3.9.34 \(\int \sqrt {e x} (a+b x^2)^2 (c+d x^2)^{3/2} \, dx\) [834]

Optimal. Leaf size=482 \[ \frac {4 c \left (221 a^2 d^2+3 b c (7 b c-34 a d)\right ) (e x)^{3/2} \sqrt {c+d x^2}}{3315 d^2 e}+\frac {8 c^2 \left (221 a^2 d^2+3 b c (7 b c-34 a d)\right ) \sqrt {e x} \sqrt {c+d x^2}}{3315 d^{5/2} \left (\sqrt {c}+\sqrt {d} x\right )}+\frac {2 \left (221 a^2 d^2+3 b c (7 b c-34 a d)\right ) (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{1989 d^2 e}-\frac {2 b (7 b c-34 a d) (e x)^{3/2} \left (c+d x^2\right )^{5/2}}{221 d^2 e}+\frac {2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{5/2}}{17 d e^3}-\frac {8 c^{9/4} \left (221 a^2 d^2+3 b c (7 b c-34 a d)\right ) \sqrt {e} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{3315 d^{11/4} \sqrt {c+d x^2}}+\frac {4 c^{9/4} \left (221 a^2 d^2+3 b c (7 b c-34 a d)\right ) \sqrt {e} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{3315 d^{11/4} \sqrt {c+d x^2}} \]

[Out]

2/1989*(221*a^2*d^2+3*b*c*(-34*a*d+7*b*c))*(e*x)^(3/2)*(d*x^2+c)^(3/2)/d^2/e-2/221*b*(-34*a*d+7*b*c)*(e*x)^(3/
2)*(d*x^2+c)^(5/2)/d^2/e+2/17*b^2*(e*x)^(7/2)*(d*x^2+c)^(5/2)/d/e^3+4/3315*c*(221*a^2*d^2+3*b*c*(-34*a*d+7*b*c
))*(e*x)^(3/2)*(d*x^2+c)^(1/2)/d^2/e+8/3315*c^2*(221*a^2*d^2+3*b*c*(-34*a*d+7*b*c))*(e*x)^(1/2)*(d*x^2+c)^(1/2
)/d^(5/2)/(c^(1/2)+x*d^(1/2))-8/3315*c^(9/4)*(221*a^2*d^2+3*b*c*(-34*a*d+7*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)))*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))*e^(1/2)*((d*x^2+c)/(c^(1/2)+x*d^(1/2))^2)^(
1/2)/d^(11/4)/(d*x^2+c)^(1/2)+4/3315*c^(9/4)*(221*a^2*d^2+3*b*c*(-34*a*d+7*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))*e^(1/2)*((d*x^2+c)/(c^(1/2)+x*d^(1/2))^2)^(
1/2)/d^(11/4)/(d*x^2+c)^(1/2)

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Rubi [A]
time = 0.33, antiderivative size = 482, 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 = {475, 470, 285, 335, 311, 226, 1210} \begin {gather*} \frac {4 c^{9/4} \sqrt {e} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (221 a^2 d^2+3 b c (7 b c-34 a d)\right ) F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{3315 d^{11/4} \sqrt {c+d x^2}}-\frac {8 c^{9/4} \sqrt {e} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (221 a^2 d^2+3 b c (7 b c-34 a d)\right ) E\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{3315 d^{11/4} \sqrt {c+d x^2}}+\frac {8 c^2 \sqrt {e x} \sqrt {c+d x^2} \left (221 a^2 d^2+3 b c (7 b c-34 a d)\right )}{3315 d^{5/2} \left (\sqrt {c}+\sqrt {d} x\right )}+\frac {2 (e x)^{3/2} \left (c+d x^2\right )^{3/2} \left (221 a^2 d^2+3 b c (7 b c-34 a d)\right )}{1989 d^2 e}+\frac {4 c (e x)^{3/2} \sqrt {c+d x^2} \left (221 a^2 d^2+3 b c (7 b c-34 a d)\right )}{3315 d^2 e}-\frac {2 b (e x)^{3/2} \left (c+d x^2\right )^{5/2} (7 b c-34 a d)}{221 d^2 e}+\frac {2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{5/2}}{17 d e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*c*(221*a^2*d^2 + 3*b*c*(7*b*c - 34*a*d))*(e*x)^(3/2)*Sqrt[c + d*x^2])/(3315*d^2*e) + (8*c^2*(221*a^2*d^2 +
3*b*c*(7*b*c - 34*a*d))*Sqrt[e*x]*Sqrt[c + d*x^2])/(3315*d^(5/2)*(Sqrt[c] + Sqrt[d]*x)) + (2*(221*a^2*d^2 + 3*
b*c*(7*b*c - 34*a*d))*(e*x)^(3/2)*(c + d*x^2)^(3/2))/(1989*d^2*e) - (2*b*(7*b*c - 34*a*d)*(e*x)^(3/2)*(c + d*x
^2)^(5/2))/(221*d^2*e) + (2*b^2*(e*x)^(7/2)*(c + d*x^2)^(5/2))/(17*d*e^3) - (8*c^(9/4)*(221*a^2*d^2 + 3*b*c*(7
*b*c - 34*a*d))*Sqrt[e]*(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])/(3315*d^(11/4)*Sqrt[c + d*x^2]) + (4*c^(9/4)*(221*a^2*d^2 + 3*b*c*(
7*b*c - 34*a*d))*Sqrt[e]*(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])/(3315*d^(11/4)*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 475

