3.458 \(\int \frac {x^2 (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{d+e x} \, dx\)

Optimal. Leaf size=452 \[ \frac {\left (-35 a^2 e^4-10 c d e x \left (9 c d^2-5 a e^2\right )-20 a c d^2 e^2+63 c^2 d^4\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{840 c^2 d^2 e^3}-\frac {\left (5 a^2 e^4+10 a c d^2 e^2+9 c^2 d^4\right ) \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2048 c^{9/2} d^{9/2} e^{11/2}}+\frac {\left (5 a^2 e^4+10 a c d^2 e^2+9 c^2 d^4\right ) \left (c d^2-a e^2\right )^3 \left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{1024 c^4 d^4 e^5}-\frac {\left (5 a^2 e^4+10 a c d^2 e^2+9 c^2 d^4\right ) \left (c d^2-a e^2\right ) \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{384 c^3 d^3 e^4}+\frac {x^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 e} \]

[Out]

-1/384*(-a*e^2+c*d^2)*(5*a^2*e^4+10*a*c*d^2*e^2+9*c^2*d^4)*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*
e*x^2)^(3/2)/c^3/d^3/e^4+1/7*x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/e+1/840*(63*c^2*d^4-20*a*c*d^2*e^2-35
*a^2*e^4-10*c*d*e*(-5*a*e^2+9*c*d^2)*x)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c^2/d^2/e^3-1/2048*(-a*e^2+c*d
^2)^5*(5*a^2*e^4+10*a*c*d^2*e^2+9*c^2*d^4)*arctanh(1/2*(2*c*d*e*x+a*e^2+c*d^2)/c^(1/2)/d^(1/2)/e^(1/2)/(a*d*e+
(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/c^(9/2)/d^(9/2)/e^(11/2)+1/1024*(-a*e^2+c*d^2)^3*(5*a^2*e^4+10*a*c*d^2*e^2+9
*c^2*d^4)*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^4/d^4/e^5

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Rubi [A]  time = 0.41, antiderivative size = 452, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {851, 832, 779, 612, 621, 206} \[ \frac {\left (5 a^2 e^4+10 a c d^2 e^2+9 c^2 d^4\right ) \left (c d^2-a e^2\right )^3 \left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{1024 c^4 d^4 e^5}-\frac {\left (5 a^2 e^4+10 a c d^2 e^2+9 c^2 d^4\right ) \left (c d^2-a e^2\right ) \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{384 c^3 d^3 e^4}+\frac {\left (-35 a^2 e^4-10 c d e x \left (9 c d^2-5 a e^2\right )-20 a c d^2 e^2+63 c^2 d^4\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{840 c^2 d^2 e^3}-\frac {\left (5 a^2 e^4+10 a c d^2 e^2+9 c^2 d^4\right ) \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2048 c^{9/2} d^{9/2} e^{11/2}}+\frac {x^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((c*d^2 - a*e^2)^3*(9*c^2*d^4 + 10*a*c*d^2*e^2 + 5*a^2*e^4)*(c*d^2 + a*e^2 + 2*c*d*e*x)*Sqrt[a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2])/(1024*c^4*d^4*e^5) - ((c*d^2 - a*e^2)*(9*c^2*d^4 + 10*a*c*d^2*e^2 + 5*a^2*e^4)*(c*d^2 +
a*e^2 + 2*c*d*e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(384*c^3*d^3*e^4) + (x^2*(a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2)^(5/2))/(7*e) + ((63*c^2*d^4 - 20*a*c*d^2*e^2 - 35*a^2*e^4 - 10*c*d*e*(9*c*d^2 - 5*a*e^2)
*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(840*c^2*d^2*e^3) - ((c*d^2 - a*e^2)^5*(9*c^2*d^4 + 10*a*c*
d^2*e^2 + 5*a^2*e^4)*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^
2)*x + c*d*e*x^2])])/(2048*c^(9/2)*d^(9/2)*e^(11/2))

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])

