3.9.36 \(\int \frac {(a+b x^2)^2 (c+d x^2)^{3/2}}{(e x)^{3/2}} \, dx\) [836]

Optimal. Leaf size=476 \[ -\frac {4 \left (3 b^2 c^2-13 a d (2 b c+9 a d)\right ) (e x)^{3/2} \sqrt {c+d x^2}}{195 d e^3}-\frac {8 c \left (3 b^2 c^2-13 a d (2 b c+9 a d)\right ) \sqrt {e x} \sqrt {c+d x^2}}{195 d^{3/2} e^2 \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {2 \left (3 b^2 c^2-13 a d (2 b c+9 a d)\right ) (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{117 c d e^3}-\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{c e \sqrt {e x}}+\frac {2 b^2 (e x)^{3/2} \left (c+d x^2\right )^{5/2}}{13 d e^3}+\frac {8 c^{5/4} \left (3 b^2 c^2-13 a d (2 b c+9 a d)\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{195 d^{7/4} e^{3/2} \sqrt {c+d x^2}}-\frac {4 c^{5/4} \left (3 b^2 c^2-13 a d (2 b c+9 a d)\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{195 d^{7/4} e^{3/2} \sqrt {c+d x^2}} \]

[Out]

-2/117*(3*b^2*c^2-13*a*d*(9*a*d+2*b*c))*(e*x)^(3/2)*(d*x^2+c)^(3/2)/c/d/e^3+2/13*b^2*(e*x)^(3/2)*(d*x^2+c)^(5/
2)/d/e^3-2*a^2*(d*x^2+c)^(5/2)/c/e/(e*x)^(1/2)-4/195*(3*b^2*c^2-13*a*d*(9*a*d+2*b*c))*(e*x)^(3/2)*(d*x^2+c)^(1
/2)/d/e^3-8/195*c*(3*b^2*c^2-13*a*d*(9*a*d+2*b*c))*(e*x)^(1/2)*(d*x^2+c)^(1/2)/d^(3/2)/e^2/(c^(1/2)+x*d^(1/2))
+8/195*c^(5/4)*(3*b^2*c^2-13*a*d*(9*a*d+2*b*c))*(cos(2*arctan(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2)))^2)^(1/2)/c
os(2*arctan(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2)))*EllipticE(sin(2*arctan(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2)))
,1/2*2^(1/2))*(c^(1/2)+x*d^(1/2))*((d*x^2+c)/(c^(1/2)+x*d^(1/2))^2)^(1/2)/d^(7/4)/e^(3/2)/(d*x^2+c)^(1/2)-4/19
5*c^(5/4)*(3*b^2*c^2-13*a*d*(9*a*d+2*b*c))*(cos(2*arctan(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2)))^2)^(1/2)/cos(2*
arctan(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2)))*EllipticF(sin(2*arctan(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2))),1/2*
2^(1/2))*(c^(1/2)+x*d^(1/2))*((d*x^2+c)/(c^(1/2)+x*d^(1/2))^2)^(1/2)/d^(7/4)/e^(3/2)/(d*x^2+c)^(1/2)

