3.9.39 \(\int \frac {(e x)^{5/2} (a+b x^2)^2}{\sqrt {c+d x^2}} \, dx\) [839]

Optimal. Leaf size=430 \[ \frac {2 \left (117 a^2 d^2+7 b c (11 b c-26 a d)\right ) e (e x)^{3/2} \sqrt {c+d x^2}}{585 d^3}-\frac {2 b (11 b c-26 a d) (e x)^{7/2} \sqrt {c+d x^2}}{117 d^2 e}+\frac {2 b^2 (e x)^{11/2} \sqrt {c+d x^2}}{13 d e^3}-\frac {2 c \left (117 a^2 d^2+7 b c (11 b c-26 a d)\right ) e^2 \sqrt {e x} \sqrt {c+d x^2}}{195 d^{7/2} \left (\sqrt {c}+\sqrt {d} x\right )}+\frac {2 c^{5/4} \left (117 a^2 d^2+7 b c (11 b c-26 a d)\right ) e^{5/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{195 d^{15/4} \sqrt {c+d x^2}}-\frac {c^{5/4} \left (117 a^2 d^2+7 b c (11 b c-26 a d)\right ) e^{5/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{195 d^{15/4} \sqrt {c+d x^2}} \]

[Out]

2/585*(117*a^2*d^2+7*b*c*(-26*a*d+11*b*c))*e*(e*x)^(3/2)*(d*x^2+c)^(1/2)/d^3-2/117*b*(-26*a*d+11*b*c)*(e*x)^(7
/2)*(d*x^2+c)^(1/2)/d^2/e+2/13*b^2*(e*x)^(11/2)*(d*x^2+c)^(1/2)/d/e^3-2/195*c*(117*a^2*d^2+7*b*c*(-26*a*d+11*b
*c))*e^2*(e*x)^(1/2)*(d*x^2+c)^(1/2)/d^(7/2)/(c^(1/2)+x*d^(1/2))+2/195*c^(5/4)*(117*a^2*d^2+7*b*c*(-26*a*d+11*
b*c))*e^(5/2)*(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))*
((d*x^2+c)/(c^(1/2)+x*d^(1/2))^2)^(1/2)/d^(15/4)/(d*x^2+c)^(1/2)-1/195*c^(5/4)*(117*a^2*d^2+7*b*c*(-26*a*d+11*
b*c))*e^(5/2)*(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^(15/4)/(d*x^2+c)^(1/2)

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Rubi [A]
time = 0.28, antiderivative size = 430, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {475, 470, 327, 335, 311, 226, 1210} \begin {gather*} -\frac {c^{5/4} e^{5/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (117 a^2 d^2+7 b c (11 b c-26 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 )}{195 d^{15/4} \sqrt {c+d x^2}}+\frac {2 c^{5/4} e^{5/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (117 a^2 d^2+7 b c (11 b c-26 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 )}{195 d^{15/4} \sqrt {c+d x^2}}-\frac {2 c e^2 \sqrt {e x} \sqrt {c+d x^2} \left (117 a^2 d^2+7 b c (11 b c-26 a d)\right )}{195 d^{7/2} \left (\sqrt {c}+\sqrt {d} x\right )}+\frac {2 e (e x)^{3/2} \sqrt {c+d x^2} \left (117 a^2 d^2+7 b c (11 b c-26 a d)\right )}{585 d^3}-\frac {2 b (e x)^{7/2} \sqrt {c+d x^2} (11 b c-26 a d)}{117 d^2 e}+\frac {2 b^2 (e x)^{11/2} \sqrt {c+d x^2}}{13 d e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((e*x)^(5/2)*(a + b*x^2)^2)/Sqrt[c + d*x^2],x]

[Out]

