimal. Leaf size=339 \[ -\frac {4 c (5 A+21 B x) \sqrt {a+c x^2}}{35 e^3 (e x)^{3/2}}+\frac {24 B c^{3/2} x \sqrt {a+c x^2}}{5 e^4 \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {2 (5 A+7 B x) \left (a+c x^2\right )^{3/2}}{35 e (e x)^{7/2}}-\frac {24 \sqrt [4]{a} B c^{5/4} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 e^4 \sqrt {e x} \sqrt {a+c x^2}}+\frac {4 \left (21 \sqrt {a} B+5 A \sqrt {c}\right ) c^{5/4} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{35 \sqrt [4]{a} e^4 \sqrt {e x} \sqrt {a+c x^2}} \]

[Out]

-2/35*(7*B*x+5*A)*(c*x^2+a)^(3/2)/e/(e*x)^(7/2)-4/35*c*(21*B*x+5*A)*(c*x^2+a)^(1/2)/e^3/(e*x)^(3/2)+24/5*B*c^(
3/2)*x*(c*x^2+a)^(1/2)/e^4/(a^(1/2)+x*c^(1/2))/(e*x)^(1/2)-24/5*a^(1/4)*B*c^(5/4)*(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)/e^4/(e*x)^(1/2)/(c*x^2+a)^(1/2
)+4/35*c^(5/4)*(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)))*Ellipti
cF(sin(2*arctan(c^(1/4)*x^(1/2)/a^(1/4))),1/2*2^(1/2))*(21*B*a^(1/2)+5*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^(1/4)/e^4/(e*x)^(1/2)/(c*x^2+a)^(1/2)

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Rubi [A]
time = 0.24, antiderivative size = 339, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {825, 856, 854, 1212, 226, 1210} \begin {gather*} \frac {4 c^{5/4} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (21 \sqrt {a} B+5 A \sqrt {c}\right ) F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{35 \sqrt [4]{a} e^4 \sqrt {e x} \sqrt {a+c x^2}}-\frac {4 c \sqrt {a+c x^2} (5 A+21 B x)}{35 e^3 (e x)^{3/2}}-\frac {2 \left (a+c x^2\right )^{3/2} (5 A+7 B x)}{35 e (e x)^{7/2}}-\frac {24 \sqrt [4]{a} B c^{5/4} \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 )}{5 e^4 \sqrt {e x} \sqrt {a+c x^2}}+\frac {24 B c^{3/2} x \sqrt {a+c x^2}}{5 e^4 \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2)^(3/2))/(e*x)^(9/2),x]

[Out]

(-4*c*(5*A + 21*B*x)*Sqrt[a + c*x^2])/(35*e^3*(e*x)^(3/2)) + (24*B*c^(3/2)*x*Sqrt[a + c*x^2])/(5*e^4*Sqrt[e*x]
*(Sqrt[a] + Sqrt[c]*x)) - (2*(5*A + 7*B*x)*(a + c*x^2)^(3/2))/(35*e*(e*x)^(7/2)) - (24*a^(1/4)*B*c^(5/4)*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])/(5*e^4*Sqrt[e*x]*Sqrt[a + c*x^2]) + (4*(21*Sqrt[a]*B + 5*A*Sqrt[c])*c^(5/4)*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])/(35*a^(1/
4)*e^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 825