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Rubi [A]
time = 0.32, antiderivative size = 476, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {473, 470, 285, 335, 311, 226, 1210} \begin {gather*} -\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{c e \sqrt {e x}}-\frac {4 c^{5/4} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (3 b^2 c^2-13 a d (9 a d+2 b c)\right ) F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{195 d^{7/4} e^{3/2} \sqrt {c+d x^2}}+\frac {8 c^{5/4} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (3 b^2 c^2-13 a d (9 a d+2 b c)\right ) E\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{195 d^{7/4} e^{3/2} \sqrt {c+d x^2}}-\frac {8 c \sqrt {e x} \sqrt {c+d x^2} \left (3 b^2 c^2-13 a d (9 a d+2 b c)\right )}{195 d^{3/2} e^2 \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {2 (e x)^{3/2} \left (c+d x^2\right )^{3/2} \left (3 b^2 c^2-13 a d (9 a d+2 b c)\right )}{117 c d e^3}-\frac {4 (e x)^{3/2} \sqrt {c+d x^2} \left (3 b^2 c^2-13 a d (9 a d+2 b c)\right )}{195 d e^3}+\frac {2 b^2 (e x)^{3/2} \left (c+d x^2\right )^{5/2}}{13 d e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*(3*b^2*c^2 - 13*a*d*(2*b*c + 9*a*d))*(e*x)^(3/2)*Sqrt[c + d*x^2])/(195*d*e^3) - (8*c*(3*b^2*c^2 - 13*a*d*(
2*b*c + 9*a*d))*Sqrt[e*x]*Sqrt[c + d*x^2])/(195*d^(3/2)*e^2*(Sqrt[c] + Sqrt[d]*x)) - (2*(3*b^2*c^2 - 13*a*d*(2
*b*c + 9*a*d))*(e*x)^(3/2)*(c + d*x^2)^(3/2))/(117*c*d*e^3) - (2*a^2*(c + d*x^2)^(5/2))/(c*e*Sqrt[e*x]) + (2*b
^2*(e*x)^(3/2)*(c + d*x^2)^(5/2))/(13*d*e^3) + (8*c^(5/4)*(3*b^2*c^2 - 13*a*d*(2*b*c + 9*a*d))*(Sqrt[c] + Sqrt
[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*EllipticE[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/
2])/(195*d^(7/4)*e^(3/2)*Sqrt[c + d*x^2]) - (4*c^(5/4)*(3*b^2*c^2 - 13*a*d*(2*b*c + 9*a*d))*(Sqrt[c] + Sqrt[d]
*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*EllipticF[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])
/(195*d^(7/4)*e^(3/2)*Sqrt[c + d*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 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 470

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

Rule 473

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^2, x_Symbol] :> Simp[c^2*(e*x)^(m
+ 1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), x] - Dist[1/(a*e^n*(m + 1)), Int[(e*x)^(m + n)*(a + b*x^n)^p*Simp[b
*c^2*n*(p + 1) + c*(b*c - 2*a*d)*(m + 1) - a*(m + 1)*d^2*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && Ne
Q[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[m, -1] && GtQ[n, 0]