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^2, x_Symbol] :> Simp[d^2*(e*x)^(
m + n + 1)*((a + b*x^n)^(p + 1)/(b*e^(n + 1)*(m + n*(p + 2) + 1))), x] + Dist[1/(b*(m + n*(p + 2) + 1)), Int[(
e*x)^m*(a + b*x^n)^p*Simp[b*c^2*(m + n*(p + 2) + 1) + d*((2*b*c - a*d)*(m + n + 1) + 2*b*c*n*(p + 1))*x^n, x],
x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && NeQ[m + n*(p + 2) + 1, 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 \sqrt {e x} \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2} \, dx &=\frac {2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{5/2}}{17 d e^3}+\frac {2 \int \sqrt {e x} \left (c+d x^2\right )^{3/2} \left (\frac {17 a^2 d}{2}-\frac {1}{2} b (7 b c-34 a d) x^2\right ) \, dx}{17 d}\\ &=-\frac {2 b (7 b c-34 a d) (e x)^{3/2} \left (c+d x^2\right )^{5/2}}{221 d^2 e}+\frac {2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{5/2}}{17 d e^3}-\frac {1}{221} \left (-221 a^2-\frac {3 b c (7 b c-34 a d)}{d^2}\right ) \int \sqrt {e x} \left (c+d x^2\right )^{3/2} \, dx\\ &=\frac {2 \left (221 a^2+\frac {3 b c (7 b c-34 a d)}{d^2}\right ) (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{1989 e}-\frac {2 b (7 b c-34 a d) (e x)^{3/2} \left (c+d x^2\right )^{5/2}}{221 d^2 e}+\frac {2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{5/2}}{17 d e^3}+\frac {1}{663} \left (2 c \left (221 a^2+\frac {3 b c (7 b c-34 a d)}{d^2}\right )\right ) \int \sqrt {e x} \sqrt {c+d x^2} \, dx\\ &=\frac {4 c \left (221 a^2+\frac {3 b c (7 b c-34 a d)}{d^2}\right ) (e x)^{3/2} \sqrt {c+d x^2}}{3315 e}+\frac {2 \left (221 a^2+\frac {3 b c (7 b c-34 a d)}{d^2}\right ) (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{1989 e}-\frac {2 b (7 b c-34 a d) (e x)^{3/2} \left (c+d x^2\right )^{5/2}}{221 d^2 e}+\frac {2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{5/2}}{17 d e^3}+\frac {\left (4 c^2 \left (221 a^2+\frac {3 b c (7 b c-34 a d)}{d^2}\right )\right ) \int \frac {\sqrt {e x}}{\sqrt {c+d x^2}} \, dx}{3315}\\ &=\frac {4 c \left (221 a^2+\frac {3 b c (7 b c-34 a d)}{d^2}\right ) (e x)^{3/2} \sqrt {c+d x^2}}{3315 e}+\frac {2 \left (221 a^2+\frac {3 b c (7 b c-34 a d)}{d^2}\right ) (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{1989 e}-\frac {2 b (7 b c-34 a d) (e x)^{3/2} \left (c+d x^2\right )^{5/2}}{221 d^2 e}+\frac {2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{5/2}}{17 d e^3}+\frac {\left (8 c^2 \left (221 a^2+\frac {3 b c (7 b c-34 a d)}{d^2}\right )\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{3315 