\(\int \frac {(a+b x^2)^{3/2} (A+B x^2)}{(e x)^{7/2}} \, dx\) [799]

Optimal result

Integrand size = 26, antiderivative size = 365 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{(e x)^{7/2}} \, dx=\frac {12 b (A b+a B) (e x)^{3/2} \sqrt {a+b x^2}}{5 a e^5}+\frac {24 \sqrt {b} (A b+a B) \sqrt {e x} \sqrt {a+b x^2}}{5 e^4 \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {2 (A b+a B) \left (a+b x^2\right )^{3/2}}{a e^3 \sqrt {e x}}-\frac {2 A \left (a+b x^2\right )^{5/2}}{5 a e (e x)^{5/2}}-\frac {24 \sqrt [4]{a} \sqrt [4]{b} (A b+a B) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{5 e^{7/2} \sqrt {a+b x^2}}+\frac {12 \sqrt [4]{a} \sqrt [4]{b} (A b+a B) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{5 e^{7/2} \sqrt {a+b x^2}} \]

[Out]

-2/5*A*(b*x^2+a)^(5/2)/a/e/(e*x)^(5/2)-2*(A*b+B*a)*(b*x^2+a)^(3/2)/a/e^3/(e*x)^(1/2)+12/5*b*(A*b+B*a)*(e*x)^(3
/2)*(b*x^2+a)^(1/2)/a/e^5+24/5*(A*b+B*a)*b^(1/2)*(e*x)^(1/2)*(b*x^2+a)^(1/2)/e^4/(a^(1/2)+x*b^(1/2))-24/5*a^(1
/4)*b^(1/4)*(A*b+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)/e^(7/2)/(b*x^2+a)^(1/2)+12/5*a^(1/4)*b^(1/4)*(A*b+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)))*Ellip
ticF(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)/e^(7/2)/(b*x^2+a)^(1/2)

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {464, 283, 285, 335, 311, 226, 1210} \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{(e x)^{7/2}} \, dx=\frac {12 \sqrt [4]{a} \sqrt [4]{b} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (a B+A b) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{5 e^{7/2} \sqrt {a+b x^2}}-\frac {24 \sqrt [4]{a} \sqrt [4]{b} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (a B+A b) E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{5 e^{7/2} \sqrt {a+b x^2}}+\frac {12 b (e x)^{3/2} \sqrt {a+b x^2} (a B+A b)}{5 a e^5}+\frac {24 \sqrt {b} \sqrt {e x} \sqrt {a+b x^2} (a B+A b)}{5 e^4 \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {2 \left (a+b x^2\right )^{3/2} (a B+A b)}{a e^3 \sqrt {e x}}-\frac {2 A \left (a+b x^2\right )^{5/2}}{5 a e (e x)^{5/2}} \]

[In]

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

[Out]

(12*b*(A*b + a*B)*(e*x)^(3/2)*Sqrt[a + b*x^2])/(5*a*e^5) + (24*Sqrt[b]*(A*b + a*B)*Sqrt[e*x]*Sqrt[a + b*x^2])/
(5*e^4*(Sqrt[a] + Sqrt[b]*x)) - (2*(A*b + a*B)*(a + b*x^2)^(3/2))/(a*e^3*Sqrt[e*x]) - (2*A*(a + b*x^2)^(5/2))/
(5*a*e*(e*x)^(5/2)) - (24*a^(1/4)*b^(1/4)*(A*b + a*B)*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b
]*x)^2]*EllipticE[2*ArcTan[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], 1/2])/(5*e^(7/2)*Sqrt[a + b*x^2]) + (12*a^(
1/4)*b^(1/4)*(A*b + a*B)*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticF[2*ArcTan[(b
^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], 1/2])/(5*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 283