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 {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{(e x)^{3/2}} \, dx &=-\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{c e \sqrt {e x}}+\frac {2 \int \sqrt {e x} \left (\frac {1}{2} a (2 b c+9 a d)+\frac {1}{2} b^2 c x^2\right ) \left (c+d x^2\right )^{3/2} \, dx}{c e^2}\\ &=-\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{c e \sqrt {e x}}+\frac {2 b^2 (e x)^{3/2} \left (c+d x^2\right )^{5/2}}{13 d e^3}-\frac {\left (4 \left (\frac {3 b^2 c^2}{4}-\frac {13}{4} a d (2 b c+9 a d)\right )\right ) \int \sqrt {e x} \left (c+d x^2\right )^{3/2} \, dx}{13 c d e^2}\\ &=-\frac {2 \left (3 b^2 c^2-13 a d (2 b c+9 a d)\right ) (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{117 c d e^3}-\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{c e \sqrt {e x}}+\frac {2 b^2 (e x)^{3/2} \left (c+d x^2\right )^{5/2}}{13 d e^3}-\frac {\left (2 \left (3 b^2 c^2-13 a d (2 b c+9 a d)\right )\right ) \int \sqrt {e x} \sqrt {c+d x^2} \, dx}{39 d e^2}\\ &=-\frac {4 \left (3 b^2 c^2-13 a d (2 b c+9 a d)\right ) (e x)^{3/2} \sqrt {c+d x^2}}{195 d e^3}-\frac {2 \left (3 b^2 c^2-13 a d (2 b c+9 a d)\right ) (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{117 c d e^3}-\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{c e \sqrt {e x}}+\frac {2 b^2 (e x)^{3/2} \left (c+d x^2\right )^{5/2}}{13 d e^3}-\frac {\left (4 c \left (3 b^2 c^2-13 a d (2 b c+9 a d)\right )\right ) \int \frac {\sqrt {e x}}{\sqrt {c+d x^2}} \, dx}{195 d e^2}\\ &=-\frac {4 \left (3 b^2 c^2-13 a d (2 b c+9 a d)\right ) (e x)^{3/2} \sqrt {c+d x^2}}{195 d e^3}-\frac {2 \left (3 b^2 c^2-13 a d (2 b c+9 a d)\right ) (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{117 c d e^3}-\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{c e \sqrt {e x}}+\frac {2 b^2 (e x)^{3/2} \left (c+d x^2\right )^{5/2}}{13 d e^3}-\frac {\left (8 c \left (3 b^2 c^2-13 a d (2 b c+9 a d)\right )\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{195 d e^3}\\ &=-\frac {4 \left (3 b^2 c^2-13 a d (2 b c+9 a d)\right ) (e x)^{3/2} \sqrt {c+d x^2}}{195 d e^3}-\frac {2 \left (3 b^2 c^2-13 a d (2 b c+9 a d)\right ) (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{117 c d e^3}-\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{c e \sqrt {e x}}+\frac {2 b^2 (e x)^{3/2} \left (c+d x^2\right )^{5/2}}{13 d e^3}-\frac {\left (8 c^{3/2} \left (3 b^2 c^2-13 a d (2 b c+9 a d)\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{195 d^{3/2} e^2}+\frac {\left (8 c^{3/2} \left (3 b^2 c^2-13 a d (2 b c+9 a d)\right )\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt {d} x^2}{\sqrt {c} e}}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{195 d^{3/2} e^2}\\ &=-\frac {4 \left (3 b^2 c^2-13 a d (2 b c+9 a d)\right ) (e x)^{3/2} \sqrt {c+d x^2}}{195 d e^3}-\frac {8 c \left (3 b^2 c^2-13 a d (2 b c+9 a d)\right ) \sqrt {e x} \sqrt {c+d x^2}}{195 d^{3/2} e^2 \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {2 \left (3 b^2 c^2-13 a d (2 b c+9 a d)\right ) (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{117 c d e^3}-\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{c e \sqrt {e x}}+\frac {2 b^2 (e x)^{3/2} \left (c+d x^2\right )^{5/2}}{13 d e^3}+\frac {8 c^{5/4} \left (3 b^2 c^2-13 a d (2 b c+9 a d)\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{195 d^{7/4} e^{3/2} \sqrt {c+d x^2}}-\frac {4 c^{5/4} \left (3 b^2 c^2-13 a d (2 b c+9 a d)\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{195 d^{7/4} e^{3/2} \sqrt {c+d 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 = 20.13, size = 161, normalized size = 0.34 \begin {gather*} \frac {x \left (2 \left (c+d x^2\right ) \left (117 a^2 d \left (-5 c+d x^2\right )+26 a b d x^2 \left (11 c+5 d x^2\right )+3 b^2 x^2 \left (4 c^2+25 c d x^2+15 d^2 x^4\right )\right )+24 c \left (-3 b^2 c^2+26 a b c d+117 a^2 d^2\right ) \sqrt {1+\frac {c}{d x^2}} x^2 \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};-\frac {c}{d x^2}\right )\right )}{585 d (e x)^{3/2} \sqrt {c+d x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(2*(c + d*x^2)*(117*a^2*d*(-5*c + d*x^2) + 26*a*b*d*x^2*(11*c + 5*d*x^2) + 3*b^2*x^2*(4*c^2 + 25*c*d*x^2 +
15*d^2*x^4)) + 24*c*(-3*b^2*c^2 + 26*a*b*c*d + 117*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x^2*Hypergeometric2F1[-1/4, 1/
2, 3/4, -(c/(d*x^2))]))/(585*d*(e*x)^(3/2)*Sqrt[c + d*x^2])

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Maple [A]
time = 0.12, size = 669, normalized size = 1.41