e}\\ &=\frac {4 c \left (221 a^2+\frac {3 b c (7 b c-34 a d)}{d^2}\right ) (e x)^{3/2} \sqrt {c+d x^2}}{3315 e}+\frac {2 \left (221 a^2+\frac {3 b c (7 b c-34 a d)}{d^2}\right ) (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{1989 e}-\frac {2 b (7 b c-34 a d) (e x)^{3/2} \left (c+d x^2\right )^{5/2}}{221 d^2 e}+\frac {2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{5/2}}{17 d e^3}+\frac {\left (8 c^{5/2} \left (221 a^2+\frac {3 b c (7 b c-34 a d)}{d^2}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{3315 \sqrt {d}}-\frac {\left (8 c^{5/2} \left (221 a^2+\frac {3 b c (7 b c-34 a d)}{d^2}\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 )}{3315 \sqrt {d}}\\ &=\frac {4 c \left (221 a^2+\frac {3 b c (7 b c-34 a d)}{d^2}\right ) (e x)^{3/2} \sqrt {c+d x^2}}{3315 e}+\frac {8 c^2 \left (221 a^2+\frac {3 b c (7 b c-34 a d)}{d^2}\right ) \sqrt {e x} \sqrt {c+d x^2}}{3315 \sqrt {d} \left (\sqrt {c}+\sqrt {d} x\right )}+\frac {2 \left (221 a^2+\frac {3 b c (7 b c-34 a d)}{d^2}\right ) (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{1989 e}-\frac {2 b (7 b c-34 a d) (e x)^{3/2} \left (c+d x^2\right )^{5/2}}{221 d^2 e}+\frac {2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{5/2}}{17 d e^3}-\frac {8 c^{9/4} \left (221 a^2+\frac {3 b c (7 b c-34 a d)}{d^2}\right ) \sqrt {e} \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 )}{3315 d^{3/4} \sqrt {c+d x^2}}+\frac {4 c^{9/4} \left (221 a^2+\frac {3 b c (7 b c-34 a d)}{d^2}\right ) \sqrt {e} \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 )}{3315 d^{3/4} \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 = 179, normalized size = 0.37 \begin {gather*} \frac {2 \sqrt {e x} \left (-x \left (c+d x^2\right ) \left (-221 a^2 d^2 \left (11 c+5 d x^2\right )-102 a b d \left (4 c^2+25 c d x^2+15 d^2 x^4\right )+b^2 \left (84 c^3-60 c^2 d x^2-855 c d^2 x^4-585 d^3 x^6\right )\right )+12 c^2 \left (21 b^2 c^2-102 a b c d+221 a^2 d^2\right ) \sqrt {1+\frac {c}{d x^2}} x \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};-\frac {c}{d x^2}\right )\right )}{9945 d^2 \sqrt {c+d x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[e*x]*(-(x*(c + d*x^2)*(-221*a^2*d^2*(11*c + 5*d*x^2) - 102*a*b*d*(4*c^2 + 25*c*d*x^2 + 15*d^2*x^4) + b
^2*(84*c^3 - 60*c^2*d*x^2 - 855*c*d^2*x^4 - 585*d^3*x^6))) + 12*c^2*(21*b^2*c^2 - 102*a*b*c*d + 221*a^2*d^2)*S
qrt[1 + c/(d*x^2)]*x*Hypergeometric2F1[-1/4, 1/2, 3/4, -(c/(d*x^2))]))/(9945*d^2*Sqrt[c + d*x^2])

