3.9.14 \(\int \frac {A+B x^2}{(e x)^{7/2} (a+b x^2)^{3/2}} \, dx\) [814]

Optimal. Leaf size=379 \[ -\frac {2 A}{5 a e (e x)^{5/2} \sqrt {a+b x^2}}-\frac {7 A b-5 a B}{5 a^2 e^3 \sqrt {e x} \sqrt {a+b x^2}}+\frac {3 (7 A b-5 a B) \sqrt {a+b x^2}}{5 a^3 e^3 \sqrt {e x}}-\frac {3 \sqrt {b} (7 A b-5 a B) \sqrt {e x} \sqrt {a+b x^2}}{5 a^3 e^4 \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {3 \sqrt [4]{b} (7 A b-5 a B) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{5 a^{11/4} e^{7/2} \sqrt {a+b x^2}}-\frac {3 \sqrt [4]{b} (7 A b-5 a B) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{10 a^{11/4} e^{7/2} \sqrt {a+b x^2}} \]

[Out]

-2/5*A/a/e/(e*x)^(5/2)/(b*x^2+a)^(1/2)+1/5*(-7*A*b+5*B*a)/a^2/e^3/(e*x)^(1/2)/(b*x^2+a)^(1/2)+3/5*(7*A*b-5*B*a
)*(b*x^2+a)^(1/2)/a^3/e^3/(e*x)^(1/2)-3/5*(7*A*b-5*B*a)*b^(1/2)*(e*x)^(1/2)*(b*x^2+a)^(1/2)/a^3/e^4/(a^(1/2)+x
*b^(1/2))+3/5*b^(1/4)*(7*A*b-5*B*a)*(cos(2*arctan(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2)))^2)^(1/2)/cos(2*arctan(
b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2)))*EllipticE(sin(2*arctan(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2))),1/2*2^(1/2)
)*(a^(1/2)+x*b^(1/2))*((b*x^2+a)/(a^(1/2)+x*b^(1/2))^2)^(1/2)/a^(11/4)/e^(7/2)/(b*x^2+a)^(1/2)-3/10*b^(1/4)*(7
*A*b-5*B*a)*(cos(2*arctan(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2)))^2)^(1/2)/cos(2*arctan(b^(1/4)*(e*x)^(1/2)/a^(1
/4)/e^(1/2)))*EllipticF(sin(2*arctan(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2))),1/2*2^(1/2))*(a^(1/2)+x*b^(1/2))*((
b*x^2+a)/(a^(1/2)+x*b^(1/2))^2)^(1/2)/a^(11/4)/e^(7/2)/(b*x^2+a)^(1/2)

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Rubi [A]
time = 0.21, antiderivative size = 379, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {464, 296, 331, 335, 311, 226, 1210} \begin {gather*} -\frac {3 \sqrt [4]{b} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (7 A b-5 a B) F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{10 a^{11/4} e^{7/2} \sqrt {a+b x^2}}+\frac {3 \sqrt [4]{b} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (7 A b-5 a B) E\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{5 a^{11/4} e^{7/2} \sqrt {a+b x^2}}-\frac {3 \sqrt {b} \sqrt {e x} \sqrt {a+b x^2} (7 A b-5 a B)}{5 a^3 e^4 \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {3 \sqrt {a+b x^2} (7 A b-5 a B)}{5 a^3 e^3 \sqrt {e x}}-\frac {7 A b-5 a B}{5 a^2 e^3 \sqrt {e x} \sqrt {a+b x^2}}-\frac {2 A}{5 a e (e x)^{5/2} \sqrt {a+b x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x^2)/((e*x)^(7/2)*(a + b*x^2)^(3/2)),x]

[Out]