(2*(117*a^2*d^2 + 7*b*c*(11*b*c - 26*a*d))*e*(e*x)^(3/2)*Sqrt[c + d*x^2])/(585*d^3) - (2*b*(11*b*c - 26*a*d)*(
e*x)^(7/2)*Sqrt[c + d*x^2])/(117*d^2*e) + (2*b^2*(e*x)^(11/2)*Sqrt[c + d*x^2])/(13*d*e^3) - (2*c*(117*a^2*d^2
+ 7*b*c*(11*b*c - 26*a*d))*e^2*Sqrt[e*x]*Sqrt[c + d*x^2])/(195*d^(7/2)*(Sqrt[c] + Sqrt[d]*x)) + (2*c^(5/4)*(11
7*a^2*d^2 + 7*b*c*(11*b*c - 26*a*d))*e^(5/2)*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*E
llipticE[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(195*d^(15/4)*Sqrt[c + d*x^2]) - (c^(5/4)*(117
*a^2*d^2 + 7*b*c*(11*b*c - 26*a*d))*e^(5/2)*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*El
lipticF[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(195*d^(15/4)*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 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 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && 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 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 \frac {(e x)^{5/2} \left (a+b x^2\right )^2}{\sqrt {c+d x^2}} \, dx &=\frac {2 b^2 (e x)^{11/2} \sqrt {c+d x^2}}{13 d e^3}+\frac {2 \int \frac {(e x)^{5/2} \left (\frac {13 a^2 d}{2}-\frac {1}{2} b (11 b c-26 a d) x^2\right )}{\sqrt {c+d x^2}} \, dx}{13 d}\\ &=-\frac {2 b (11 b c-26 a d) (e x)^{7/2} \sqrt {c+d x^2}}{117 d^2 e}+\frac {2 b^2 (e x)^{11/2} \sqrt {c+d x^2}}{13 d e^3}-\frac {1}{117} \left (-117 a^2-\frac {7 b c (11 b c-26 a d)}{d^2}\right ) \int \frac {(e x)^{5/2}}{\sqrt {c+d x^2}} \, dx\\ &=\frac {2 \left (117 a^2+\frac {7 b c (11 b c-26 a d)}{d^2}\right ) e (e x)^{3/2} \sqrt {c+d x^2}}{585 d}-\frac {2 b (11 b c-26 a d) (e x)^{7/2} \sqrt {c+d x^2}}{117 d^2 e}+\frac {2 b^2 (e x)^{11/2} \sqrt {c+d x^2}}{13 d e^3}-\frac {\left (c \left (117 a^2+\frac {7 b c (11 b c-26 a d)}{d^2}\right ) e^2\right ) \int \frac {\sqrt {e x}}{\sqrt {c+d x^2}} \, dx}{195 d}\\ &=\frac {2 \left (117 a^2+\frac {7 b c (11 b c-26 a d)}{d^2}\right ) e (e x)^{3/2} \sqrt {c+d x^2}}{585 d}-\frac {2 b (11 b c-26 a d) (e x)^{7/2} \sqrt {c+d x^2}}{117 d^2 e}+\frac {2 b^2 (e x)^{11/2} \sqrt {c+d x^2}}{13 d e^3}-\frac {\left (2 c \left (117 a^2+\frac {7 b c (11 b c-26 a d)}{d^2}\right ) e\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{195 d}\\ &=\frac {2 \left (117 a^2+\frac {7 b c (11 b c-26 a d)}{d^2}\right ) e (e x)^{3/2} \sqrt {c+d x^2}}{585 d}-\frac {2 b (11 b c-26 a d) (e x)^{7/2} \sqrt {c+d x^2}}{117 d^2 e}+\frac {2 b^2 (e x)^{11/2} \sqrt {c+d x^2}}{13 d e^3}-\frac {\left (2 c^{3/2} \left (117 a^2+\frac {7 b c (11 b c-26 a d)}{d^2}\right ) e^2\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}}+\frac {\left (2 c^{3/2} \left (117 a^2+\frac {7 b c (11 b c-26 a d)}{d^2}\right ) e^2\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}}\\ &=\frac {2 \left (117 a^2+\frac {7 b c (11 b c-26 a d)}{d^2}\right ) e (e x)^{3/2} \sqrt {c+d x^2}}{585 d}-\frac {2 b (11 b c-26 a d) (e x)^{7/2} \sqrt {c+d x^2}}{117 d^2 e}+\frac {2 b^2 (e x)^{11/2} \sqrt {c+d x^2}}{13 d e^3}-\frac {2 c \left (117 a^2+\frac {7 b c (11 b c-26 a d)}{d^2}\right ) e^2 \sqrt {e x} \sqrt {c+d x^2}}{195 d^{3/2} \left (\sqrt {c}+\sqrt {d} x\right )}+\frac {2 c^{5/4} \left (117 a^2+\frac {7 b c (11 b c-26 a d)}{d^2}\right ) e^{5/2} \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} \sqrt {c+d x^2}}-\frac {c^{5/4} \left (117 a^2+\frac {7 b c (11 b c-26 a d)}{d^2}\right ) e^{5/2} \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} \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 = 20.12, size = 143, normalized size = 0.33 \begin {gather*} \frac {2 e (e x)^{3/2} \left (\left (c+d x^2\right ) \left (117 a^2 d^2+26 a b d \left (-7 c+5 d x^2\right )+b^2 \left (77 c^2-55 c d x^2+45 d^2 x^4\right )\right )-3 c \left (77 b^2 c^2-182 a b c d+117 a^2 d^2\right ) \sqrt {1+\frac {c}{d x^2}} \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};-\frac {c}{d x^2}\right )\right )}{585 d^3 \sqrt {c+d x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((e*x)^(5/2)*(a + b*x^2)^2)/Sqrt[c + d*x^2],x]

