\(\int (d+e x^2)^{5/2} (a+b x^2+c x^4) \, dx\) [276]

Optimal result

Integrand size = 24, antiderivative size = 215 \[ \int \left (d+e x^2\right )^{5/2} \left (a+b x^2+c x^4\right ) \, dx=\frac {d^2 \left (3 c d^2-10 b d e+80 a e^2\right ) x \sqrt {d+e x^2}}{256 e^2}+\frac {d \left (3 c d^2-10 b d e+80 a e^2\right ) x \left (d+e x^2\right )^{3/2}}{384 e^2}+\frac {\left (3 c d^2-10 b d e+80 a e^2\right ) x \left (d+e x^2\right )^{5/2}}{480 e^2}-\frac {(3 c d-10 b e) x \left (d+e x^2\right )^{7/2}}{80 e^2}+\frac {c x^3 \left (d+e x^2\right )^{7/2}}{10 e}+\frac {d^3 \left (3 c d^2-10 b d e+80 a e^2\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{256 e^{5/2}} \]

[Out]

1/384*d*(80*a*e^2-10*b*d*e+3*c*d^2)*x*(e*x^2+d)^(3/2)/e^2+1/480*(80*a*e^2-10*b*d*e+3*c*d^2)*x*(e*x^2+d)^(5/2)/
e^2-1/80*(-10*b*e+3*c*d)*x*(e*x^2+d)^(7/2)/e^2+1/10*c*x^3*(e*x^2+d)^(7/2)/e+1/256*d^3*(80*a*e^2-10*b*d*e+3*c*d
^2)*arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/e^(5/2)+1/256*d^2*(80*a*e^2-10*b*d*e+3*c*d^2)*x*(e*x^2+d)^(1/2)/e^2

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1173, 396, 201, 223, 212} \[ \int \left (d+e x^2\right )^{5/2} \left (a+b x^2+c x^4\right ) \, dx=\frac {d^3 \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \left (80 a e^2-10 b d e+3 c d^2\right )}{256 e^{5/2}}+\frac {x \left (d+e x^2\right )^{5/2} \left (80 a e^2-10 b d e+3 c d^2\right )}{480 e^2}+\frac {d x \left (d+e x^2\right )^{3/2} \left (80 a e^2-10 b d e+3 c d^2\right )}{384 e^2}+\frac {d^2 x \sqrt {d+e x^2} \left (80 a e^2-10 b d e+3 c d^2\right )}{256 e^2}-\frac {x \left (d+e x^2\right )^{7/2} (3 c d-10 b e)}{80 e^2}+\frac {c x^3 \left (d+e x^2\right )^{7/2}}{10 e} \]

[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 201

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 212

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

Rule 223

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 396

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 1173

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]

Rubi steps \begin{align*} \text {integral}& = \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 ) \text {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] (verified)

Time = 0.32 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.84 \[ \int \left (d+e x^2\right )^{5/2} \left (a+b x^2+c x^4\right ) \, dx=\frac {\sqrt {e} x \sqrt {d+e x^2} \left (c \left (-45 d^4+30 d^3 e x^2+744 d^2 e^2 x^4+1008 d e^3 x^6+384 e^4 x^8\right )+10 e \left (8 a e \left (33 d^2+26 d e x^2+8 e^2 x^4\right )+b \left (15 d^3+118 d^2 e x^2+136 d e^2 x^4+48 e^3 x^6\right )\right )\right )-15 \left (3 c d^5-10 d^3 e (b d-8 a e)\right ) \log \left (-\sqrt {e} x+\sqrt {d+e x^2}\right )}{3840 e^{5/2}} \]

[In]

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

[Out]

(Sqrt[e]*x*Sqrt[d + e*x^2]*(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*(3*
c*d^5 - 10*d^3*e*(b*d - 8*a*e))*Log[-(Sqrt[e]*x) + Sqrt[d + e*x^2]])/(3840*e^(5/2))

Maple [A] (verified)