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/(e^2*(m + 1)*(m + 2)*(c*d^2 + a*e^2)))*((d*g - e*f*(m + 2))*(c*d^2 + a*e^2) - 2*c*d^2*
p*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 + a*e^2) + 2*c*d*p*(e*f - d*g))*x), x] - Dist[p/(e^2*(m + 1)*(m + 2)*(c*d^
2 + a*e^2)), Int[(d + e*x)^(m + 2)*(a + c*x^2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) - c*(2*c*d*(d*g*(2*p +
1) - e*f*(m + 2*p + 2)) - 2*a*e^2*g*(m + 1))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e
^2, 0] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 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 {(A+B x) \left (a+c x^2\right )^{3/2}}{(e x)^{9/2}} \, dx &=-\frac {2 (5 A+7 B x) \left (a+c x^2\right )^{3/2}}{35 e (e x)^{7/2}}-\frac {6 \int \frac {\left (-5 a A c e^2-7 a B c e^2 x\right ) \sqrt {a+c x^2}}{(e x)^{5/2}} \, dx}{35 a e^4}\\ &=-\frac {4 c (5 A+21 B x) \sqrt {a+c x^2}}{35 e^3 (e x)^{3/2}}-\frac {2 (5 A+7 B x) \left (a+c x^2\right )^{3/2}}{35 e (e x)^{7/2}}+\frac {4 \int \frac {5 a^2 A c^2 e^4+21 a^2 B c^2 e^4 x}{\sqrt {e x} \sqrt {a+c x^2}} \, dx}{35 a^2 e^8}\\ &=-\frac {4 c (5 A+21 B x) \sqrt {a+c x^2}}{35 e^3 (e x)^{3/2}}-\frac {2 (5 A+7 B x) \left (a+c x^2\right )^{3/2}}{35 e (e x)^{7/2}}+\frac {\left (4 \sqrt {x}\right ) \int \frac {5 a^2 A c^2 e^4+21 a^2 B c^2 e^4 x}{\sqrt {x} \sqrt {a+c x^2}} \, dx}{35 a^2 e^8 \sqrt {e x}}\\ &=-\frac {4 c (5 A+21 B x) \sqrt {a+c x^2}}{35 e^3 (e x)^{3/2}}-\frac {2 (5 A+7 B x) \left (a+c x^2\right )^{3/2}}{35 e (e x)^{7/2}}+\frac {\left (8 \sqrt {x}\right ) \text {Subst}\left (\int \frac {5 a^2 A c^2 e^4+21 a^2 B c^2 e^4 x^2}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{35 a^2 e^8 \sqrt {e x}}\\ &=-\frac {4 c (5 A+21 B x) \sqrt {a+c x^2}}{35 e^3 (e x)^{3/2}}-\frac {2 (5 A+7 B x) \left (a+c x^2\right )^{3/2}}{35 e (e x)^{7/2}}-\frac {\left (24 \sqrt {a} B c^{3/2} \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 )}{5 e^4 \sqrt {e x}}+\frac {\left (8 \left (21 \sqrt {a} B+5 A \sqrt {c}\right ) c^{3/2} \sqrt {x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{35 e^4 \sqrt {e x}}\\ &=-\frac {4 c (5 A+21 B x) \sqrt {a+c x^2}}{35 e^3 (e x)^{3/2}}+\frac {24 B c^{3/2} x \sqrt {a+c x^2}}{5 e^4 \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {2 (5 A+7 B x) \left (a+c x^2\right )^{3/2}}{35 e (e x)^{7/2}}-\frac {24 \sqrt [4]{a} B c^{5/4} \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 )}{5 e^4 \sqrt {e x} \sqrt {a+c x^2}}+\frac {4 \left (21 \sqrt {a} B+5 A \sqrt {c}\right ) c^{5/4} \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 )}{35 \sqrt [4]{a} e^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 = 10.04, size = 89, normalized size = 0.26 \begin {gather*} -\frac {2 a \sqrt {e x} \sqrt {a+c x^2} \left (5 A \, _2F_1\left (-\frac {7}{4},-\frac {3}{2};-\frac {3}{4};-\frac {c x^2}{a}\right )+7 B x \, _2F_1\left (-\frac {3}{2},-\frac {5}{4};-\frac {1}{4};-\frac {c x^2}{a}\right )\right )}{35 e^5 x^4 \sqrt {1+\frac {c x^2}{a}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + c*x^2)^(3/2))/(e*x)^(9/2),x]

[Out]

(-2*a*Sqrt[e*x]*Sqrt[a + c*x^2]*(5*A*Hypergeometric2F1[-7/4, -3/2, -3/4, -((c*x^2)/a)] + 7*B*x*Hypergeometric2
F1[-3/2, -5/4, -1/4, -((c*x^2)/a)]))/(35*e^5*x^4*Sqrt[1 + (c*x^2)/a])

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Maple [A]
time = 0.61, size = 337, normalized size = 0.99

method	result	size
default	\(\frac {\frac {4 A \sqrt {2}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a c}\, c \,x^{3}}{7}-\frac {12 B \sqrt {2}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a c \,x^{3}}{5}+\frac {24 B \sqrt {2}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \EllipticE \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a c \,x^{3}}{5}-\frac {14 B \,c^{2} x^{5}}{5}-\frac {6 A \,c^{2} x^{4}}{7}-\frac {16 a B c \,x^{3}}{5}-\frac {8 a A c \,x^{2}}{7}-\frac {2 a^{2} B x}{5}-\frac {2 a^{2} A}{7}}{x^{3} \sqrt {c \,x^{2}+a}\, e^{4} \sqrt {e x}}\)	\(337\)
risch	\(-\frac {2 \sqrt {c \,x^{2}+a}\, \left (49 B c \,x^{3}+15 A c \,x^{2}+7 B a x +5 A a \right )}{35 x^{3} e^{4} \sqrt {e x}}+\frac {4 c^{2} \left (\frac {21 B \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{c \sqrt {c e \,x^{3}+a e x}}+\frac {5 A \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c \sqrt {c e \,x^{3}+a e x}}\right ) \sqrt {\left (c \,x^{2}+a \right ) e x}}{35 e^{4} \sqrt {e x}\, \sqrt {c \,x^{2}+a}}\)	\(355\)
elliptic	\(\frac {\sqrt {\left (c \,x^{2}+a \right ) e x}\, \left (-\frac {2 a A \sqrt {c e \,x^{3}+a e x}}{7 e^{5} x^{4}}-\frac {2 B a \sqrt {c e \,x^{3}+a e x}}{5 e^{5} x^{3}}-\frac {6 A c \sqrt {c e \,x^{3}+a e x}}{7 e^{5} x^{2}}-\frac {14 \left (c e \,x^{2}+a e \right ) B c}{5 e^{5} \sqrt {x \left (c e \,x^{2}+a e \right )}}+\frac {4 A c \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{7 e^{4} \sqrt {c e \,x^{3}+a e x}}+\frac {12 B c \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{5 e^{4} \sqrt {c e \,x^{3}+a e x}}\right )}{\sqrt {e x}\, \sqrt {c \,x^{2}+a}}\)	\(403\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+a)^(3/2)/(e*x)^(9/2),x,method=_RETURNVERBOSE)

