3.5.71 \(\int \frac {A+B x}{(e x)^{3/2} (a+c x^2)^{3/2}} \, dx\) [471]

Optimal. Leaf size=327 \[ \frac {A+B x}{a e \sqrt {e x} \sqrt {a+c x^2}}-\frac {3 A \sqrt {a+c x^2}}{a^2 e \sqrt {e x}}+\frac {3 A \sqrt {c} x \sqrt {a+c x^2}}{a^2 e \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {3 A \sqrt [4]{c} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{a^{7/4} e \sqrt {e x} \sqrt {a+c x^2}}+\frac {\left (\sqrt {a} B+3 A \sqrt {c}\right ) \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{7/4} \sqrt [4]{c} e \sqrt {e x} \sqrt {a+c x^2}} \]

[Out]

(B*x+A)/a/e/(e*x)^(1/2)/(c*x^2+a)^(1/2)-3*A*(c*x^2+a)^(1/2)/a^2/e/(e*x)^(1/2)+3*A*x*c^(1/2)*(c*x^2+a)^(1/2)/a^
2/e/(a^(1/2)+x*c^(1/2))/(e*x)^(1/2)-3*A*c^(1/4)*(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)/a^(7/4)/e/(e*x)^(1/2)/(c*x^2+a)^(1/2)+1/2*(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))*(B*a^(1/2)+3*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^(7/4)/c^(1/4)/e/(e*x)^(1/2)/(c*x^2+a)^(1/2)

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Rubi [A]
time = 0.23, antiderivative size = 327, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {837, 849, 856, 854, 1212, 226, 1210} \begin {gather*} \frac {\sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (\sqrt {a} B+3 A \sqrt {c}\right ) F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{7/4} \sqrt [4]{c} e \sqrt {e x} \sqrt {a+c x^2}}-\frac {3 A \sqrt [4]{c} \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 )}{a^{7/4} e \sqrt {e x} \sqrt {a+c x^2}}-\frac {3 A \sqrt {a+c x^2}}{a^2 e \sqrt {e x}}+\frac {3 A \sqrt {c} x \sqrt {a+c x^2}}{a^2 e \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {A+B x}{a e \sqrt {e x} \sqrt {a+c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(A + B*x)/(a*e*Sqrt[e*x]*Sqrt[a + c*x^2]) - (3*A*Sqrt[a + c*x^2])/(a^2*e*Sqrt[e*x]) + (3*A*Sqrt[c]*x*Sqrt[a +
c*x^2])/(a^2*e*Sqrt[e*x]*(Sqrt[a] + Sqrt[c]*x)) - (3*A*c^(1/4)*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])/(a^(7/4)*e*Sqrt[e*x]*Sqrt[a + c*
x^2]) + ((Sqrt[a]*B + 3*A*Sqrt[c])*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*Ell
ipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(2*a^(7/4)*c^(1/4)*e*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 837

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(
m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] +
Dist[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^
2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g},
x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 849

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*f - d*g)*
(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/((m + 1)*(c*d^2 + a*e^2))), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[
(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; Fr
eeQ[{a, c, d, e, f, g, p}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || Integer
sQ[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}{(e x)^{3/2} \left (a+c x^2\right )^{3/2}} \, dx &=\frac {A+B x}{a e \sqrt {e x} \sqrt {a+c x^2}}-\frac {\int \frac {-\frac {3}{2} a A c e^2-\frac {1}{2} a B c e^2 x}{(e x)^{3/2} \sqrt {a+c x^2}} \, dx}{a^2 c e^2}\\ &=\frac {A+B x}{a e \sqrt {e x} \sqrt {a+c x^2}}-\frac {3 A \sqrt {a+c x^2}}{a^2 e \sqrt {e x}}+\frac {2 \int \frac {\frac {1}{4} a^2 B c e^3+\frac {3}{4} a A c^2 e^3 x}{\sqrt {e x} \sqrt {a+c x^2}} \, dx}{a^3 c e^4}\\ &=\frac {A+B x}{a e \sqrt {e x} \sqrt {a+c x^2}}-\frac {3 A \sqrt {a+c x^2}}{a^2 e \sqrt {e x}}+\frac {\left (2 \sqrt {x}\right ) \int \frac {\frac {1}{4} a^2 B c e^3+\frac {3}{4} a A c^2 e^3 x}{\sqrt {x} \sqrt {a+c x^2}} \, dx}{a^3 c e^4 \sqrt {e x}}\\ &=\frac {A+B x}{a e \sqrt {e x} \sqrt {a+c x^2}}-\frac {3 A \sqrt {a+c x^2}}{a^2 e \sqrt {e x}}+\frac {\left (4 \sqrt {x}\right ) \text {Subst}\left (\int \frac {\frac {1}{4} a^2 B c e^3+\frac {3}{4} a A c^2 e^3 x^2}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{a^3 c e^4 \sqrt {e x}}\\ &=\frac {A+B x}{a e \sqrt {e x} \sqrt {a+c x^2}}-\frac {3 A \sqrt {a+c x^2}}{a^2 e \sqrt {e x}}+\frac {\left (\left (\sqrt {a} B+3 A \sqrt {c}\right ) \sqrt {x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{a^{3/2} e \sqrt {e x}}-\frac {\left (3 A \sqrt {c} \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 )}{a^{3/2} e \sqrt {e x}}\\ &=\frac {A+B x}{a e \sqrt {e x} \sqrt {a+c x^2}}-\frac {3 A \sqrt {a+c x^2}}{a^2 e \sqrt {e x}}+\frac {3 A \sqrt {c} x \sqrt {a+c x^2}}{a^2 e \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {3 A \sqrt [4]{c} \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 )}{a^{7/4} e \sqrt {e x} \sqrt {a+c x^2}}+\frac {\left (\sqrt {a} B+3 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 )}{2 a^{7/4} \sqrt [4]{c} e \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 = 10.05, size = 100, normalized size = 0.31 \begin {gather*} \frac {x \left (A+B x-3 A \sqrt {1+\frac {c x^2}{a}} \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};-\frac {c x^2}{a}\right )+B x \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 )\right )}{a (e x)^{3/2} \sqrt {a+c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.58, size = 304, normalized size = 0.93