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

Rule 285

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^p/(c*(m + n
*p + 1))), x] + Dist[a*n*(p/(m + n*p + 1)), Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x]
&& IGtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && 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 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*} \text {integral}& = -\frac {2 A \left (a+b x^2\right )^{5/2}}{5 a e (e x)^{5/2}}+\frac {(A b+a B) \int \frac {\left (a+b x^2\right )^{3/2}}{(e x)^{3/2}} \, dx}{a e^2} \\ & = -\frac {2 (A b+a B) \left (a+b x^2\right )^{3/2}}{a e^3 \sqrt {e x}}-\frac {2 A \left (a+b x^2\right )^{5/2}}{5 a e (e x)^{5/2}}+\frac {(6 b (A b+a B)) \int \sqrt {e x} \sqrt {a+b x^2} \, dx}{a e^4} \\ & = \frac {12 b (A b+a B) (e x)^{3/2} \sqrt {a+b x^2}}{5 a e^5}-\frac {2 (A b+a B) \left (a+b x^2\right )^{3/2}}{a e^3 \sqrt {e x}}-\frac {2 A \left (a+b x^2\right )^{5/2}}{5 a e (e x)^{5/2}}+\frac {(12 b (A b+a B)) \int \frac {\sqrt {e x}}{\sqrt {a+b x^2}} \, dx}{5 e^4} \\ & = \frac {12 b (A b+a B) (e x)^{3/2} \sqrt {a+b x^2}}{5 a e^5}-\frac {2 (A b+a B) \left (a+b x^2\right )^{3/2}}{a e^3 \sqrt {e x}}-\frac {2 A \left (a+b x^2\right )^{5/2}}{5 a e (e x)^{5/2}}+\frac {(24 b (A b+a B)) \text {Subst}\left (\int \frac {x^2}{\sqrt {a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{5 e^5} \\ & = \frac {12 b (A b+a B) (e x)^{3/2} \sqrt {a+b x^2}}{5 a e^5}-\frac {2 (A b+a B) \left (a+b x^2\right )^{3/2}}{a e^3 \sqrt {e x}}-\frac {2 A \left (a+b x^2\right )^{5/2}}{5 a e (e x)^{5/2}}+\frac {\left (24 \sqrt {a} \sqrt {b} (A b+a B)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{5 e^4}-\frac {\left (24 \sqrt {a} \sqrt {b} (A b+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 e^4} \\ & = \frac {12 b (A b+a B) (e x)^{3/2} \sqrt {a+b x^2}}{5 a e^5}+\frac {24 \sqrt {b} (A b+a B) \sqrt {e x} \sqrt {a+b x^2}}{5 e^4 \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {2 (A b+a B) \left (a+b x^2\right )^{3/2}}{a e^3 \sqrt {e x}}-\frac {2 A \left (a+b x^2\right )^{5/2}}{5 a e (e x)^{5/2}}-\frac {24 \sqrt [4]{a} \sqrt [4]{b} (A b+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 e^{7/2} \sqrt {a+b x^2}}+\frac {12 \sqrt [4]{a} \sqrt [4]{b} (A b+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 )}{5 e^{7/2} \sqrt {a+b x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.05 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.23 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{(e x)^{7/2}} \, dx=\frac {2 x \sqrt {a+b x^2} \left (-\frac {A \left (a+b x^2\right )^2}{a}-\frac {5 (A b+a B) x^2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {1}{4},\frac {3}{4},-\frac {b x^2}{a}\right )}{\sqrt {1+\frac {b x^2}{a}}}\right )}{5 (e x)^{7/2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 3.07 (sec) , antiderivative size = 242, normalized size of antiderivative = 0.66