Time = 0.35 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.69

method	result	size
pseudoelliptic	\(\frac {\frac {5 d^{3} \left (a \,e^{2}-\frac {1}{8} b d e +\frac {3}{80} c \,d^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}}{x \sqrt {e}}\right )}{16}+\frac {11 \left (d^{2} \left (\frac {31}{110} c \,x^{4}+\frac {59}{132} b \,x^{2}+a \right ) e^{\frac {5}{2}}+\frac {26 d \left (\frac {63}{130} c \,x^{4}+\frac {17}{26} b \,x^{2}+a \right ) x^{2} e^{\frac {7}{2}}}{33}+\frac {8 \left (\frac {3}{5} c \,x^{4}+\frac {3}{4} b \,x^{2}+a \right ) x^{4} e^{\frac {9}{2}}}{33}+\frac {5 \left (\left (\frac {c \,x^{2}}{5}+b \right ) e^{\frac {3}{2}}-\frac {3 c d \sqrt {e}}{10}\right ) d^{3}}{88}\right ) \sqrt {e \,x^{2}+d}\, x}{16}}{e^{\frac {5}{2}}}\)	\(149\)
risch	\(\frac {x \left (384 e^{4} c \,x^{8}+480 e^{4} b \,x^{6}+1008 d \,e^{3} c \,x^{6}+640 a \,e^{4} x^{4}+1360 b d \,e^{3} x^{4}+744 c \,d^{2} e^{2} x^{4}+2080 d \,e^{3} a \,x^{2}+1180 e^{2} d^{2} b \,x^{2}+30 d^{3} e c \,x^{2}+2640 e^{2} d^{2} a +150 d^{3} e b -45 d^{4} c \right ) \sqrt {e \,x^{2}+d}}{3840 e^{2}}+\frac {d^{3} \left (80 a \,e^{2}-10 b d e +3 c \,d^{2}\right ) \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{256 e^{\frac {5}{2}}}\)	\(173\)
default	\(a \left (\frac {x \left (e \,x^{2}+d \right )^{\frac {5}{2}}}{6}+\frac {5 d \left (\frac {x \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{4}+\frac {3 d \left (\frac {x \sqrt {e \,x^{2}+d}}{2}+\frac {d \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{2 \sqrt {e}}\right )}{4}\right )}{6}\right )+c \left (\frac {x^{3} \left (e \,x^{2}+d \right )^{\frac {7}{2}}}{10 e}-\frac {3 d \left (\frac {x \left (e \,x^{2}+d \right )^{\frac {7}{2}}}{8 e}-\frac {d \left (\frac {x \left (e \,x^{2}+d \right )^{\frac {5}{2}}}{6}+\frac {5 d \left (\frac {x \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{4}+\frac {3 d \left (\frac {x \sqrt {e \,x^{2}+d}}{2}+\frac {d \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{2 \sqrt {e}}\right )}{4}\right )}{6}\right )}{8 e}\right )}{10 e}\right )+b \left (\frac {x \left (e \,x^{2}+d \right )^{\frac {7}{2}}}{8 e}-\frac {d \left (\frac {x \left (e \,x^{2}+d \right )^{\frac {5}{2}}}{6}+\frac {5 d \left (\frac {x \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{4}+\frac {3 d \left (\frac {x \sqrt {e \,x^{2}+d}}{2}+\frac {d \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{2 \sqrt {e}}\right )}{4}\right )}{6}\right )}{8 e}\right )\)	\(277\)

[In]

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

[Out]

11/16/e^(5/2)*(5/11*d^3*(a*e^2-1/8*b*d*e+3/80*c*d^2)*arctanh((e*x^2+d)^(1/2)/x/e^(1/2))+(d^2*(31/110*c*x^4+59/
132*b*x^2+a)*e^(5/2)+26/33*d*(63/130*c*x^4+17/26*b*x^2+a)*x^2*e^(7/2)+8/33*(3/5*c*x^4+3/4*b*x^2+a)*x^4*e^(9/2)
+5/88*((1/5*c*x^2+b)*e^(3/2)-3/10*c*d*e^(1/2))*d^3)*(e*x^2+d)^(1/2)*x)

