3.483 \(\int \frac {A+B x}{(e x)^{5/2} (a+c x^2)^{5/2}} \, dx\)

Optimal. Leaf size=402 \[ \frac {\sqrt [4]{c} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (7 \sqrt {a} B-5 A \sqrt {c}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{4 a^{13/4} e^2 \sqrt {e x} \sqrt {a+c x^2}}-\frac {7 B \sqrt [4]{c} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{11/4} e^2 \sqrt {e x} \sqrt {a+c x^2}}-\frac {5 A \sqrt {a+c x^2}}{2 a^3 e (e x)^{3/2}}-\frac {7 B \sqrt {a+c x^2}}{2 a^3 e^2 \sqrt {e x}}+\frac {7 B \sqrt {c} x \sqrt {a+c x^2}}{2 a^3 e^2 \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {9 A+7 B x}{6 a^2 e (e x)^{3/2} \sqrt {a+c x^2}}+\frac {A+B x}{3 a e (e x)^{3/2} \left (a+c x^2\right )^{3/2}} \]

[Out]

1/3*(B*x+A)/a/e/(e*x)^(3/2)/(c*x^2+a)^(3/2)+1/6*(7*B*x+9*A)/a^2/e/(e*x)^(3/2)/(c*x^2+a)^(1/2)-5/2*A*(c*x^2+a)^
(1/2)/a^3/e/(e*x)^(3/2)-7/2*B*(c*x^2+a)^(1/2)/a^3/e^2/(e*x)^(1/2)+7/2*B*x*c^(1/2)*(c*x^2+a)^(1/2)/a^3/e^2/(a^(
1/2)+x*c^(1/2))/(e*x)^(1/2)-7/2*B*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^(11/4)/e^2/(e*x)^(1/2)/(c*x^2+a)^(1/2)+1/4*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)))*EllipticF(sin(2*arctan(c^(1/4)*x^(1/2)/
a^(1/4))),1/2*2^(1/2))*(7*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^(13/4)/e^2/(e*x)^(1/2)/(c*x^2+a)^(1/2)

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Rubi [A]  time = 0.50, antiderivative size = 402, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {823, 835, 842, 840, 1198, 220, 1196} \[ \frac {\sqrt [4]{c} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (7 \sqrt {a} B-5 A \sqrt {c}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{4 a^{13/4} e^2 \sqrt {e x} \sqrt {a+c x^2}}+\frac {9 A+7 B x}{6 a^2 e (e x)^{3/2} \sqrt {a+c x^2}}-\frac {5 A \sqrt {a+c x^2}}{2 a^3 e (e x)^{3/2}}-\frac {7 B \sqrt {a+c x^2}}{2 a^3 e^2 \sqrt {e x}}+\frac {7 B \sqrt {c} x \sqrt {a+c x^2}}{2 a^3 e^2 \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {7 B \sqrt [4]{c} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{11/4} e^2 \sqrt {e x} \sqrt {a+c x^2}}+\frac {A+B x}{3 a e (e x)^{3/2} \left (a+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(A + B*x)/(3*a*e*(e*x)^(3/2)*(a + c*x^2)^(3/2)) + (9*A + 7*B*x)/(6*a^2*e*(e*x)^(3/2)*Sqrt[a + c*x^2]) - (5*A*S
qrt[a + c*x^2])/(2*a^3*e*(e*x)^(3/2)) - (7*B*Sqrt[a + c*x^2])/(2*a^3*e^2*Sqrt[e*x]) + (7*B*Sqrt[c]*x*Sqrt[a +
c*x^2])/(2*a^3*e^2*Sqrt[e*x]*(Sqrt[a] + Sqrt[c]*x)) - (7*B*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])/(2*a^(11/4)*e^2*Sqrt[e*x]*Sq
rt[a + c*x^2]) + ((7*Sqrt[a]*B - 5*A*Sqrt[c])*c^(1/4)*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])/(4*a^(13/4)*e^2*Sqrt[e*x]*Sqrt[a + c*x^2]
)

Rule 220

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)]*EllipticF[2*ArcTan[q*x], 1/2])/(2*q*Sqrt[a + b*x^4]), x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 823

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] + Di
st[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 835

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 840

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 842

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 1196

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)]*EllipticE[2*ArcTan[
q*x], 1/2])/(q*Sqrt[a + c*x^4]), x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e}, x] && PosQ[c/a]