method	result	size
default	\(-\frac {3 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a c -6 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \EllipticE \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a c -B \sqrt {-a c}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a +6 A \,c^{2} x^{2}-2 a B c x +4 A a c}{2 \sqrt {c \,x^{2}+a}\, c e \sqrt {e x}\, a^{2}}\)	\(304\)
elliptic	\(\frac {\sqrt {\left (c \,x^{2}+a \right ) e x}\, \left (-\frac {2 \left (c e \,x^{2}+a e \right ) A}{e^{2} a^{2} \sqrt {x \left (c e \,x^{2}+a e \right )}}-\frac {2 c e x \left (\frac {A x}{2 e^{2} a^{2}}-\frac {B}{2 e^{2} a c}\right )}{\sqrt {\left (x^{2}+\frac {a}{c}\right ) c e x}}+\frac {B \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{2 a e c \sqrt {c e \,x^{3}+a e x}}+\frac {3 A \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{2 a^{2} e \sqrt {c e \,x^{3}+a e x}}\right )}{\sqrt {e x}\, \sqrt {c \,x^{2}+a}}\)	\(386\)
risch	\(-\frac {2 A \sqrt {c \,x^{2}+a}}{a^{2} e \sqrt {e x}}+\frac {\left (\frac {A \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{\sqrt {c e \,x^{3}+a e x}}-a \left (-\frac {2 x c e \left (-\frac {A x}{2 e a}+\frac {B}{2 e c}\right )}{\sqrt {\left (x^{2}+\frac {a}{c}\right ) c e x}}-\frac {B \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{2 c \sqrt {c e \,x^{3}+a e x}}-\frac {A \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{2 a \sqrt {c e \,x^{3}+a e x}}\right )\right ) \sqrt {\left (c \,x^{2}+a \right ) e x}}{a^{2} e \sqrt {e x}\, \sqrt {c \,x^{2}+a}}\)	\(532\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(3*A*((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))*2^(1/2)*a*c-6*A*((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))*2^(1/2)*a*c-B*(-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))*2^(1/2)*a+6*A*c^2*x^2-2*a*B*c*x+4*A*a*c)/(c*x^2+a)^(1/2)/c/e/(e*x)^(1/2)/a^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)/(e*x)^(3/2)/(c*x^2+a)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.52, size = 121, normalized size = 0.37 \begin {gather*} \frac {{\left ({\left (B a c x^{3} + B a^{2} x\right )} \sqrt {c} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) - 3 \, {\left (A c^{2} x^{3} + A a c x\right )} \sqrt {c} {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) - {\left (3 \, A c^{2} x^{2} - B a c x + 2 \, A a c\right )} \sqrt {c x^{2} + a} \sqrt {x}\right )} e^{\left (-\frac {3}{2}\right )}}{a^{2} c^{2} x^{3} + a^{3} c x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((B*a*c*x^3 + B*a^2*x)*sqrt(c)*weierstrassPInverse(-4*a/c, 0, x) - 3*(A*c^2*x^3 + A*a*c*x)*sqrt(c)*weierstrass
Zeta(-4*a/c, 0, weierstrassPInverse(-4*a/c, 0, x)) - (3*A*c^2*x^2 - B*a*c*x + 2*A*a*c)*sqrt(c*x^2 + a)*sqrt(x)
)*e^(-3/2)/(a^2*c^2*x^3 + a^3*c*x)

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Sympy [C] Result contains complex when optimal does not.
time = 12.12, size = 97, normalized size = 0.30 \begin {gather*} \frac {A \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {3}{2} \\ \frac {3}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} e^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} + \frac {B \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {3}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} e^{\frac {3}{2}} \Gamma \left (\frac {5}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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)/(e*x)^(3/2)/(c*x^2+a)^(3/2),x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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