3.276 \(\int (d+e x^2)^{5/2} (a+b x^2+c x^4) \, dx\)

Optimal. Leaf size=215 \[ \frac{x \left (d+e x^2\right )^{5/2} \left (80 a e^2-10 b d e+3 c d^2\right )}{480 e^2}+\frac{d x \left (d+e x^2\right )^{3/2} \left (80 a e^2-10 b d e+3 c d^2\right )}{384 e^2}+\frac{d^2 x \sqrt{d+e x^2} \left (80 a e^2-10 b d e+3 c d^2\right )}{256 e^2}+\frac{d^3 \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right ) \left (80 a e^2-10 b d e+3 c d^2\right )}{256 e^{5/2}}-\frac{x \left (d+e x^2\right )^{7/2} (3 c d-10 b e)}{80 e^2}+\frac{c x^3 \left (d+e x^2\right )^{7/2}}{10 e} \]

[Out]

(d^2*(3*c*d^2 - 10*b*d*e + 80*a*e^2)*x*Sqrt[d + e*x^2])/(256*e^2) + (d*(3*c*d^2 - 10*b*d*e + 80*a*e^2)*x*(d +
e*x^2)^(3/2))/(384*e^2) + ((3*c*d^2 - 10*b*d*e + 80*a*e^2)*x*(d + e*x^2)^(5/2))/(480*e^2) - ((3*c*d - 10*b*e)*
x*(d + e*x^2)^(7/2))/(80*e^2) + (c*x^3*(d + e*x^2)^(7/2))/(10*e) + (d^3*(3*c*d^2 - 10*b*d*e + 80*a*e^2)*ArcTan
h[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(256*e^(5/2))

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Rubi [A]  time = 0.160864, antiderivative size = 215, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {1159, 388, 195, 217, 206} \[ \frac{x \left (d+e x^2\right )^{5/2} \left (80 a e^2-10 b d e+3 c d^2\right )}{480 e^2}+\frac{d x \left (d+e x^2\right )^{3/2} \left (80 a e^2-10 b d e+3 c d^2\right )}{384 e^2}+\frac{d^2 x \sqrt{d+e x^2} \left (80 a e^2-10 b d e+3 c d^2\right )}{256 e^2}+\frac{d^3 \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right ) \left (80 a e^2-10 b d e+3 c d^2\right )}{256 e^{5/2}}-\frac{x \left (d+e x^2\right )^{7/2} (3 c d-10 b e)}{80 e^2}+\frac{c x^3 \left (d+e x^2\right )^{7/2}}{10 e} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x^2)^(5/2)*(a + b*x^2 + c*x^4),x]

[Out]

(d^2*(3*c*d^2 - 10*b*d*e + 80*a*e^2)*x*Sqrt[d + e*x^2])/(256*e^2) + (d*(3*c*d^2 - 10*b*d*e + 80*a*e^2)*x*(d +
e*x^2)^(3/2))/(384*e^2) + ((3*c*d^2 - 10*b*d*e + 80*a*e^2)*x*(d + e*x^2)^(5/2))/(480*e^2) - ((3*c*d - 10*b*e)*
x*(d + e*x^2)^(7/2))/(80*e^2) + (c*x^3*(d + e*x^2)^(7/2))/(10*e) + (d^3*(3*c*d^2 - 10*b*d*e + 80*a*e^2)*ArcTan
h[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(256*e^(5/2))

Rule 1159

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[(c^p*x^(4*p - 1)*
(d + e*x^2)^(q + 1))/(e*(4*p + 2*q + 1)), x] + Dist[1/(e*(4*p + 2*q + 1)), Int[(d + e*x^2)^q*ExpandToSum[e*(4*
p + 2*q + 1)*(a + b*x^2 + c*x^4)^p - d*c^p*(4*p - 1)*x^(4*p - 2) - e*c^p*(4*p + 2*q + 1)*x^(4*p), x], x], x] /
; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] &&  !LtQ[
q, -1]