method	result	size
risch	\(-\frac {2 \sqrt {d \,x^{2}+c}\, \left (-45 b^{2} d^{2} x^{6}-130 a b \,d^{2} x^{4}-75 b^{2} c d \,x^{4}-117 a^{2} d^{2} x^{2}-286 a b c d \,x^{2}-12 b^{2} c^{2} x^{2}+585 a^{2} c d \right )}{585 d e \sqrt {e x}}+\frac {4 c \left (117 a^{2} d^{2}+26 a b c d -3 b^{2} c^{2}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right ) \sqrt {e x \left (d \,x^{2}+c \right )}}{195 d^{2} \sqrt {d e \,x^{3}+c e x}\, e \sqrt {e x}\, \sqrt {d \,x^{2}+c}}\)	\(307\)
elliptic	\(\frac {\sqrt {e x \left (d \,x^{2}+c \right )}\, \left (-\frac {2 \left (d e \,x^{2}+c e \right ) c \,a^{2}}{e^{2} \sqrt {x \left (d e \,x^{2}+c e \right )}}+\frac {2 b^{2} d \,x^{5} \sqrt {d e \,x^{3}+c e x}}{13 e^{2}}+\frac {2 \left (\frac {2 b d \left (a d +b c \right )}{e}-\frac {11 b^{2} d c}{13 e}\right ) x^{3} \sqrt {d e \,x^{3}+c e x}}{9 d e}+\frac {2 \left (\frac {a^{2} d^{2}+4 a b c d +b^{2} c^{2}}{e}-\frac {7 \left (\frac {2 b d \left (a d +b c \right )}{e}-\frac {11 b^{2} d c}{13 e}\right ) c}{9 d}\right ) x \sqrt {d e \,x^{3}+c e x}}{5 d e}+\frac {\left (\frac {2 a c \left (a d +b c \right )}{e}+\frac {c d \,a^{2}}{e}-\frac {3 \left (\frac {a^{2} d^{2}+4 a b c d +b^{2} c^{2}}{e}-\frac {7 \left (\frac {2 b d \left (a d +b c \right )}{e}-\frac {11 b^{2} d c}{13 e}\right ) c}{9 d}\right ) c}{5 d}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right )}{d \sqrt {d e \,x^{3}+c e x}}\right )}{\sqrt {e x}\, \sqrt {d \,x^{2}+c}}\)	\(461\)
default	\(\frac {\frac {2 b^{2} d^{4} x^{8}}{13}+\frac {4 a b \,d^{4} x^{6}}{9}+\frac {16 b^{2} c \,d^{3} x^{6}}{39}+\frac {24 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticE \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c^{2} d^{2}}{5}+\frac {16 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticE \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{3} d}{15}-\frac {8 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticE \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{4}}{65}-\frac {12 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c^{2} d^{2}}{5}-\frac {8 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{3} d}{15}+\frac {4 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{4}}{65}+\frac {2 a^{2} d^{4} x^{4}}{5}+\frac {64 a b c \,d^{3} x^{4}}{45}+\frac {58 b^{2} c^{2} d^{2} x^{4}}{195}-\frac {8 a^{2} c \,d^{3} x^{2}}{5}+\frac {44 a b \,c^{2} d^{2} x^{2}}{45}+\frac {8 b^{2} c^{3} d \,x^{2}}{195}-2 a^{2} c^{2} d^{2}}{\sqrt {d \,x^{2}+c}\, d^{2} e \sqrt {e x}}\)	\(669\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/585*(45*b^2*d^4*x^8+130*a*b*d^4*x^6+120*b^2*c*d^3*x^6+1404*((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*2^(1/2)*(
(-d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*(-x/(-c*d)^(1/2)*d)^(1/2)*EllipticE(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^
(1/2),1/2*2^(1/2))*a^2*c^2*d^2+312*((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(-c*d)^(1/2))/(-c*d)
^(1/2))^(1/2)*(-x/(-c*d)^(1/2)*d)^(1/2)*EllipticE(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*a*b*c^3
*d-36*((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*(-x/(-c*d)^(1/2
)*d)^(1/2)*EllipticE(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*b^2*c^4-702*((d*x+(-c*d)^(1/2))/(-c*
d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*(-x/(-c*d)^(1/2)*d)^(1/2)*EllipticF(((d*x+(-c
*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*a^2*c^2*d^2-156*((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*2^(1/2)*((
-d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*(-x/(-c*d)^(1/2)*d)^(1/2)*EllipticF(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(
1/2),1/2*2^(1/2))*a*b*c^3*d+18*((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(-c*d)^(1/2))/(-c*d)^(1/
2))^(1/2)*(-x/(-c*d)^(1/2)*d)^(1/2)*EllipticF(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*b^2*c^4+117
*a^2*d^4*x^4+416*a*b*c*d^3*x^4+87*b^2*c^2*d^2*x^4-468*a^2*c*d^3*x^2+286*a*b*c^2*d^2*x^2+12*b^2*c^3*d*x^2-585*a
^2*c^2*d^2)/(d*x^2+c)^(1/2)/d^2/e/(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^2+a)^2*(d*x^2+c)^(3/2)/(e*x)^(3/2),x, algorithm="maxima")