Rule 612

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p +
1)), x] - Dist[(p*(b^2 - 4*a*c))/(2*c*(2*p + 1)), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 779

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((b
*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x)*(a + b*x + c*x^2)^(p + 1))/(2*c^2*(p + 1)*(2*p + 3
)), x] + Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)), Int[(a
+ b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

Rule 832

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m
- 1)*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m*(c*e*f + c*d*g - b*e*g) + e*(p
+ 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
&&  !(IGtQ[m, 0] && EqQ[f, 0])

Rule 851

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :>
Int[((f + g*x)^n*(a + b*x + c*x^2)^(m + p))/(a/d + (c*x)/e)^m, x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] &&
NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[m, 0] && In
tegerQ[n] && (LtQ[n, 0] || GtQ[p, 0])

Rubi steps

\begin {align*} \int \frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx &=\int x^2 (a e+c d x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx\\ &=\frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 e}+\frac {\int x \left (-2 a c d^2 e-\frac {1}{2} c d \left (9 c d^2-5 a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx}{7 c d e}\\ &=\frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 e}+\frac {\left (63 c^2 d^4-20 a c d^2 e^2-35 a^2 e^4-10 c d e \left (9 c d^2-5 a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{840 c^2 d^2 e^3}-\frac {\left (\left (c d^2-a e^2\right ) \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right )\right ) \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx}{48 c^2 d^2 e^3}\\ &=-\frac {\left (c d^2-a e^2\right ) \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right ) \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{384 c^3 d^3 e^4}+\frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 e}+\frac {\left (63 c^2 d^4-20 a c d^2 e^2-35 a^2 e^4-10 c d e \left (9 c d^2-5 a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{840 c^2 d^2 e^3}+\frac {\left (\left (c d^2-a e^2\right )^3 \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right )\right ) \int \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{256 c^3 d^3 e^4}\\ &=\frac {\left (c d^2-a e^2\right )^3 \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right ) \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{1024 c^4 d^4 e^5}-\frac {\left (c d^2-a e^2\right ) \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right ) \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{384 c^3 d^3 e^4}+\frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 e}+\frac {\left (63 c^2 d^4-20 a c d^2 e^2-35 a^2 e^4-10 c d e \left (9 c d^2-5 a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{840 c^2 d^2 e^3}-\frac {\left (\left (c d^2-a e^2\right )^5 \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right )\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{2048 c^4 d^4 e^5}\\ &=\frac {\left (c d^2-a e^2\right )^3 \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right ) \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{1024 c^4 d^4 e^5}-\frac {\left (c d^2-a e^2\right ) \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right ) \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{384 c^3 d^3 e^4}+\frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 e}+\frac {\left (63 c^2 d^4-20 a c d^2 e^2-35 a^2 e^4-10 c d e \left (9 c d^2-5 a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{840 c^2 d^2 e^3}-\frac {\left (\left (c d^2-a e^2\right )^5 \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 c d e-x^2} \, dx,x,\frac {c d^2+a e^2+2 c d e x}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{1024 c^4 d^4 e^5}\\ &=\frac {\left (c d^2-a e^2\right )^3 \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right ) \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{1024 c^4 d^4 e^5}-\frac {\left (c d^2-a e^2\right ) \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right ) \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{384 c^3 d^3 e^4}+\frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 e}+\frac {\left (63 c^2 d^4-20 a c d^2 e^2-35 a^2 e^4-10 c d e \left (9 c d^2-5 a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{840 c^2 d^2 e^3}-\frac {\left (c d^2-a e^2\right )^5 \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right ) \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{2048 c^{9/2} d^{9/2} e^{11/2}}\\ \end {align*}