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Maple [A]
time = 0.11, size = 699, normalized size = 1.45

method	result	size
risch	\(\frac {2 x^{2} \left (585 b^{2} x^{6} d^{3}+1530 a b \,d^{3} x^{4}+855 b^{2} c \,d^{2} x^{4}+1105 a^{2} d^{3} x^{2}+2550 a b c \,d^{2} x^{2}+60 b^{2} c^{2} d \,x^{2}+2431 a^{2} c \,d^{2}+408 a b \,c^{2} d -84 b^{2} c^{3}\right ) \sqrt {d \,x^{2}+c}\, e}{9945 d^{2} \sqrt {e x}}+\frac {4 c^{2} \left (221 a^{2} d^{2}-102 a b c d +21 b^{2} c^{2}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right ) e \sqrt {e x \left (d \,x^{2}+c \right )}}{3315 d^{3} \sqrt {d e \,x^{3}+c e x}\, \sqrt {e x}\, \sqrt {d \,x^{2}+c}}\)	\(331\)
elliptic	\(\frac {\sqrt {e x \left (d \,x^{2}+c \right )}\, \sqrt {e x}\, \left (\frac {2 b^{2} d \,x^{7} \sqrt {d e \,x^{3}+c e x}}{17}+\frac {2 \left (2 b d \left (a d +b c \right ) e -\frac {15 b^{2} d c e}{17}\right ) x^{5} \sqrt {d e \,x^{3}+c e x}}{13 d e}+\frac {2 \left (\left (a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) e -\frac {11 \left (2 b d \left (a d +b c \right ) e -\frac {15 b^{2} d c e}{17}\right ) c}{13 d}\right ) x^{3} \sqrt {d e \,x^{3}+c e x}}{9 d e}+\frac {2 \left (2 a c \left (a d +b c \right ) e -\frac {7 \left (\left (a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) e -\frac {11 \left (2 b d \left (a d +b c \right ) e -\frac {15 b^{2} d c e}{17}\right ) c}{13 d}\right ) c}{9 d}\right ) x \sqrt {d e \,x^{3}+c e x}}{5 d e}+\frac {\left (a^{2} c^{2} e -\frac {3 \left (2 a c \left (a d +b c \right ) e -\frac {7 \left (\left (a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) e -\frac {11 \left (2 b d \left (a d +b c \right ) e -\frac {15 b^{2} d c e}{17}\right ) c}{13 d}\right ) c}{9 d}\right ) c}{5 d}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right )}{d \sqrt {d e \,x^{3}+c e x}}\right )}{e x \sqrt {d \,x^{2}+c}}\)	\(513\)
default	\(\frac {2 \sqrt {e x}\, \left (585 b^{2} d^{5} x^{10}+1530 a b \,d^{5} x^{8}+1440 b^{2} c \,d^{4} x^{8}+1105 a^{2} d^{5} x^{6}+4080 a b c \,d^{4} x^{6}+915 b^{2} c^{2} d^{3} x^{6}+2652 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticE \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c^{3} d^{2}-1224 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticE \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{4} d +252 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticE \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{5}-1326 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c^{3} d^{2}+612 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{4} d -126 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{5}+3536 a^{2} c \,d^{4} x^{4}+2958 a b \,c^{2} d^{3} x^{4}-24 b^{2} c^{3} d^{2} x^{4}+2431 a^{2} c^{2} d^{3} x^{2}+408 a b \,c^{3} d^{2} x^{2}-84 b^{2} c^{4} d \,x^{2}\right )}{9945 \sqrt {d \,x^{2}+c}\, d^{3} x}\)	\(699\)

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)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/9945/(d*x^2+c)^(1/2)*(e*x)^(1/2)/d^3*(585*b^2*d^5*x^10+1530*a*b*d^5*x^8+1440*b^2*c*d^4*x^8+1105*a^2*d^5*x^6+
4080*a*b*c*d^4*x^6+915*b^2*c^2*d^3*x^6+2652*((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^3*d^2-1224*((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^4*d+252*((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)*Elli
pticE(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*b^2*c^5-1326*((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^3*d^2+612*((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^4*d-126*((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^5+3536*a^2*c*d^4*x
^4+2958*a*b*c^2*d^3*x^4-24*b^2*c^3*d^2*x^4+2431*a^2*c^2*d^3*x^2+408*a*b*c^3*d^2*x^2-84*b^2*c^4*d*x^2)/x

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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)^(1/2),x, algorithm="maxima")