[Out]

(2*e*(e*x)^(3/2)*((c + d*x^2)*(117*a^2*d^2 + 26*a*b*d*(-7*c + 5*d*x^2) + b^2*(77*c^2 - 55*c*d*x^2 + 45*d^2*x^4
)) - 3*c*(77*b^2*c^2 - 182*a*b*c*d + 117*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*Hypergeometric2F1[-1/4, 1/2, 3/4, -(c/(d
*x^2))]))/(585*d^3*Sqrt[c + d*x^2])

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Maple [A]
time = 0.11, size = 661, normalized size = 1.54

method	result	size
risch	\(\frac {2 x^{2} \left (45 b^{2} x^{4} d^{2}+130 a b \,d^{2} x^{2}-55 b^{2} c d \,x^{2}+117 a^{2} d^{2}-182 a b c d +77 b^{2} c^{2}\right ) \sqrt {d \,x^{2}+c}\, e^{3}}{585 d^{3} \sqrt {e x}}-\frac {c \left (117 a^{2} d^{2}-182 a b c d +77 b^{2} c^{2}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right ) e^{3} \sqrt {e x \left (d \,x^{2}+c \right )}}{195 d^{4} \sqrt {d e \,x^{3}+c e x}\, \sqrt {e x}\, \sqrt {d \,x^{2}+c}}\)	\(294\)
elliptic	\(\frac {\sqrt {e x \left (d \,x^{2}+c \right )}\, \sqrt {e x}\, \left (\frac {2 b^{2} e^{2} x^{5} \sqrt {d e \,x^{3}+c e x}}{13 d}+\frac {2 \left (2 a b \,e^{3}-\frac {11 b^{2} e^{3} c}{13 d}\right ) x^{3} \sqrt {d e \,x^{3}+c e x}}{9 d e}+\frac {2 \left (a^{2} e^{3}-\frac {7 \left (2 a b \,e^{3}-\frac {11 b^{2} e^{3} c}{13 d}\right ) c}{9 d}\right ) x \sqrt {d e \,x^{3}+c e x}}{5 d e}-\frac {3 \left (a^{2} e^{3}-\frac {7 \left (2 a b \,e^{3}-\frac {11 b^{2} e^{3} c}{13 d}\right ) c}{9 d}\right ) c \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right )}{5 d^{2} \sqrt {d e \,x^{3}+c e x}}\right )}{e x \sqrt {d \,x^{2}+c}}\)	\(357\)
default	\(-\frac {e^{2} \sqrt {e x}\, \left (-90 b^{2} d^{4} x^{8}-260 a b \,d^{4} x^{6}+20 b^{2} c \,d^{3} x^{6}+702 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticE \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c^{2} d^{2}-1092 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticE \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{3} d +462 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticE \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{4}-351 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c^{2} d^{2}+546 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{3} d -231 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{4}-234 a^{2} d^{4} x^{4}+104 a b c \,d^{3} x^{4}-44 b^{2} c^{2} d^{2} x^{4}-234 a^{2} c \,d^{3} x^{2}+364 a b \,c^{2} d^{2} x^{2}-154 b^{2} c^{3} d \,x^{2}\right )}{585 x \sqrt {d \,x^{2}+c}\, d^{4}}\)	\(661\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^(5/2)*(b*x^2+a)^2/(d*x^2+c)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/585*e^2/x*(e*x)^(1/2)/(d*x^2+c)^(1/2)/d^4*(-90*b^2*d^4*x^8-260*a*b*d^4*x^6+20*b^2*c*d^3*x^6+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)*Ellipt
icE(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*a^2*c^2*d^2-1092*((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+462*((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^4-351*((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+546*((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)*Ellip
ticF(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*a*b*c^3*d-231*((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^4-234*a^2*d^4*x^4+104*a*b*c*d^3*x^4-44*b^2*c^2*d^2*x^4-234*a^2*c*d^3*x^2+3
64*a*b*c^2*d^2*x^2-154*b^2*c^3*d*x^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((e*x)^(5/2)*(b*x^2+a)^2/(d*x^2+c)^(1/2),x, algorithm="maxima")