[Out]

e^(-3/2)*integrate((b*x^2 + a)^2*(d*x^2 + c)^(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 = 0.32, size = 146, normalized size = 0.31 \begin {gather*} \frac {2 \, {\left (12 \, {\left (3 \, b^{2} c^{3} - 26 \, a b c^{2} d - 117 \, a^{2} c d^{2}\right )} \sqrt {d} x {\rm weierstrassZeta}\left (-\frac {4 \, c}{d}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right )\right ) + {\left (45 \, b^{2} d^{3} x^{6} - 585 \, a^{2} c d^{2} + 5 \, {\left (15 \, b^{2} c d^{2} + 26 \, a b d^{3}\right )} x^{4} + {\left (12 \, b^{2} c^{2} d + 286 \, a b c d^{2} + 117 \, a^{2} d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {x}\right )} e^{\left (-\frac {3}{2}\right )}}{585 \, d^{2} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/585*(12*(3*b^2*c^3 - 26*a*b*c^2*d - 117*a^2*c*d^2)*sqrt(d)*x*weierstrassZeta(-4*c/d, 0, weierstrassPInverse(
-4*c/d, 0, x)) + (45*b^2*d^3*x^6 - 585*a^2*c*d^2 + 5*(15*b^2*c*d^2 + 26*a*b*d^3)*x^4 + (12*b^2*c^2*d + 286*a*b
*c*d^2 + 117*a^2*d^3)*x^2)*sqrt(d*x^2 + c)*sqrt(x))*e^(-3/2)/(d^2*x)

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Sympy [C] Result contains complex when optimal does not.
time = 10.15, size = 309, normalized size = 0.65 \begin {gather*} \frac {a^{2} c^{\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 {d x^{2} e^{i \pi }}{c}} \right )}}{2 e^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} + \frac {a^{2} \sqrt {c} d 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 {d x^{2} e^{i \pi }}{c}} \right )}}{2 e^{\frac {3}{2}} \Gamma \left (\frac {7}{4}\right )} + \frac {a b c^{\frac {3}{2}} 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 {d x^{2} e^{i \pi }}{c}} \right )}}{e^{\frac {3}{2}} \Gamma \left (\frac {7}{4}\right )} + \frac {a b \sqrt {c} d x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{e^{\frac {3}{2}} \Gamma \left (\frac {11}{4}\right )} + \frac {b^{2} c^{\frac {3}{2}} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 e^{\frac {3}{2}} \Gamma \left (\frac {11}{4}\right )} + \frac {b^{2} \sqrt {c} d x^{\frac {11}{2}} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 e^{\frac {3}{2}} \Gamma \left (\frac {15}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**2*(d*x**2+c)**(3/2)/(e*x)**(3/2),x)

[Out]

a**2*c**(3/2)*gamma(-1/4)*hyper((-1/2, -1/4), (3/4,), d*x**2*exp_polar(I*pi)/c)/(2*e**(3/2)*sqrt(x)*gamma(3/4)
) + a**2*sqrt(c)*d*x**(3/2)*gamma(3/4)*hyper((-1/2, 3/4), (7/4,), d*x**2*exp_polar(I*pi)/c)/(2*e**(3/2)*gamma(
7/4)) + a*b*c**(3/2)*x**(3/2)*gamma(3/4)*hyper((-1/2, 3/4), (7/4,), d*x**2*exp_polar(I*pi)/c)/(e**(3/2)*gamma(
7/4)) + a*b*sqrt(c)*d*x**(7/2)*gamma(7/4)*hyper((-1/2, 7/4), (11/4,), d*x**2*exp_polar(I*pi)/c)/(e**(3/2)*gamm
a(11/4)) + b**2*c**(3/2)*x**(7/2)*gamma(7/4)*hyper((-1/2, 7/4), (11/4,), d*x**2*exp_polar(I*pi)/c)/(2*e**(3/2)
*gamma(11/4)) + b**2*sqrt(c)*d*x**(11/2)*gamma(11/4)*hyper((-1/2, 11/4), (15/4,), d*x**2*exp_polar(I*pi)/c)/(2
*e**(3/2)*gamma(15/4))

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x^2)^2*(c + d*x^2)^(3/2))/(e*x)^(3/2),x)

[Out]

int(((a + b*x^2)^2*(c + d*x^2)^(3/2))/(e*x)^(3/2), x)

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