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Mathematica [A]  time = 5.71, size = 562, normalized size = 1.24 \[ \frac {((d+e x) (a e+c d x))^{3/2} \left (\frac {7 \sqrt {c d} \left (5 a^2 e^4+10 a c d^2 e^2+9 c^2 d^4\right ) \left (15 \sqrt {e} \sqrt {c d} \left (c d^2-a e^2\right )^{11/2} (a e+c d x) \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}}-15 \sqrt {c} \sqrt {d} \left (c d^2-a e^2\right )^6 \sqrt {a e+c d x} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d} \sqrt {c d^2-a e^2}}\right )-10 e^{3/2} \sqrt {c d} \left (c d^2-a e^2\right )^{9/2} (a e+c d x)^2 \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}}+8 e^{5/2} \sqrt {c d} \left (c d^2-a e^2\right )^{7/2} (a e+c d x)^3 \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}}+16 e^{7/2} \sqrt {c d} \left (c d^2-a e^2\right )^{3/2} (a e+c d x)^4 \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}} \left (c d (11 d+8 e x)-3 a e^2\right )\right )}{15360 c^3 d^3 e^{9/2} \left (c d^2-a e^2\right )^{5/2} \left (\frac {c d (d+e x)}{c d^2-a e^2}\right )^{3/2} (a e+c d x)^2}-\frac {(d+e x) \left (7 a e^2+9 c d^2\right ) (a e+c d x)^2}{12 c d e}+x (d+e x) (a e+c d x)^2\right )}{7 c d e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(((a*e + c*d*x)*(d + e*x))^(3/2)*(-1/12*((9*c*d^2 + 7*a*e^2)*(a*e + c*d*x)^2*(d + e*x))/(c*d*e) + x*(a*e + c*d
*x)^2*(d + e*x) + (7*Sqrt[c*d]*(9*c^2*d^4 + 10*a*c*d^2*e^2 + 5*a^2*e^4)*(15*Sqrt[c*d]*Sqrt[e]*(c*d^2 - a*e^2)^
(11/2)*(a*e + c*d*x)*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)] - 10*Sqrt[c*d]*e^(3/2)*(c*d^2 - a*e^2)^(9/2)*(a*e +
c*d*x)^2*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)] + 8*Sqrt[c*d]*e^(5/2)*(c*d^2 - a*e^2)^(7/2)*(a*e + c*d*x)^3*Sq
rt[(c*d*(d + e*x))/(c*d^2 - a*e^2)] + 16*Sqrt[c*d]*e^(7/2)*(c*d^2 - a*e^2)^(3/2)*(a*e + c*d*x)^4*Sqrt[(c*d*(d
+ e*x))/(c*d^2 - a*e^2)]*(-3*a*e^2 + c*d*(11*d + 8*e*x)) - 15*Sqrt[c]*Sqrt[d]*(c*d^2 - a*e^2)^6*Sqrt[a*e + c*d
*x]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])]))/(15360*c^3*d^3*e^(9
/2)*(c*d^2 - a*e^2)^(5/2)*(a*e + c*d*x)^2*((c*d*(d + e*x))/(c*d^2 - a*e^2))^(3/2))))/(7*c*d*e)