(-2*A)/(5*a*e*(e*x)^(5/2)*Sqrt[a + b*x^2]) - (7*A*b - 5*a*B)/(5*a^2*e^3*Sqrt[e*x]*Sqrt[a + b*x^2]) + (3*(7*A*b
- 5*a*B)*Sqrt[a + b*x^2])/(5*a^3*e^3*Sqrt[e*x]) - (3*Sqrt[b]*(7*A*b - 5*a*B)*Sqrt[e*x]*Sqrt[a + b*x^2])/(5*a^
3*e^4*(Sqrt[a] + Sqrt[b]*x)) + (3*b^(1/4)*(7*A*b - 5*a*B)*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sq
rt[b]*x)^2]*EllipticE[2*ArcTan[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], 1/2])/(5*a^(11/4)*e^(7/2)*Sqrt[a + b*x^
2]) - (3*b^(1/4)*(7*A*b - 5*a*B)*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticF[2*A
rcTan[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], 1/2])/(10*a^(11/4)*e^(7/2)*Sqrt[a + b*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 296

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-(c*x)^(m + 1))*((a + b*x^n)^(p + 1)/
(a*c*n*(p + 1))), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; Free
Q[{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 311

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, Dist[1/q, Int[1/Sqrt[a + b*x^4], x],
x] - Dist[1/q, Int[(1 - q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 464

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[c*(e*x)^(m +
1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

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]

Rubi steps

\begin {align*} \int \frac {A+B x^2}{(e x)^{7/2} \left (a+b x^2\right )^{3/2}} \, dx &=-\frac {2 A}{5 a e (e x)^{5/2} \sqrt {a+b x^2}}-\frac {(7 A b-5 a B) \int \frac {1}{(e x)^{3/2} \left (a+b x^2\right )^{3/2}} \, dx}{5 a e^2}\\ &=-\frac {2 A}{5 a e (e x)^{5/2} \sqrt {a+b x^2}}-\frac {7 A b-5 a B}{5 a^2 e^3 \sqrt {e x} \sqrt {a+b x^2}}-\frac {(3 (7 A b-5 a B)) \int \frac {1}{(e x)^{3/2} \sqrt {a+b x^2}} \, dx}{10 a^2 e^2}\\ &=-\frac {2 A}{5 a e (e x)^{5/2} \sqrt {a+b x^2}}-\frac {7 A b-5 a B}{5 a^2 e^3 \sqrt {e x} \sqrt {a+b x^2}}+\frac {3 (7 A b-5 a B) \sqrt {a+b x^2}}{5 a^3 e^3 \sqrt {e x}}-\frac {(3 b (7 A b-5 a B)) \int \frac {\sqrt {e x}}{\sqrt {a+b x^2}} \, dx}{10 a^3 e^4}\\ &=-\frac {2 A}{5 a e (e x)^{5/2} \sqrt {a+b x^2}}-\frac {7 A b-5 a B}{5 a^2 e^3 \sqrt {e x} \sqrt {a+b x^2}}+\frac {3 (7 A b-5 a B) \sqrt {a+b x^2}}{5 a^3 e^3 \sqrt {e x}}-\frac {(3 b (7 A b-5 a B)) \text {Subst}\left (\int \frac {x^2}{\sqrt {a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{5 a^3 e^5}\\ &=-\frac {2 A}{5 a e (e x)^{5/2} \sqrt {a+b x^2}}-\frac {7 A b-5 a B}{5 a^2 e^3 \sqrt {e x} \sqrt {a+b x^2}}+\frac {3 (7 A b-5 a B) \sqrt {a+b x^2}}{5 a^3 e^3 \sqrt {e x}}-\frac {\left (3 \sqrt {b} (7 A b-5 a B)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{5 a^{5/2} e^4}+\frac {\left (3 \sqrt {b} (7 A b-5 a B)\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a} e}}{\sqrt {a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{5 a^{5/2} e^4}\\ &=-\frac {2 A}{5 a e (e x)^{5/2} \sqrt {a+b x^2}}-\frac {7 A b-5 a B}{5 a^2 e^3 \sqrt {e x} \sqrt {a+b x^2}}+\frac {3 (7 A b-5 a B) \sqrt {a+b x^2}}{5 a^3 e^3 \sqrt {e x}}-\frac {3 \sqrt {b} (7 A b-5 a B) \sqrt {e x} \sqrt {a+b x^2}}{5 a^3 e^4 \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {3 \sqrt [4]{b} (7 A b-5 a B) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{5 a^{11/4} e^{7/2} \sqrt {a+b x^2}}-\frac {3 \sqrt [4]{b} (7 A b-5 a B) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{10 a^{11/4} e^{7/2} \sqrt {a+b 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 = 78, normalized size = 0.21 \begin {gather*} \frac {x \left (-2 a A+2 (7 A b-5 a B) x^2 \sqrt {1+\frac {b x^2}{a}} \, _2F_1\left (-\frac {1}{4},\frac {3}{2};\frac {3}{4};-\frac {b x^2}{a}\right )\right )}{5 a^2 (e x)^{7/2} \sqrt {a+b x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x^2)/((e*x)^(7/2)*(a + b*x^2)^(3/2)),x]