Rule 388

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

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \left (d+e x^2\right )^{5/2} \left (a+b x^2+c x^4\right ) \, dx &=\frac{c x^3 \left (d+e x^2\right )^{7/2}}{10 e}+\frac{\int \left (d+e x^2\right )^{5/2} \left (10 a e-(3 c d-10 b e) x^2\right ) \, dx}{10 e}\\ &=-\frac{(3 c d-10 b e) x \left (d+e x^2\right )^{7/2}}{80 e^2}+\frac{c x^3 \left (d+e x^2\right )^{7/2}}{10 e}-\frac{1}{80} \left (-80 a-\frac{d (3 c d-10 b e)}{e^2}\right ) \int \left (d+e x^2\right )^{5/2} \, dx\\ &=\frac{1}{480} \left (80 a+\frac{d (3 c d-10 b e)}{e^2}\right ) x \left (d+e x^2\right )^{5/2}-\frac{(3 c d-10 b e) x \left (d+e x^2\right )^{7/2}}{80 e^2}+\frac{c x^3 \left (d+e x^2\right )^{7/2}}{10 e}+\frac{1}{96} \left (d \left (80 a+\frac{d (3 c d-10 b e)}{e^2}\right )\right ) \int \left (d+e x^2\right )^{3/2} \, dx\\ &=\frac{1}{384} d \left (80 a+\frac{d (3 c d-10 b e)}{e^2}\right ) x \left (d+e x^2\right )^{3/2}+\frac{1}{480} \left (80 a+\frac{d (3 c d-10 b e)}{e^2}\right ) x \left (d+e x^2\right )^{5/2}-\frac{(3 c d-10 b e) x \left (d+e x^2\right )^{7/2}}{80 e^2}+\frac{c x^3 \left (d+e x^2\right )^{7/2}}{10 e}+\frac{1}{128} \left (d^2 \left (80 a+\frac{d (3 c d-10 b e)}{e^2}\right )\right ) \int \sqrt{d+e x^2} \, dx\\ &=\frac{1}{256} d^2 \left (80 a+\frac{d (3 c d-10 b e)}{e^2}\right ) x \sqrt{d+e x^2}+\frac{1}{384} d \left (80 a+\frac{d (3 c d-10 b e)}{e^2}\right ) x \left (d+e x^2\right )^{3/2}+\frac{1}{480} \left (80 a+\frac{d (3 c d-10 b e)}{e^2}\right ) x \left (d+e x^2\right )^{5/2}-\frac{(3 c d-10 b e) x \left (d+e x^2\right )^{7/2}}{80 e^2}+\frac{c x^3 \left (d+e x^2\right )^{7/2}}{10 e}+\frac{1}{256} \left (d^3 \left (80 a+\frac{d (3 c d-10 b e)}{e^2}\right )\right ) \int \frac{1}{\sqrt{d+e x^2}} \, dx\\ &=\frac{1}{256} d^2 \left (80 a+\frac{d (3 c d-10 b e)}{e^2}\right ) x \sqrt{d+e x^2}+\frac{1}{384} d \left (80 a+\frac{d (3 c d-10 b e)}{e^2}\right ) x \left (d+e x^2\right )^{3/2}+\frac{1}{480} \left (80 a+\frac{d (3 c d-10 b e)}{e^2}\right ) x \left (d+e x^2\right )^{5/2}-\frac{(3 c d-10 b e) x \left (d+e x^2\right )^{7/2}}{80 e^2}+\frac{c x^3 \left (d+e x^2\right )^{7/2}}{10 e}+\frac{1}{256} \left (d^3 \left (80 a+\frac{d (3 c d-10 b e)}{e^2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-e x^2} \, dx,x,\frac{x}{\sqrt{d+e x^2}}\right )\\ &=\frac{1}{256} d^2 \left (80 a+\frac{d (3 c d-10 b e)}{e^2}\right ) x \sqrt{d+e x^2}+\frac{1}{384} d \left (80 a+\frac{d (3 c d-10 b e)}{e^2}\right ) x \left (d+e x^2\right )^{3/2}+\frac{1}{480} \left (80 a+\frac{d (3 c d-10 b e)}{e^2}\right ) x \left (d+e x^2\right )^{5/2}-\frac{(3 c d-10 b e) x \left (d+e x^2\right )^{7/2}}{80 e^2}+\frac{c x^3 \left (d+e x^2\right )^{7/2}}{10 e}+\frac{d^3 \left (3 c d^2-10 b d e+80 a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{256 e^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.402441, size = 190, normalized size = 0.88 \[ \frac{\sqrt{d+e x^2} \left (\sqrt{e} x \left (10 e \left (8 a e \left (33 d^2+26 d e x^2+8 e^2 x^4\right )+b \left (118 d^2 e x^2+15 d^3+136 d e^2 x^4+48 e^3 x^6\right )\right )+c \left (744 d^2 e^2 x^4+30 d^3 e x^2-45 d^4+1008 d e^3 x^6+384 e^4 x^8\right )\right )+\frac{15 d^{5/2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (10 e (8 a e-b d)+3 c d^2\right )}{\sqrt{\frac{e x^2}{d}+1}}\right )}{3840 e^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x^2)^(5/2)*(a + b*x^2 + c*x^4),x]