method	result	size
risch	\(-\frac {2 \sqrt {b \,x^{2}+a}\, \left (-b B \,x^{4}+7 A b \,x^{2}+5 B a \,x^{2}+A a \right )}{5 x^{2} e^{3} \sqrt {e x}}+\frac {12 \left (A b +B a \right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right ) \sqrt {\left (b \,x^{2}+a \right ) e x}}{5 \sqrt {b e \,x^{3}+a e x}\, e^{3} \sqrt {e x}\, \sqrt {b \,x^{2}+a}}\)	\(242\)
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) e x}\, \left (-\frac {2 a A \sqrt {b e \,x^{3}+a e x}}{5 e^{4} x^{3}}-\frac {2 \left (b e \,x^{2}+a e \right ) \left (7 A b +5 B a \right )}{5 e^{4} \sqrt {x \left (b e \,x^{2}+a e \right )}}+\frac {2 b B x \sqrt {b e \,x^{3}+a e x}}{5 e^{4}}+\frac {\left (\frac {b \left (A b +2 B a \right )}{e^{3}}+\frac {b \left (7 A b +5 B a \right )}{5 e^{3}}-\frac {3 b B a}{5 e^{3}}\right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{b \sqrt {b e \,x^{3}+a e x}}\right )}{\sqrt {e x}\, \sqrt {b \,x^{2}+a}}\)	\(307\)
default	\(\frac {\frac {24 A \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, E\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a b \,x^{2}}{5}-\frac {12 A \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a b \,x^{2}}{5}+\frac {24 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, E\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} x^{2}}{5}-\frac {12 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} x^{2}}{5}+\frac {2 b^{2} B \,x^{6}}{5}-\frac {14 A \,b^{2} x^{4}}{5}-\frac {8 B a b \,x^{4}}{5}-\frac {16 a A b \,x^{2}}{5}-2 a^{2} B \,x^{2}-\frac {2 a^{2} A}{5}}{x^{2} \sqrt {b \,x^{2}+a}\, e^{3} \sqrt {e x}}\)	\(422\)

[In]

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

[Out]

-2/5*(b*x^2+a)^(1/2)*(-B*b*x^4+7*A*b*x^2+5*B*a*x^2+A*a)/x^2/e^3/(e*x)^(1/2)+12/5*(A*b+B*a)*(-a*b)^(1/2)*((x+(-
a*b)^(1/2)/b)/(-a*b)^(1/2)*b)^(1/2)*(-2*(x-(-a*b)^(1/2)/b)/(-a*b)^(1/2)*b)^(1/2)*(-x/(-a*b)^(1/2)*b)^(1/2)/(b*
e*x^3+a*e*x)^(1/2)*(-2*(-a*b)^(1/2)/b*EllipticE(((x+(-a*b)^(1/2)/b)/(-a*b)^(1/2)*b)^(1/2),1/2*2^(1/2))+(-a*b)^
(1/2)/b*EllipticF(((x+(-a*b)^(1/2)/b)/(-a*b)^(1/2)*b)^(1/2),1/2*2^(1/2)))/e^3*((b*x^2+a)*e*x)^(1/2)/(e*x)^(1/2
)/(b*x^2+a)^(1/2)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.09 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.23 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{(e x)^{7/2}} \, dx=-\frac {2 \, {\left (12 \, {\left (B a + A b\right )} \sqrt {b e} x^{3} {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) - {\left (B b x^{4} - {\left (5 \, B a + 7 \, A b\right )} x^{2} - A a\right )} \sqrt {b x^{2} + a} \sqrt {e x}\right )}}{5 \, e^{4} x^{3}} \]

[In]

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

[Out]

-2/5*(12*(B*a + A*b)*sqrt(b*e)*x^3*weierstrassZeta(-4*a/b, 0, weierstrassPInverse(-4*a/b, 0, x)) - (B*b*x^4 -
(5*B*a + 7*A*b)*x^2 - A*a)*sqrt(b*x^2 + a)*sqrt(e*x))/(e^4*x^3)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 23.28 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.58 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{(e x)^{7/2}} \, dx=\frac {A a^{\frac {3}{2}} \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, - \frac {1}{2} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {7}{2}} x^{\frac {5}{2}} \Gamma \left (- \frac {1}{4}\right )} + \frac {A \sqrt {a} b \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {7}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} + \frac {B a^{\frac {3}{2}} \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {7}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} + \frac {B \sqrt {a} b x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right )} \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{(e x)^{7/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}}}{\left (e x\right )^{\frac {7}{2}}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{(e x)^{7/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}}}{\left (e x\right )^{\frac {7}{2}}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{(e x)^{7/2}} \, dx=\int \frac {\left (B\,x^2+A\right )\,{\left (b\,x^2+a\right )}^{3/2}}{{\left (e\,x\right )}^{7/2}} \,d x \]

[In]

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

[Out]

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