[Out]

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

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Maple [A]
time = 0.14, size = 417, normalized size = 1.10

method	result	size
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) e x}\, \left (\frac {b \,x^{2} \left (A b -B a \right )}{e^{3} a^{3} \sqrt {\left (x^{2}+\frac {a}{b}\right ) b e x}}-\frac {2 A \sqrt {b e \,x^{3}+a e x}}{5 a^{2} e^{4} x^{3}}+\frac {2 \left (b e \,x^{2}+a e \right ) \left (8 A b -5 B a \right )}{5 a^{3} e^{4} \sqrt {x \left (b e \,x^{2}+a e \right )}}+\frac {\left (-\frac {b \left (A b -B a \right )}{2 a^{3} e^{3}}-\frac {b \left (8 A b -5 B a \right )}{5 a^{3} e^{3}}\right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{b \sqrt {b e \,x^{3}+a e x}}\right )}{\sqrt {e x}\, \sqrt {b \,x^{2}+a}}\)	\(324\)
default	\(-\frac {42 A \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticE \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a b \,x^{2}-21 A \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a b \,x^{2}-30 B \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticE \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} x^{2}+15 B \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} x^{2}-42 A \,b^{2} x^{4}+30 B a b \,x^{4}-28 a A b \,x^{2}+20 B \,a^{2} x^{2}+4 a^{2} A}{10 x^{2} \sqrt {b \,x^{2}+a}\, e^{3} \sqrt {e x}\, a^{3}}\)	\(417\)
risch	\(-\frac {2 \sqrt {b \,x^{2}+a}\, \left (-8 A b \,x^{2}+5 B a \,x^{2}+A a \right )}{5 a^{3} x^{2} e^{3} \sqrt {e x}}-\frac {b^{2} \left (\frac {\left (8 A b -5 B a \right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{b^{2} \sqrt {b e \,x^{3}+a e x}}-\frac {5 \left (A b -B a \right ) a \left (\frac {x^{2}}{a \sqrt {\left (x^{2}+\frac {a}{b}\right ) b e x}}-\frac {\sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{2 a b \sqrt {b e \,x^{3}+a e x}}\right )}{b}\right ) \sqrt {\left (b \,x^{2}+a \right ) e x}}{5 a^{3} e^{3} \sqrt {e x}\, \sqrt {b \,x^{2}+a}}\)	\(451\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x^2+A)/(e*x)^(7/2)/(b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/10/x^2*(42*A*((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*2^(1/2)*((-b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-x*b
/(-a*b)^(1/2))^(1/2)*EllipticE(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*a*b*x^2-21*A*((b*x+(-a*b)^
(1/2))/(-a*b)^(1/2))^(1/2)*2^(1/2)*((-b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-x*b/(-a*b)^(1/2))^(1/2)*Elliptic
F(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*a*b*x^2-30*B*((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*2^
(1/2)*((-b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-x*b/(-a*b)^(1/2))^(1/2)*EllipticE(((b*x+(-a*b)^(1/2))/(-a*b)^
(1/2))^(1/2),1/2*2^(1/2))*a^2*x^2+15*B*((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*2^(1/2)*((-b*x+(-a*b)^(1/2))/(-
a*b)^(1/2))^(1/2)*(-x*b/(-a*b)^(1/2))^(1/2)*EllipticF(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*a^2
*x^2-42*A*b^2*x^4+30*B*a*b*x^4-28*a*A*b*x^2+20*B*a^2*x^2+4*a^2*A)/(b*x^2+a)^(1/2)/e^3/(e*x)^(1/2)/a^3