[Out]

(Sqrt[d + e*x^2]*(Sqrt[e]*x*(c*(-45*d^4 + 30*d^3*e*x^2 + 744*d^2*e^2*x^4 + 1008*d*e^3*x^6 + 384*e^4*x^8) + 10*
e*(8*a*e*(33*d^2 + 26*d*e*x^2 + 8*e^2*x^4) + b*(15*d^3 + 118*d^2*e*x^2 + 136*d*e^2*x^4 + 48*e^3*x^6))) + (15*d
^(5/2)*(3*c*d^2 + 10*e*(-(b*d) + 8*a*e))*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[1 + (e*x^2)/d]))/(3840*e^(5/2))

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Maple [A]  time = 0.008, size = 283, normalized size = 1.3 \begin{align*}{\frac{c{x}^{3}}{10\,e} \left ( e{x}^{2}+d \right ) ^{{\frac{7}{2}}}}-{\frac{3\,cdx}{80\,{e}^{2}} \left ( e{x}^{2}+d \right ) ^{{\frac{7}{2}}}}+{\frac{c{d}^{2}x}{160\,{e}^{2}} \left ( e{x}^{2}+d \right ) ^{{\frac{5}{2}}}}+{\frac{c{d}^{3}x}{128\,{e}^{2}} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}+{\frac{3\,c{d}^{4}x}{256\,{e}^{2}}\sqrt{e{x}^{2}+d}}+{\frac{3\,c{d}^{5}}{256}\ln \left ( x\sqrt{e}+\sqrt{e{x}^{2}+d} \right ){e}^{-{\frac{5}{2}}}}+{\frac{bx}{8\,e} \left ( e{x}^{2}+d \right ) ^{{\frac{7}{2}}}}-{\frac{bdx}{48\,e} \left ( e{x}^{2}+d \right ) ^{{\frac{5}{2}}}}-{\frac{5\,b{d}^{2}x}{192\,e} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{d}^{3}bx}{128\,e}\sqrt{e{x}^{2}+d}}-{\frac{5\,{d}^{4}b}{128}\ln \left ( x\sqrt{e}+\sqrt{e{x}^{2}+d} \right ){e}^{-{\frac{3}{2}}}}+{\frac{ax}{6} \left ( e{x}^{2}+d \right ) ^{{\frac{5}{2}}}}+{\frac{5\,adx}{24} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}+{\frac{5\,a{d}^{2}x}{16}\sqrt{e{x}^{2}+d}}+{\frac{5\,a{d}^{3}}{16}\ln \left ( x\sqrt{e}+\sqrt{e{x}^{2}+d} \right ){\frac{1}{\sqrt{e}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)^(5/2)*(c*x^4+b*x^2+a),x)