Fricas [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.72 \[ \int \left (d+e x^2\right )^{5/2} \left (a+b x^2+c x^4\right ) \, dx=\left [\frac {15 \, {\left (3 \, c d^{5} - 10 \, b d^{4} e + 80 \, a d^{3} e^{2}\right )} \sqrt {e} \log \left (-2 \, e x^{2} - 2 \, \sqrt {e x^{2} + d} \sqrt {e} x - d\right ) + 2 \, {\left (384 \, c e^{5} x^{9} + 48 \, {\left (21 \, c d e^{4} + 10 \, b e^{5}\right )} x^{7} + 8 \, {\left (93 \, c d^{2} e^{3} + 170 \, b d e^{4} + 80 \, a e^{5}\right )} x^{5} + 10 \, {\left (3 \, c d^{3} e^{2} + 118 \, b d^{2} e^{3} + 208 \, a d e^{4}\right )} x^{3} - 15 \, {\left (3 \, c d^{4} e - 10 \, b d^{3} e^{2} - 176 \, a d^{2} e^{3}\right )} x\right )} \sqrt {e x^{2} + d}}{7680 \, e^{3}}, -\frac {15 \, {\left (3 \, c d^{5} - 10 \, b d^{4} e + 80 \, a d^{3} e^{2}\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) - {\left (384 \, c e^{5} x^{9} + 48 \, {\left (21 \, c d e^{4} + 10 \, b e^{5}\right )} x^{7} + 8 \, {\left (93 \, c d^{2} e^{3} + 170 \, b d e^{4} + 80 \, a e^{5}\right )} x^{5} + 10 \, {\left (3 \, c d^{3} e^{2} + 118 \, b d^{2} e^{3} + 208 \, a d e^{4}\right )} x^{3} - 15 \, {\left (3 \, c d^{4} e - 10 \, b d^{3} e^{2} - 176 \, a d^{2} e^{3}\right )} x\right )} \sqrt {e x^{2} + d}}{3840 \, e^{3}}\right ] \]

[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]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 439 vs. \(2 (209) = 418\).

Time = 0.52 (sec) , antiderivative size = 439, normalized size of antiderivative = 2.04 \[ \int \left (d+e x^2\right )^{5/2} \left (a+b x^2+c x^4\right ) \, dx=\begin {cases} \sqrt {d + e x^{2}} \left (\frac {c e^{2} x^{9}}{10} + \frac {x^{7} \left (b e^{3} + \frac {21 c d e^{2}}{10}\right )}{8 e} + \frac {x^{5} \left (a e^{3} + 3 b d e^{2} + 3 c d^{2} e - \frac {7 d \left (b e^{3} + \frac {21 c d e^{2}}{10}\right )}{8 e}\right )}{6 e} + \frac {x^{3} \cdot \left (3 a d e^{2} + 3 b d^{2} e + c d^{3} - \frac {5 d \left (a e^{3} + 3 b d e^{2} + 3 c d^{2} e - \frac {7 d \left (b e^{3} + \frac {21 c d e^{2}}{10}\right )}{8 e}\right )}{6 e}\right )}{4 e} + \frac {x \left (3 a d^{2} e + b d^{3} - \frac {3 d \left (3 a d e^{2} + 3 b d^{2} e + c d^{3} - \frac {5 d \left (a e^{3} + 3 b d e^{2} + 3 c d^{2} e - \frac {7 d \left (b e^{3} + \frac {21 c d e^{2}}{10}\right )}{8 e}\right )}{6 e}\right )}{4 e}\right )}{2 e}\right ) + \left (a d^{3} - \frac {d \left (3 a d^{2} e + b d^{3} - \frac {3 d \left (3 a d e^{2} + 3 b d^{2} e + c d^{3} - \frac {5 d \left (a e^{3} + 3 b d e^{2} + 3 c d^{2} e - \frac {7 d \left (b e^{3} + \frac {21 c d e^{2}}{10}\right )}{8 e}\right )}{6 e}\right )}{4 e}\right )}{2 e}\right ) \left (\begin {cases} \frac {\log {\left (2 \sqrt {e} \sqrt {d + e x^{2}} + 2 e x \right )}}{\sqrt {e}} & \text {for}\: d \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {e x^{2}}} & \text {otherwise} \end {cases}\right ) & \text {for}\: e \neq 0 \\d^{\frac {5}{2}} \left (a x + \frac {b x^{3}}{3} + \frac {c x^{5}}{5}\right ) & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((sqrt(d + e*x**2)*(c*e**2*x**9/10 + x**7*(b*e**3 + 21*c*d*e**2/10)/(8*e) + x**5*(a*e**3 + 3*b*d*e**2
+ 3*c*d**2*e - 7*d*(b*e**3 + 21*c*d*e**2/10)/(8*e))/(6*e) + x**3*(3*a*d*e**2 + 3*b*d**2*e + c*d**3 - 5*d*(a*e
**3 + 3*b*d*e**2 + 3*c*d**2*e - 7*d*(b*e**3 + 21*c*d*e**2/10)/(8*e))/(6*e))/(4*e) + x*(3*a*d**2*e + b*d**3 - 3
*d*(3*a*d*e**2 + 3*b*d**2*e + c*d**3 - 5*d*(a*e**3 + 3*b*d*e**2 + 3*c*d**2*e - 7*d*(b*e**3 + 21*c*d*e**2/10)/(
8*e))/(6*e))/(4*e))/(2*e)) + (a*d**3 - d*(3*a*d**2*e + b*d**3 - 3*d*(3*a*d*e**2 + 3*b*d**2*e + c*d**3 - 5*d*(a
*e**3 + 3*b*d*e**2 + 3*c*d**2*e - 7*d*(b*e**3 + 21*c*d*e**2/10)/(8*e))/(6*e))/(4*e))/(2*e))*Piecewise((log(2*s
qrt(e)*sqrt(d + e*x**2) + 2*e*x)/sqrt(e), Ne(d, 0)), (x*log(x)/sqrt(e*x**2), True)), Ne(e, 0)), (d**(5/2)*(a*x
+ b*x**3/3 + c*x**5/5), True))