[Out]

e^(1/2)*integrate((b*x^2 + a)^2*(d*x^2 + c)^(3/2)*sqrt(x), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.35, size = 173, normalized size = 0.36 \begin {gather*} -\frac {2 \, {\left (12 \, {\left (21 \, b^{2} c^{4} - 102 \, a b c^{3} d + 221 \, a^{2} c^{2} d^{2}\right )} \sqrt {d} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (-\frac {4 \, c}{d}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right )\right ) - {\left (585 \, b^{2} d^{4} x^{7} + 45 \, {\left (19 \, b^{2} c d^{3} + 34 \, a b d^{4}\right )} x^{5} + 5 \, {\left (12 \, b^{2} c^{2} d^{2} + 510 \, a b c d^{3} + 221 \, a^{2} d^{4}\right )} x^{3} - {\left (84 \, b^{2} c^{3} d - 408 \, a b c^{2} d^{2} - 2431 \, a^{2} c d^{3}\right )} x\right )} \sqrt {d x^{2} + c} \sqrt {x} e^{\frac {1}{2}}\right )}}{9945 \, d^{3}} \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)^(1/2),x, algorithm="fricas")

[Out]

-2/9945*(12*(21*b^2*c^4 - 102*a*b*c^3*d + 221*a^2*c^2*d^2)*sqrt(d)*e^(1/2)*weierstrassZeta(-4*c/d, 0, weierstr
assPInverse(-4*c/d, 0, x)) - (585*b^2*d^4*x^7 + 45*(19*b^2*c*d^3 + 34*a*b*d^4)*x^5 + 5*(12*b^2*c^2*d^2 + 510*a
*b*c*d^3 + 221*a^2*d^4)*x^3 - (84*b^2*c^3*d - 408*a*b*c^2*d^2 - 2431*a^2*c*d^3)*x)*sqrt(d*x^2 + c)*sqrt(x)*e^(
1/2))/d^3

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Sympy [C] Result contains complex when optimal does not.
time = 7.09, size = 304, normalized size = 0.63 \begin {gather*} \frac {a^{2} c^{\frac {3}{2}} \left (e x\right )^{\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 \Gamma \left (\frac {7}{4}\right )} + \frac {a^{2} \sqrt {c} d \left (e x\right )^{\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^{3} \Gamma \left (\frac {11}{4}\right )} + \frac {a b c^{\frac {3}{2}} \left (e x\right )^{\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^{3} \Gamma \left (\frac {11}{4}\right )} + \frac {a b \sqrt {c} d \left (e x\right )^{\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 )}}{e^{5} \Gamma \left (\frac {15}{4}\right )} + \frac {b^{2} c^{\frac {3}{2}} \left (e x\right )^{\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^{5} \Gamma \left (\frac {15}{4}\right )} + \frac {b^{2} \sqrt {c} d \left (e x\right )^{\frac {15}{2}} \Gamma \left (\frac {15}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {15}{4} \\ \frac {19}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 e^{7} \Gamma \left (\frac {19}{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)**(1/2),x)

[Out]

a**2*c**(3/2)*(e*x)**(3/2)*gamma(3/4)*hyper((-1/2, 3/4), (7/4,), d*x**2*exp_polar(I*pi)/c)/(2*e*gamma(7/4)) +
a**2*sqrt(c)*d*(e*x)**(7/2)*gamma(7/4)*hyper((-1/2, 7/4), (11/4,), d*x**2*exp_polar(I*pi)/c)/(2*e**3*gamma(11/
4)) + a*b*c**(3/2)*(e*x)**(7/2)*gamma(7/4)*hyper((-1/2, 7/4), (11/4,), d*x**2*exp_polar(I*pi)/c)/(e**3*gamma(1
1/4)) + a*b*sqrt(c)*d*(e*x)**(11/2)*gamma(11/4)*hyper((-1/2, 11/4), (15/4,), d*x**2*exp_polar(I*pi)/c)/(e**5*g
amma(15/4)) + b**2*c**(3/2)*(e*x)**(11/2)*gamma(11/4)*hyper((-1/2, 11/4), (15/4,), d*x**2*exp_polar(I*pi)/c)/(
2*e**5*gamma(15/4)) + b**2*sqrt(c)*d*(e*x)**(15/2)*gamma(15/4)*hyper((-1/2, 15/4), (19/4,), d*x**2*exp_polar(I
*pi)/c)/(2*e**7*gamma(19/4))

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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)^(1/2),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^2*(d*x^2 + c)^(3/2)*sqrt(x)*e^(1/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^(1/2)*(a + b*x^2)^2*(c + d*x^2)^(3/2),x)

[Out]

int((e*x)^(1/2)*(a + b*x^2)^2*(c + d*x^2)^(3/2), x)

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