3.447 \(\int \frac {x^2 (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{d+e x} \, dx\)

Optimal. Leaf size=352 \[ \frac {\left (-15 a^2 e^4-6 c d e x \left (7 c d^2-3 a e^2\right )-12 a c d^2 e^2+35 c^2 d^4\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{240 c^2 d^2 e^3}+\frac {\left (3 a^2 e^4+6 a c d^2 e^2+7 c^2 d^4\right ) \left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{256 c^{7/2} d^{7/2} e^{9/2}}-\frac {\left (3 a^2 e^4+6 a c d^2 e^2+7 c^2 d^4\right ) \left (c d^2-a e^2\right ) \left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{128 c^3 d^3 e^4}+\frac {x^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 e} \]

[Out]

1/5*x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/e+1/240*(35*c^2*d^4-12*a*c*d^2*e^2-15*a^2*e^4-6*c*d*e*(-3*a*e^
2+7*c*d^2)*x)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c^2/d^2/e^3+1/256*(-a*e^2+c*d^2)^3*(3*a^2*e^4+6*a*c*d^2*
e^2+7*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^(7/2)/d^(7/2)/e^(9/2)-1/128*(-a*e^2+c*d^2)*(3*a^2*e^4+6*a*c*d^2*e^2+7*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^3/d^3/e^4

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Rubi [A]  time = 0.33, antiderivative size = 352, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \[ -\frac {\left (3 a^2 e^4+6 a c d^2 e^2+7 c^2 d^4\right ) \left (c d^2-a e^2\right ) \left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{128 c^3 d^3 e^4}+\frac {\left (-15 a^2 e^4-6 c d e x \left (7 c d^2-3 a e^2\right )-12 a c d^2 e^2+35 c^2 d^4\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{240 c^2 d^2 e^3}+\frac {\left (3 a^2 e^4+6 a c d^2 e^2+7 c^2 d^4\right ) \left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{256 c^{7/2} d^{7/2} e^{9/2}}+\frac {x^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((c*d^2 - a*e^2)*(7*c^2*d^4 + 6*a*c*d^2*e^2 + 3*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])/(128*c^3*d^3*e^4) + (x^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(5*e) + ((35*c^2*
d^4 - 12*a*c*d^2*e^2 - 15*a^2*e^4 - 6*c*d*e*(7*c*d^2 - 3*a*e^2)*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/
2))/(240*c^2*d^2*e^3) + ((c*d^2 - a*e^2)^3*(7*c^2*d^4 + 6*a*c*d^2*e^2 + 3*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])])/(256*c^(7/2)*d^(7/2)*e^(9/2
))

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 )^{3/2}}{d+e x} \, dx &=\int x^2 (a e+c d x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx\\ &=\frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 e}+\frac {\int x \left (-2 a c d^2 e-\frac {1}{2} c d \left (7 c d^2-3 a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{5 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 )^{3/2}}{5 e}+\frac {\left (35 c^2 d^4-12 a c d^2 e^2-15 a^2 e^4-6 c d e \left (7 c d^2-3 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}}{240 c^2 d^2 e^3}-\frac {\left (\left (c d^2-a e^2\right ) \left (7 c^2 d^4+6 a c d^2 e^2+3 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}{32 c^2 d^2 e^3}\\ &=-\frac {\left (c d^2-a e^2\right ) \left (7 c^2 d^4+6 a c d^2 e^2+3 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}}{128 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 )^{3/2}}{5 e}+\frac {\left (35 c^2 d^4-12 a c d^2 e^2-15 a^2 e^4-6 c d e \left (7 c d^2-3 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}}{240 c^2 d^2 e^3}+\frac {\left (\left (c d^2-a e^2\right )^3 \left (7 c^2 d^4+6 a c d^2 e^2+3 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}{256 c^3 d^3 e^4}\\ &=-\frac {\left (c d^2-a e^2\right ) \left (7 c^2 d^4+6 a c d^2 e^2+3 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}}{128 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 )^{3/2}}{5 e}+\frac {\left (35 c^2 d^4-12 a c d^2 e^2-15 a^2 e^4-6 c d e \left (7 c d^2-3 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}}{240 c^2 d^2 e^3}+\frac {\left (\left (c d^2-a e^2\right )^3 \left (7 c^2 d^4+6 a c d^2 e^2+3 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 )}{128 c^3 d^3 e^4}\\ &=-\frac {\left (c d^2-a e^2\right ) \left (7 c^2 d^4+6 a c d^2 e^2+3 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}}{128 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 )^{3/2}}{5 e}+\frac {\left (35 c^2 d^4-12 a c d^2 e^2-15 a^2 e^4-6 c d e \left (7 c d^2-3 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}}{240 c^2 d^2 e^3}+\frac {\left (c d^2-a e^2\right )^3 \left (7 c^2 d^4+6 a c d^2 e^2+3 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 )}{256 c^{7/2} d^{7/2} e^{9/2}}\\ \end {align*}