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fricas [A]  time = 0.84, size = 1272, normalized size = 2.81 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/430080*(105*(9*c^7*d^14 - 35*a*c^6*d^12*e^2 + 45*a^2*c^5*d^10*e^4 - 15*a^3*c^4*d^8*e^6 - 5*a^4*c^3*d^6*e^8
- 9*a^5*c^2*d^4*e^10 + 15*a^6*c*d^2*e^12 - 5*a^7*e^14)*sqrt(c*d*e)*log(8*c^2*d^2*e^2*x^2 + c^2*d^4 + 6*a*c*d^
2*e^2 + a^2*e^4 + 4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*c*d*e*x + c*d^2 + a*e^2)*sqrt(c*d*e) + 8*(c
^2*d^3*e + a*c*d*e^3)*x) - 4*(15360*c^7*d^7*e^7*x^6 + 945*c^7*d^13*e - 3360*a*c^6*d^11*e^3 + 3689*a^2*c^5*d^9*
e^5 - 600*a^3*c^4*d^7*e^7 - 525*a^4*c^3*d^5*e^9 + 1400*a^5*c^2*d^3*e^11 - 525*a^6*c*d*e^13 + 1280*(15*c^7*d^8*
e^6 + 29*a*c^6*d^6*e^8)*x^5 + 128*(3*c^7*d^9*e^5 + 380*a*c^6*d^7*e^7 + 185*a^2*c^5*d^5*e^9)*x^4 - 16*(27*c^7*d
^10*e^4 - 93*a*c^6*d^8*e^6 - 2095*a^2*c^5*d^6*e^8 - 15*a^3*c^4*d^4*e^10)*x^3 + 8*(63*c^7*d^11*e^3 - 218*a*c^6*
d^9*e^5 + 228*a^2*c^5*d^7*e^7 + 90*a^3*c^4*d^5*e^9 - 35*a^4*c^3*d^3*e^11)*x^2 - 2*(315*c^7*d^12*e^2 - 1099*a*c
^6*d^10*e^4 + 1166*a^2*c^5*d^8*e^6 - 150*a^3*c^4*d^6*e^8 + 455*a^4*c^3*d^4*e^10 - 175*a^5*c^2*d^2*e^12)*x)*sqr
t(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^5*d^5*e^6), 1/215040*(105*(9*c^7*d^14 - 35*a*c^6*d^12*e^2 + 45*a^
2*c^5*d^10*e^4 - 15*a^3*c^4*d^8*e^6 - 5*a^4*c^3*d^6*e^8 - 9*a^5*c^2*d^4*e^10 + 15*a^6*c*d^2*e^12 - 5*a^7*e^14)
*sqrt(-c*d*e)*arctan(1/2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*c*d*e*x + c*d^2 + a*e^2)*sqrt(-c*d*e)/
(c^2*d^2*e^2*x^2 + a*c*d^2*e^2 + (c^2*d^3*e + a*c*d*e^3)*x)) + 2*(15360*c^7*d^7*e^7*x^6 + 945*c^7*d^13*e - 336
0*a*c^6*d^11*e^3 + 3689*a^2*c^5*d^9*e^5 - 600*a^3*c^4*d^7*e^7 - 525*a^4*c^3*d^5*e^9 + 1400*a^5*c^2*d^3*e^11 -
525*a^6*c*d*e^13 + 1280*(15*c^7*d^8*e^6 + 29*a*c^6*d^6*e^8)*x^5 + 128*(3*c^7*d^9*e^5 + 380*a*c^6*d^7*e^7 + 185
*a^2*c^5*d^5*e^9)*x^4 - 16*(27*c^7*d^10*e^4 - 93*a*c^6*d^8*e^6 - 2095*a^2*c^5*d^6*e^8 - 15*a^3*c^4*d^4*e^10)*x
^3 + 8*(63*c^7*d^11*e^3 - 218*a*c^6*d^9*e^5 + 228*a^2*c^5*d^7*e^7 + 90*a^3*c^4*d^5*e^9 - 35*a^4*c^3*d^3*e^11)*
x^2 - 2*(315*c^7*d^12*e^2 - 1099*a*c^6*d^10*e^4 + 1166*a^2*c^5*d^8*e^6 - 150*a^3*c^4*d^6*e^8 + 455*a^4*c^3*d^4
*e^10 - 175*a^5*c^2*d^2*e^12)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^5*d^5*e^6)]