[Out]

1/10*c*x^3*(e*x^2+d)^(7/2)/e-3/80*c*d/e^2*x*(e*x^2+d)^(7/2)+1/160*c*d^2/e^2*x*(e*x^2+d)^(5/2)+1/128*c*d^3/e^2*
x*(e*x^2+d)^(3/2)+3/256*c*d^4/e^2*x*(e*x^2+d)^(1/2)+3/256*c*d^5/e^(5/2)*ln(x*e^(1/2)+(e*x^2+d)^(1/2))+1/8*b*x*
(e*x^2+d)^(7/2)/e-1/48*b*d/e*x*(e*x^2+d)^(5/2)-5/192*b*d^2/e*x*(e*x^2+d)^(3/2)-5/128*b*d^3/e*x*(e*x^2+d)^(1/2)
-5/128*b*d^4/e^(3/2)*ln(x*e^(1/2)+(e*x^2+d)^(1/2))+1/6*a*x*(e*x^2+d)^(5/2)+5/24*a*d*x*(e*x^2+d)^(3/2)+5/16*a*d
^2*x*(e*x^2+d)^(1/2)+5/16*a*d^3/e^(1/2)*ln(x*e^(1/2)+(e*x^2+d)^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^(5/2)*(c*x^4+b*x^2+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 5.98159, size = 882, normalized size = 4.1 \begin{align*} \left [\frac{15 \,{\left (3 \, c d^{5} - 10 \, b d^{4} e + 80 \, a d^{3} e^{2}\right )} \sqrt{e} \log \left (-2 \, e x^{2} - 2 \, \sqrt{e x^{2} + d} \sqrt{e} x - d\right ) + 2 \,{\left (384 \, c e^{5} x^{9} + 48 \,{\left (21 \, c d e^{4} + 10 \, b e^{5}\right )} x^{7} + 8 \,{\left (93 \, c d^{2} e^{3} + 170 \, b d e^{4} + 80 \, a e^{5}\right )} x^{5} + 10 \,{\left (3 \, c d^{3} e^{2} + 118 \, b d^{2} e^{3} + 208 \, a d e^{4}\right )} x^{3} - 15 \,{\left (3 \, c d^{4} e - 10 \, b d^{3} e^{2} - 176 \, a d^{2} e^{3}\right )} x\right )} \sqrt{e x^{2} + d}}{7680 \, e^{3}}, -\frac{15 \,{\left (3 \, c d^{5} - 10 \, b d^{4} e + 80 \, a d^{3} e^{2}\right )} \sqrt{-e} \arctan \left (\frac{\sqrt{-e} x}{\sqrt{e x^{2} + d}}\right ) -{\left (384 \, c e^{5} x^{9} + 48 \,{\left (21 \, c d e^{4} + 10 \, b e^{5}\right )} x^{7} + 8 \,{\left (93 \, c d^{2} e^{3} + 170 \, b d e^{4} + 80 \, a e^{5}\right )} x^{5} + 10 \,{\left (3 \, c d^{3} e^{2} + 118 \, b d^{2} e^{3} + 208 \, a d e^{4}\right )} x^{3} - 15 \,{\left (3 \, c d^{4} e - 10 \, b d^{3} e^{2} - 176 \, a d^{2} e^{3}\right )} x\right )} \sqrt{e x^{2} + d}}{3840 \, e^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^(5/2)*(c*x^4+b*x^2+a),x, algorithm="fricas")

[Out]