[Out]

e^(5/2)*integrate((b*x^2 + a)^2*x^(5/2)/sqrt(d*x^2 + c), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.29, size = 133, normalized size = 0.31 \begin {gather*} \frac {2 \, {\left (3 \, {\left (77 \, b^{2} c^{3} - 182 \, a b c^{2} d + 117 \, a^{2} c d^{2}\right )} \sqrt {d} e^{\frac {5}{2}} {\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^{5} - 5 \, {\left (11 \, b^{2} c d^{2} - 26 \, a b d^{3}\right )} x^{3} + {\left (77 \, b^{2} c^{2} d - 182 \, a b c d^{2} + 117 \, a^{2} d^{3}\right )} x\right )} \sqrt {d x^{2} + c} \sqrt {x} e^{\frac {5}{2}}\right )}}{585 \, d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^(5/2)*(b*x^2+a)^2/(d*x^2+c)^(1/2),x, algorithm="fricas")

[Out]

2/585*(3*(77*b^2*c^3 - 182*a*b*c^2*d + 117*a^2*c*d^2)*sqrt(d)*e^(5/2)*weierstrassZeta(-4*c/d, 0, weierstrassPI
nverse(-4*c/d, 0, x)) + (45*b^2*d^3*x^5 - 5*(11*b^2*c*d^2 - 26*a*b*d^3)*x^3 + (77*b^2*c^2*d - 182*a*b*c*d^2 +
117*a^2*d^3)*x)*sqrt(d*x^2 + c)*sqrt(x)*e^(5/2))/d^4

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Sympy [C] Result contains complex when optimal does not.
time = 31.38, size = 144, normalized size = 0.33 \begin {gather*} \frac {a^{2} e^{\frac {5}{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 \sqrt {c} \Gamma \left (\frac {11}{4}\right )} + \frac {a b e^{\frac {5}{2}} 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 )}}{\sqrt {c} \Gamma \left (\frac {15}{4}\right )} + \frac {b^{2} e^{\frac {5}{2}} x^{\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 \sqrt {c} \Gamma \left (\frac {19}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**(5/2)*(b*x**2+a)**2/(d*x**2+c)**(1/2),x)

[Out]

a**2*e**(5/2)*x**(7/2)*gamma(7/4)*hyper((1/2, 7/4), (11/4,), d*x**2*exp_polar(I*pi)/c)/(2*sqrt(c)*gamma(11/4))
+ a*b*e**(5/2)*x**(11/2)*gamma(11/4)*hyper((1/2, 11/4), (15/4,), d*x**2*exp_polar(I*pi)/c)/(sqrt(c)*gamma(15/
4)) + b**2*e**(5/2)*x**(15/2)*gamma(15/4)*hyper((1/2, 15/4), (19/4,), d*x**2*exp_polar(I*pi)/c)/(2*sqrt(c)*gam
ma(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((e*x)^(5/2)*(b*x^2+a)^2/(d*x^2+c)^(1/2),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^2*x^(5/2)*e^(5/2)/sqrt(d*x^2 + c), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (e\,x\right )}^{5/2}\,{\left (b\,x^2+a\right )}^2}{\sqrt {d\,x^2+c}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((e*x)^(5/2)*(a + b*x^2)^2)/(c + d*x^2)^(1/2),x)

[Out]

int(((e*x)^(5/2)*(a + b*x^2)^2)/(c + d*x^2)^(1/2), x)

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