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, replacing 0 by ` u`, a substitution variable should perhaps be purged.Warning, replacing 0 by ` u`, a sub
stitution variable should perhaps be purged.Warning, replacing 0 by ` u`, a substitution variable should perha
ps be purged.Warning, replacing 0 by ` u`, a substitution variable should perhaps be purged.Warning, replacing
0 by ` u`, a substitution variable should perhaps be purged.Warning, replacing 0 by ` u`, a substitution vari
able should perhaps be purged.Warning, replacing 0 by ` u`, a substitution variable should perhaps be purged.W
arning, replacing 0 by ` u`, a substitution variable should perhaps be purged.Warning, replacing 0 by ` u`, a
substitution variable should perhaps be purged.Warning, replacing 0 by ` u`, a substitution variable should pe
rhaps be purged.Warning, replacing 0 by ` u`, a substitution variable should perhaps be purged.Warning, replac
ing 0 by ` u`, a substitution variable should perhaps be purged.Warning, replacing 0 by ` u`, a substitution v
ariable should perhaps be purged.Warning, replacing 0 by ` u`, a substitution variable should perhaps be purge
d.Warning, replacing 0 by ` u`, a substitution variable should perhaps be purged.Warning, replacing 0 by ` u`,
a substitution variable should perhaps be purged.Warning, replacing 0 by ` u`, a substitution variable should
perhaps be purged.Warning, replacing 0 by ` u`, a substitution variable should perhaps be purged.Warning, rep
lacing 0 by ` u`, a substitution variable should perhaps be purged.Warning, replacing 0 by ` u`, a substitutio
n variable should perhaps be purged.Warning, replacing 0 by ` u`, a substitution variable should perhaps be pu
rged.Warning, replacing 0 by ` u`, a substitution variable should perhaps be purged.Warning, replacing 0 by `
u`, a substitution variable should perhaps be purged.Evaluation time: 0.52Error: Bad Argument Type

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maple [B]  time = 0.02, size = 2731, normalized size = 6.04 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