[1/7680*(15*(3*c*d^5 - 10*b*d^4*e + 80*a*d^3*e^2)*sqrt(e)*log(-2*e*x^2 - 2*sqrt(e*x^2 + d)*sqrt(e)*x - d) + 2*
(384*c*e^5*x^9 + 48*(21*c*d*e^4 + 10*b*e^5)*x^7 + 8*(93*c*d^2*e^3 + 170*b*d*e^4 + 80*a*e^5)*x^5 + 10*(3*c*d^3*
e^2 + 118*b*d^2*e^3 + 208*a*d*e^4)*x^3 - 15*(3*c*d^4*e - 10*b*d^3*e^2 - 176*a*d^2*e^3)*x)*sqrt(e*x^2 + d))/e^3
, -1/3840*(15*(3*c*d^5 - 10*b*d^4*e + 80*a*d^3*e^2)*sqrt(-e)*arctan(sqrt(-e)*x/sqrt(e*x^2 + d)) - (384*c*e^5*x
^9 + 48*(21*c*d*e^4 + 10*b*e^5)*x^7 + 8*(93*c*d^2*e^3 + 170*b*d*e^4 + 80*a*e^5)*x^5 + 10*(3*c*d^3*e^2 + 118*b*
d^2*e^3 + 208*a*d*e^4)*x^3 - 15*(3*c*d^4*e - 10*b*d^3*e^2 - 176*a*d^2*e^3)*x)*sqrt(e*x^2 + d))/e^3]

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Sympy [B]  time = 55.0993, size = 505, normalized size = 2.35 \begin{align*} \frac{a d^{\frac{5}{2}} x \sqrt{1 + \frac{e x^{2}}{d}}}{2} + \frac{3 a d^{\frac{5}{2}} x}{16 \sqrt{1 + \frac{e x^{2}}{d}}} + \frac{35 a d^{\frac{3}{2}} e x^{3}}{48 \sqrt{1 + \frac{e x^{2}}{d}}} + \frac{17 a \sqrt{d} e^{2} x^{5}}{24 \sqrt{1 + \frac{e x^{2}}{d}}} + \frac{5 a d^{3} \operatorname{asinh}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{16 \sqrt{e}} + \frac{a e^{3} x^{7}}{6 \sqrt{d} \sqrt{1 + \frac{e x^{2}}{d}}} + \frac{5 b d^{\frac{7}{2}} x}{128 e \sqrt{1 + \frac{e x^{2}}{d}}} + \frac{133 b d^{\frac{5}{2}} x^{3}}{384 \sqrt{1 + \frac{e x^{2}}{d}}} + \frac{127 b d^{\frac{3}{2}} e x^{5}}{192 \sqrt{1 + \frac{e x^{2}}{d}}} + \frac{23 b \sqrt{d} e^{2} x^{7}}{48 \sqrt{1 + \frac{e x^{2}}{d}}} - \frac{5 b d^{4} \operatorname{asinh}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{128 e^{\frac{3}{2}}} + \frac{b e^{3} x^{9}}{8 \sqrt{d} \sqrt{1 + \frac{e x^{2}}{d}}} - \frac{3 c d^{\frac{9}{2}} x}{256 e^{2} \sqrt{1 + \frac{e x^{2}}{d}}} - \frac{c d^{\frac{7}{2}} x^{3}}{256 e \sqrt{1 + \frac{e x^{2}}{d}}} + \frac{129 c d^{\frac{5}{2}} x^{5}}{640 \sqrt{1 + \frac{e x^{2}}{d}}} + \frac{73 c d^{\frac{3}{2}} e x^{7}}{160 \sqrt{1 + \frac{e x^{2}}{d}}} + \frac{29 c \sqrt{d} e^{2} x^{9}}{80 \sqrt{1 + \frac{e x^{2}}{d}}} + \frac{3 c d^{5} \operatorname{asinh}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{256 e^{\frac{5}{2}}} + \frac{c e^{3} x^{11}}{10 \sqrt{d} \sqrt{1 + \frac{e x^{2}}{d}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d)**(5/2)*(c*x**4+b*x**2+a),x)