Rule 1198

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)^{5/2} \left (a+c x^2\right )^{5/2}} \, dx &=\frac {A+B x}{3 a e (e x)^{3/2} \left (a+c x^2\right )^{3/2}}-\frac {\int \frac {-\frac {9}{2} a A c e^2-\frac {7}{2} a B c e^2 x}{(e x)^{5/2} \left (a+c x^2\right )^{3/2}} \, dx}{3 a^2 c e^2}\\ &=\frac {A+B x}{3 a e (e x)^{3/2} \left (a+c x^2\right )^{3/2}}+\frac {9 A+7 B x}{6 a^2 e (e x)^{3/2} \sqrt {a+c x^2}}+\frac {\int \frac {\frac {45}{4} a^2 A c^2 e^4+\frac {21}{4} a^2 B c^2 e^4 x}{(e x)^{5/2} \sqrt {a+c x^2}} \, dx}{3 a^4 c^2 e^4}\\ &=\frac {A+B x}{3 a e (e x)^{3/2} \left (a+c x^2\right )^{3/2}}+\frac {9 A+7 B x}{6 a^2 e (e x)^{3/2} \sqrt {a+c x^2}}-\frac {5 A \sqrt {a+c x^2}}{2 a^3 e (e x)^{3/2}}-\frac {2 \int \frac {-\frac {63}{8} a^3 B c^2 e^5+\frac {45}{8} a^2 A c^3 e^5 x}{(e x)^{3/2} \sqrt {a+c x^2}} \, dx}{9 a^5 c^2 e^6}\\ &=\frac {A+B x}{3 a e (e x)^{3/2} \left (a+c x^2\right )^{3/2}}+\frac {9 A+7 B x}{6 a^2 e (e x)^{3/2} \sqrt {a+c x^2}}-\frac {5 A \sqrt {a+c x^2}}{2 a^3 e (e x)^{3/2}}-\frac {7 B \sqrt {a+c x^2}}{2 a^3 e^2 \sqrt {e x}}+\frac {4 \int \frac {-\frac {45}{16} a^3 A c^3 e^6+\frac {63}{16} a^3 B c^3 e^6 x}{\sqrt {e x} \sqrt {a+c x^2}} \, dx}{9 a^6 c^2 e^8}\\ &=\frac {A+B x}{3 a e (e x)^{3/2} \left (a+c x^2\right )^{3/2}}+\frac {9 A+7 B x}{6 a^2 e (e x)^{3/2} \sqrt {a+c x^2}}-\frac {5 A \sqrt {a+c x^2}}{2 a^3 e (e x)^{3/2}}-\frac {7 B \sqrt {a+c x^2}}{2 a^3 e^2 \sqrt {e x}}+\frac {\left (4 \sqrt {x}\right ) \int \frac {-\frac {45}{16} a^3 A c^3 e^6+\frac {63}{16} a^3 B c^3 e^6 x}{\sqrt {x} \sqrt {a+c x^2}} \, dx}{9 a^6 c^2 e^8 \sqrt {e x}}\\ &=\frac {A+B x}{3 a e (e x)^{3/2} \left (a+c x^2\right )^{3/2}}+\frac {9 A+7 B x}{6 a^2 e (e x)^{3/2} \sqrt {a+c x^2}}-\frac {5 A \sqrt {a+c x^2}}{2 a^3 e (e x)^{3/2}}-\frac {7 B \sqrt {a+c x^2}}{2 a^3 e^2 \sqrt {e x}}+\frac {\left (8 \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {-\frac {45}{16} a^3 A c^3 e^6+\frac {63}{16} a^3 B c^3 e^6 x^2}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{9 a^6 c^2 e^8 \sqrt {e x}}\\ &=\frac {A+B x}{3 a e (e x)^{3/2} \left (a+c x^2\right )^{3/2}}+\frac {9 A+7 B x}{6 a^2 e (e x)^{3/2} \sqrt {a+c x^2}}-\frac {5 A \sqrt {a+c x^2}}{2 a^3 e (e x)^{3/2}}-\frac {7 B \sqrt {a+c x^2}}{2 a^3 e^2 \sqrt {e x}}-\frac {\left (7 B \sqrt {c} \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{2 a^{5/2} e^2 \sqrt {e x}}+\frac {\left (\left (7 \sqrt {a} B-5 A \sqrt {c}\right ) \sqrt {c} \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{2 a^3 e^2 \sqrt {e x}}\\ &=\frac {A+B x}{3 a e (e x)^{3/2} \left (a+c x^2\right )^{3/2}}+\frac {9 A+7 B x}{6 a^2 e (e x)^{3/2} \sqrt {a+c x^2}}-\frac {5 A \sqrt {a+c x^2}}{2 a^3 e (e x)^{3/2}}-\frac {7 B \sqrt {a+c x^2}}{2 a^3 e^2 \sqrt {e x}}+\frac {7 B \sqrt {c} x \sqrt {a+c x^2}}{2 a^3 e^2 \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {7 B \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 )}{2 a^{11/4} e^2 \sqrt {e x} \sqrt {a+c x^2}}+\frac {\left (7 \sqrt {a} B-5 A \sqrt {c}\right ) \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}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{4 a^{13/4} e^2 \sqrt {e x} \sqrt {a+c x^2}}\\ \end {align*}