Maxima [F(-2)]

Exception generated. \[ \int \left (d+e x^2\right )^{5/2} \left (a+b x^2+c x^4\right ) \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.91 \[ \int \left (d+e x^2\right )^{5/2} \left (a+b x^2+c x^4\right ) \, dx=\frac {1}{3840} \, {\left (2 \, {\left (4 \, {\left (6 \, {\left (8 \, c e^{2} x^{2} + \frac {21 \, c d e^{9} + 10 \, b e^{10}}{e^{8}}\right )} x^{2} + \frac {93 \, c d^{2} e^{8} + 170 \, b d e^{9} + 80 \, a e^{10}}{e^{8}}\right )} x^{2} + \frac {5 \, {\left (3 \, c d^{3} e^{7} + 118 \, b d^{2} e^{8} + 208 \, a d e^{9}\right )}}{e^{8}}\right )} x^{2} - \frac {15 \, {\left (3 \, c d^{4} e^{6} - 10 \, b d^{3} e^{7} - 176 \, a d^{2} e^{8}\right )}}{e^{8}}\right )} \sqrt {e x^{2} + d} x - \frac {{\left (3 \, c d^{5} - 10 \, b d^{4} e + 80 \, a d^{3} e^{2}\right )} \log \left ({\left | -\sqrt {e} x + \sqrt {e x^{2} + d} \right |}\right )}{256 \, e^{\frac {5}{2}}} \]

[In]

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

[Out]

1/3840*(2*(4*(6*(8*c*e^2*x^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(e*x^2 + d)*x - 1/256*(3*c*d^5 - 10*b*d^4*e + 80*a*d^3*e^2)*log(abs(-sqrt(e)*x + sqrt(e*x^2 + d))
)/e^(5/2)

Mupad [F(-1)]

Timed out. \[ \int \left (d+e x^2\right )^{5/2} \left (a+b x^2+c x^4\right ) \, dx=\int {\left (e\,x^2+d\right )}^{5/2}\,\left (c\,x^4+b\,x^2+a\right ) \,d x \]

[In]

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

[Out]

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