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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^2+A)/(e*x)^(7/2)/(b*x^2+a)^(3/2),x, algorithm="maxima")

[Out]

e^(-7/2)*integrate((B*x^2 + A)/((b*x^2 + a)^(3/2)*x^(7/2)), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.26, size = 132, normalized size = 0.35 \begin {gather*} -\frac {{\left (3 \, {\left ({\left (5 \, B a b - 7 \, A b^{2}\right )} x^{5} + {\left (5 \, B a^{2} - 7 \, A a b\right )} x^{3}\right )} \sqrt {b} {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) + {\left (3 \, {\left (5 \, B a b - 7 \, A b^{2}\right )} x^{4} + 2 \, A a^{2} + 2 \, {\left (5 \, B a^{2} - 7 \, A a b\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {x}\right )} e^{\left (-\frac {7}{2}\right )}}{5 \, {\left (a^{3} b x^{5} + a^{4} x^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/(e*x)^(7/2)/(b*x^2+a)^(3/2),x, algorithm="fricas")

[Out]

-1/5*(3*((5*B*a*b - 7*A*b^2)*x^5 + (5*B*a^2 - 7*A*a*b)*x^3)*sqrt(b)*weierstrassZeta(-4*a/b, 0, weierstrassPInv
erse(-4*a/b, 0, x)) + (3*(5*B*a*b - 7*A*b^2)*x^4 + 2*A*a^2 + 2*(5*B*a^2 - 7*A*a*b)*x^2)*sqrt(b*x^2 + a)*sqrt(x
))*e^(-7/2)/(a^3*b*x^5 + a^4*x^3)

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Sympy [C] Result contains complex when optimal does not.
time = 63.73, size = 104, normalized size = 0.27 \begin {gather*} \frac {A \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, \frac {3}{2} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} e^{\frac {7}{2}} x^{\frac {5}{2}} \Gamma \left (- \frac {1}{4}\right )} + \frac {B \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 {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} e^{\frac {7}{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**2+A)/(e*x)**(7/2)/(b*x**2+a)**(3/2),x)

[Out]

A*gamma(-5/4)*hyper((-5/4, 3/2), (-1/4,), b*x**2*exp_polar(I*pi)/a)/(2*a**(3/2)*e**(7/2)*x**(5/2)*gamma(-1/4))
+ B*gamma(-1/4)*hyper((-1/4, 3/2), (3/4,), b*x**2*exp_polar(I*pi)/a)/(2*a**(3/2)*e**(7/2)*sqrt(x)*gamma(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^2+A)/(e*x)^(7/2)/(b*x^2+a)^(3/2),x, algorithm="giac")

[Out]

integrate((B*x^2 + A)*e^(-7/2)/((b*x^2 + a)^(3/2)*x^(7/2)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {B\,x^2+A}{{\left (e\,x\right )}^{7/2}\,{\left (b\,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^2)/((e*x)^(7/2)*(a + b*x^2)^(3/2)),x)

[Out]

int((A + B*x^2)/((e*x)^(7/2)*(a + b*x^2)^(3/2)), x)

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