195/2048/e*c*d^6*ln((c*d*e*x+1/2*a*e^2+1/2*c*d^2)/(c*d*e)^(1/2)+(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2))/(c*d*
e)^(1/2)*a^2-5/512*e^6/c^3/d^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x*a^5-3/64*d*e^2*a^3/c*((x+d/e)^2*c*d*e
+(a*e^2-c*d^2)*(x+d/e))^(1/2)*x-9/64*d^5/e^2*a*c*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2)*x-15/128*d^6/e*
a^2*c*ln((1/2*a*e^2-1/2*c*d^2+(x+d/e)*c*d*e)/(c*d*e)^(1/2)+((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(c*d
*e)^(1/2)+15/256*d^8/e^3*a*c^2*ln((1/2*a*e^2-1/2*c*d^2+(x+d/e)*c*d*e)/(c*d*e)^(1/2)+((x+d/e)^2*c*d*e+(a*e^2-c*
d^2)*(x+d/e))^(1/2))/(c*d*e)^(1/2)-15/256*d^2*e^3*a^4/c*ln((1/2*a*e^2-1/2*c*d^2+(x+d/e)*c*d*e)/(c*d*e)^(1/2)+(
(x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(c*d*e)^(1/2)+5/512*e^4/c^2/d*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^
(1/2)*x*a^4+15/256*e^2/c*d*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x*a^3-85/2048/e^3*c^2*d^8*ln((c*d*e*x+1/2*a
*e^2+1/2*c*d^2)/(c*d*e)^(1/2)+(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2))/(c*d*e)^(1/2)*a+5/192*e^3/c^2/d^2*(c*d*
e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*x*a^3+5/2048*e^9/c^4/d^4*ln((c*d*e*x+1/2*a*e^2+1/2*c*d^2)/(c*d*e)^(1/2)+(c*
d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2))/(c*d*e)^(1/2)*a^7-15/2048*e^7/c^3/d^2*ln((c*d*e*x+1/2*a*e^2+1/2*c*d^2)/(
c*d*e)^(1/2)+(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2))/(c*d*e)^(1/2)*a^6+55/512/e^2*c*d^5*(c*d*e*x^2+a*d*e+(a*e
^2+c*d^2)*x)^(1/2)*x*a+125/2048*e^3/c*d^2*ln((c*d*e*x+1/2*a*e^2+1/2*c*d^2)/(c*d*e)^(1/2)+(c*d*e*x^2+a*d*e+(a*e
^2+c*d^2)*x)^(1/2))/(c*d*e)^(1/2)*a^4-1/16*d^5/e^4*c*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(3/2)+3/128*d^8/e
^5*c^2*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2)+9/64*d^3*a^2*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2
)*x-3/128*e^3*a^4/c^2*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2)+1/16*d*a^2/c*((x+d/e)^2*c*d*e+(a*e^2-c*d^2
)*(x+d/e))^(3/2)-1/6/e/c*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*a-15/1024/e*d^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2
)*x)^(1/2)*a^2-5/192/e^2*d^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*a-15/1024/e^5*c^2*d^8*(c*d*e*x^2+a*d*e+(a
*e^2+c*d^2)*x)^(1/2)+5/128/e^4*c*d^5*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)+1/7/e^2*(c*d*e*x^2+a*d*e+(a*e^2+c
*d^2)*x)^(7/2)/c/d+35/1024*e^3/c^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a^4-5/96/c*d*(c*d*e*x^2+a*d*e+(a*e^
2+c*d^2)*x)^(3/2)*a^2-35/256*d^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x*a^2-1/4/e^2*d*(c*d*e*x^2+a*d*e+(a*e
^2+c*d^2)*x)^(5/2)*x+1/5*d^2/e^3*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(5/2)-1/8/e^3*d^2*(c*d*e*x^2+a*d*e+(a
*e^2+c*d^2)*x)^(5/2)-1/12/c/d*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*x*a-25/192/e*d^2*(c*d*e*x^2+a*d*e+(a*e^2
+c*d^2)*x)^(3/2)*x*a+15/2048/e^5*c^3*d^10*ln((c*d*e*x+1/2*a*e^2+1/2*c*d^2)/(c*d*e)^(1/2)+(c*d*e*x^2+a*d*e+(a*e
^2+c*d^2)*x)^(1/2))/(c*d*e)^(1/2)+5/384*e^4/c^3/d^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*a^4+3/256*e^5*a^5/
c^2*ln((1/2*a*e^2-1/2*c*d^2+(x+d/e)*c*d*e)/(c*d*e)^(1/2)+((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(c*d*e
)^(1/2)+1/8*d^2/e*a*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(3/2)*x+3/64*d^2*e*a^3/c*((x+d/e)^2*c*d*e+(a*e^2-c
*d^2)*(x+d/e))^(1/2)-1/8*d^4/e^3*c*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(3/2)*x-3/64*d^6/e^3*a*c*((x+d/e)^2
*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2)+15/128*d^4*e*a^3*ln((1/2*a*e^2-1/2*c*d^2+(x+d/e)*c*d*e)/(c*d*e)^(1/2)+((x+
d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(c*d*e)^(1/2)+3/64*d^7/e^4*c^2*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/
e))^(1/2)*x-3/256*d^10/e^5*c^3*ln((1/2*a*e^2-1/2*c*d^2+(x+d/e)*c*d*e)/(c*d*e)^(1/2)+((x+d/e)^2*c*d*e+(a*e^2-c*
d^2)*(x+d/e))^(1/2))/(c*d*e)^(1/2)-15/512/e^4*c^2*d^7*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x-5/1024*e^7/c^4
/d^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a^6+5/128/e^3*c*d^6*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a-5/1
28*e/c*d^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a^3+5/64/e^3*c*d^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*
x-1/24*e/c^2/d^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*a^2+5/192*e^2/c^2/d*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)
^(3/2)*a^3-15/2048*e^5/c^2*ln((c*d*e*x+1/2*a*e^2+1/2*c*d^2)/(c*d*e)^(1/2)+(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1
/2))/(c*d*e)^(1/2)*a^5-225/2048*e*d^4*ln((c*d*e*x+1/2*a*e^2+1/2*c*d^2)/(c*d*e)^(1/2)+(c*d*e*x^2+a*d*e+(a*e^2+c
*d^2)*x)^(1/2))/(c*d*e)^(1/2)*a^3+5/192*e/c*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*x*a^2

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d),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(a*e^2-c*d^2>0)', see `assume?`
for more details)Is a*e^2-c*d^2 zero or nonzero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^2\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{d+e\,x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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