[Out]

2/35/x^3*(10*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)*c*x^3-42*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*x^3+84*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*c*x^3-49*B*c^2*x^5-15*A*c^2*x^4-56*a*B*c*x^3-20*a*A*c*x^2-7*a^2*B*x-5*a^
2*A)/(c*x^2+a)^(1/2)/e^4/(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)^(3/2)/(e*x)^(9/2),x, algorithm="maxima")

[Out]

e^(-9/2)*integrate((c*x^2 + a)^(3/2)*(B*x + A)/x^(9/2), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.56, size = 90, normalized size = 0.27 \begin {gather*} \frac {2 \, {\left (20 \, A c^{\frac {3}{2}} x^{4} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) - 84 \, B c^{\frac {3}{2}} x^{4} {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) - {\left (49 \, B c x^{3} + 15 \, A c x^{2} + 7 \, B a x + 5 \, A a\right )} \sqrt {c x^{2} + a} \sqrt {x}\right )} e^{\left (-\frac {9}{2}\right )}}{35 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^(3/2)/(e*x)^(9/2),x, algorithm="fricas")

[Out]

2/35*(20*A*c^(3/2)*x^4*weierstrassPInverse(-4*a/c, 0, x) - 84*B*c^(3/2)*x^4*weierstrassZeta(-4*a/c, 0, weierst
rassPInverse(-4*a/c, 0, x)) - (49*B*c*x^3 + 15*A*c*x^2 + 7*B*a*x + 5*A*a)*sqrt(c*x^2 + a)*sqrt(x))*e^(-9/2)/x^
4

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Sympy [C] Result contains complex when optimal does not.
time = 95.93, size = 219, normalized size = 0.65 \begin {gather*} \frac {A a^{\frac {3}{2}} \Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{4}, - \frac {1}{2} \\ - \frac {3}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {9}{2}} x^{\frac {7}{2}} \Gamma \left (- \frac {3}{4}\right )} + \frac {A \sqrt {a} c \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {9}{2}} x^{\frac {3}{2}} \Gamma \left (\frac {1}{4}\right )} + \frac {B a^{\frac {3}{2}} \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, - \frac {1}{2} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {9}{2}} x^{\frac {5}{2}} \Gamma \left (- \frac {1}{4}\right )} + \frac {B \sqrt {a} c \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {9}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+a)**(3/2)/(e*x)**(9/2),x)

[Out]

A*a**(3/2)*gamma(-7/4)*hyper((-7/4, -1/2), (-3/4,), c*x**2*exp_polar(I*pi)/a)/(2*e**(9/2)*x**(7/2)*gamma(-3/4)
) + A*sqrt(a)*c*gamma(-3/4)*hyper((-3/4, -1/2), (1/4,), c*x**2*exp_polar(I*pi)/a)/(2*e**(9/2)*x**(3/2)*gamma(1
/4)) + B*a**(3/2)*gamma(-5/4)*hyper((-5/4, -1/2), (-1/4,), c*x**2*exp_polar(I*pi)/a)/(2*e**(9/2)*x**(5/2)*gamm
a(-1/4)) + B*sqrt(a)*c*gamma(-1/4)*hyper((-1/2, -1/4), (3/4,), c*x**2*exp_polar(I*pi)/a)/(2*e**(9/2)*sqrt(x)*g
amma(3/4))

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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)^(3/2)/(e*x)^(9/2),x, algorithm="giac")

[Out]

integrate((c*x^2 + a)^(3/2)*(B*x + A)*e^(-9/2)/x^(9/2), 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 )}^{3/2}\,\left (A+B\,x\right )}{{\left (e\,x\right )}^{9/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + c*x^2)^(3/2)*(A + B*x))/(e*x)^(9/2),x)

[Out]

int(((a + c*x^2)^(3/2)*(A + B*x))/(e*x)^(9/2), x)

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