[Out]

a*d**(5/2)*x*sqrt(1 + e*x**2/d)/2 + 3*a*d**(5/2)*x/(16*sqrt(1 + e*x**2/d)) + 35*a*d**(3/2)*e*x**3/(48*sqrt(1 +
e*x**2/d)) + 17*a*sqrt(d)*e**2*x**5/(24*sqrt(1 + e*x**2/d)) + 5*a*d**3*asinh(sqrt(e)*x/sqrt(d))/(16*sqrt(e))
+ a*e**3*x**7/(6*sqrt(d)*sqrt(1 + e*x**2/d)) + 5*b*d**(7/2)*x/(128*e*sqrt(1 + e*x**2/d)) + 133*b*d**(5/2)*x**3
/(384*sqrt(1 + e*x**2/d)) + 127*b*d**(3/2)*e*x**5/(192*sqrt(1 + e*x**2/d)) + 23*b*sqrt(d)*e**2*x**7/(48*sqrt(1
+ e*x**2/d)) - 5*b*d**4*asinh(sqrt(e)*x/sqrt(d))/(128*e**(3/2)) + b*e**3*x**9/(8*sqrt(d)*sqrt(1 + e*x**2/d))
- 3*c*d**(9/2)*x/(256*e**2*sqrt(1 + e*x**2/d)) - c*d**(7/2)*x**3/(256*e*sqrt(1 + e*x**2/d)) + 129*c*d**(5/2)*x
**5/(640*sqrt(1 + e*x**2/d)) + 73*c*d**(3/2)*e*x**7/(160*sqrt(1 + e*x**2/d)) + 29*c*sqrt(d)*e**2*x**9/(80*sqrt
(1 + e*x**2/d)) + 3*c*d**5*asinh(sqrt(e)*x/sqrt(d))/(256*e**(5/2)) + c*e**3*x**11/(10*sqrt(d)*sqrt(1 + e*x**2/
d))

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Giac [A]  time = 1.21445, size = 243, normalized size = 1.13 \begin{align*} -\frac{1}{256} \,{\left (3 \, c d^{5} - 10 \, b d^{4} e + 80 \, a d^{3} e^{2}\right )} e^{\left (-\frac{5}{2}\right )} \log \left ({\left | -x e^{\frac{1}{2}} + \sqrt{x^{2} e + d} \right |}\right ) + \frac{1}{3840} \,{\left (2 \,{\left (4 \,{\left (6 \,{\left (8 \, c x^{2} e^{2} +{\left (21 \, c d e^{9} + 10 \, b e^{10}\right )} e^{\left (-8\right )}\right )} x^{2} +{\left (93 \, c d^{2} e^{8} + 170 \, b d e^{9} + 80 \, a e^{10}\right )} e^{\left (-8\right )}\right )} x^{2} + 5 \,{\left (3 \, c d^{3} e^{7} + 118 \, b d^{2} e^{8} + 208 \, a d e^{9}\right )} e^{\left (-8\right )}\right )} x^{2} - 15 \,{\left (3 \, c d^{4} e^{6} - 10 \, b d^{3} e^{7} - 176 \, a d^{2} e^{8}\right )} e^{\left (-8\right )}\right )} \sqrt{x^{2} e + d} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^(5/2)*(c*x^4+b*x^2+a),x, algorithm="giac")

[Out]

-1/256*(3*c*d^5 - 10*b*d^4*e + 80*a*d^3*e^2)*e^(-5/2)*log(abs(-x*e^(1/2) + sqrt(x^2*e + d))) + 1/3840*(2*(4*(6
*(8*c*x^2*e^2 + (21*c*d*e^9 + 10*b*e^10)*e^(-8))*x^2 + (93*c*d^2*e^8 + 170*b*d*e^9 + 80*a*e^10)*e^(-8))*x^2 +
5*(3*c*d^3*e^7 + 118*b*d^2*e^8 + 208*a*d*e^9)*e^(-8))*x^2 - 15*(3*c*d^4*e^6 - 10*b*d^3*e^7 - 176*a*d^2*e^8)*e^
(-8))*sqrt(x^2*e + d)*x