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Mathematica [C]  time = 0.09, size = 137, normalized size = 0.34 \[ \frac {x \left (-15 A \left (a+c x^2\right ) \sqrt {\frac {c x^2}{a}+1} \, _2F_1\left (-\frac {3}{4},\frac {1}{2};\frac {1}{4};-\frac {c x^2}{a}\right )+11 a A-21 B x \left (a+c x^2\right ) \sqrt {\frac {c x^2}{a}+1} \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};-\frac {c x^2}{a}\right )+9 a B x+9 A c x^2+7 B c x^3\right )}{6 a^2 (e x)^{5/2} \left (a+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(11*a*A + 9*a*B*x + 9*A*c*x^2 + 7*B*c*x^3 - 15*A*(a + c*x^2)*Sqrt[1 + (c*x^2)/a]*Hypergeometric2F1[-3/4, 1/
2, 1/4, -((c*x^2)/a)] - 21*B*x*(a + c*x^2)*Sqrt[1 + (c*x^2)/a]*Hypergeometric2F1[-1/4, 1/2, 3/4, -((c*x^2)/a)]
))/(6*a^2*(e*x)^(5/2)*(a + c*x^2)^(3/2))

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fricas [F]  time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{2} + a} {\left (B x + A\right )} \sqrt {e x}}{c^{3} e^{3} x^{9} + 3 \, a c^{2} e^{3} x^{7} + 3 \, a^{2} c e^{3} x^{5} + a^{3} e^{3} x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(c*x^2 + a)*(B*x + A)*sqrt(e*x)/(c^3*e^3*x^9 + 3*a*c^2*e^3*x^7 + 3*a^2*c*e^3*x^5 + a^3*e^3*x^3),
x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {B x + A}{{\left (c x^{2} + a\right )}^{\frac {5}{2}} \left (e x\right )^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.12, size = 602, normalized size = 1.50 \[ -\frac {42 B \,c^{2} x^{5}+30 A \,c^{2} x^{4}-42 \sqrt {2}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, B a c \,x^{3} \EllipticE \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )+21 \sqrt {2}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, B a c \,x^{3} \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )+15 \sqrt {2}\, \sqrt {-a c}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, A c \,x^{3} \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )+70 B a c \,x^{3}+42 A a c \,x^{2}-42 \sqrt {2}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, B \,a^{2} x \EllipticE \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )+21 \sqrt {2}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, B \,a^{2} x \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )+15 \sqrt {-a c}\, \sqrt {2}\, \sqrt {-\frac {c x}{\sqrt {-a c}}}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, A a x \EllipticF \left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )+24 B \,a^{2} x +8 A \,a^{2}}{12 \sqrt {e x}\, \left (c \,x^{2}+a \right )^{\frac {3}{2}} a^{3} e^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(15*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)*(-1/(-a*c)^(1/2)*c*x)^(1/2)*EllipticF(((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2),1/2*2^(1/2))*x^3*c+21*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)*(-1/(-a*c)^(1/2)*c*x)^(1/
2)*EllipticF(((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2),1/2*2^(1/2))*x^3*a*c-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)*(-1/(-a*c)^(1/2)*c*x)^(1/2)*EllipticE(((c*x+(-a*c)^(
1/2))/(-a*c)^(1/2))^(1/2),1/2*2^(1/2))*x^3*a*c+15*A*(-a*c)^(1/2)*2^(1/2)*(-1/(-a*c)^(1/2)*c*x)^(1/2)*EllipticF
(((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2),1/2*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+21*B*2^(1/2)*(-1/(-a*c)^(1/2)*c*x)^(1/2)*EllipticF(((c*x+(-a*c)^(1/2))/(-a*c)^(1/
2))^(1/2),1/2*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^2-
42*B*2^(1/2)*(-1/(-a*c)^(1/2)*c*x)^(1/2)*EllipticE(((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2),1/2*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^2+42*B*c^2*x^5+30*A*c^2*x^4+70*
B*a*c*x^3+42*A*a*c*x^2+24*B*a^2*x+8*A*a^2)/x/e^2/(e*x)^(1/2)/a^3/(c*x^2+a)^(3/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {B x + A}{{\left (c x^{2} + a\right )}^{\frac {5}{2}} \left (e x\right )^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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