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Mathematica [A]  time = 2.77, size = 497, normalized size = 1.41 \[ \frac {\sqrt {(d+e x) (a e+c d x)} \left (\frac {\frac {5 \left (3 a^2 e^4+6 a c d^2 e^2+7 c^2 d^4\right ) \left (8 c^3 d^3 e^3 \sqrt {c d} \sqrt {c d^2-a e^2} (a e+c d x)^3 \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}}-c d \left (c d^2-a e^2\right ) \left (-3 c^{5/2} d^{5/2} \sqrt {e} \left (c d^2-a e^2\right )^2 \sqrt {a e+c d x} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d} \sqrt {c d^2-a e^2}}\right )-2 e^2 (c d)^{5/2} \sqrt {c d^2-a e^2} (a e+c d x)^2 \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}}+3 e (c d)^{5/2} \left (c d^2-a e^2\right )^{3/2} (a e+c d x) \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}}\right )\right )}{\sqrt {c d} \sqrt {c d^2-a e^2} \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}}}-48 c^4 d^4 e^3 (d+e x) \left (5 a e^2+7 c d^2\right ) (a e+c d x)^3}{384 c^5 d^5 e^4 (a e+c d x)}+x (d+e x) (a e+c d x)^2\right )}{5 c d e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[(a*e + c*d*x)*(d + e*x)]*(x*(a*e + c*d*x)^2*(d + e*x) + (-48*c^4*d^4*e^3*(7*c*d^2 + 5*a*e^2)*(a*e + c*d*
x)^3*(d + e*x) + (5*(7*c^2*d^4 + 6*a*c*d^2*e^2 + 3*a^2*e^4)*(8*c^3*d^3*Sqrt[c*d]*e^3*Sqrt[c*d^2 - a*e^2]*(a*e
+ c*d*x)^3*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)] - c*d*(c*d^2 - a*e^2)*(3*(c*d)^(5/2)*e*(c*d^2 - a*e^2)^(3/2)*
(a*e + c*d*x)*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)] - 2*(c*d)^(5/2)*e^2*Sqrt[c*d^2 - a*e^2]*(a*e + c*d*x)^2*Sq
rt[(c*d*(d + e*x))/(c*d^2 - a*e^2)] - 3*c^(5/2)*d^(5/2)*Sqrt[e]*(c*d^2 - a*e^2)^2*Sqrt[a*e + c*d*x]*ArcSinh[(S
qrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])])))/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2]*S
qrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]))/(384*c^5*d^5*e^4*(a*e + c*d*x))))/(5*c*d*e)

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fricas [A]  time = 0.85, size = 846, normalized size = 2.40 \[ \left [-\frac {15 \, {\left (7 \, c^{5} d^{10} - 15 \, a c^{4} d^{8} e^{2} + 6 \, a^{2} c^{3} d^{6} e^{4} + 2 \, a^{3} c^{2} d^{4} e^{6} + 3 \, a^{4} c d^{2} e^{8} - 3 \, a^{5} e^{10}\right )} \sqrt {c d e} \log \left (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 + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {c d e} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right ) - 4 \, {\left (384 \, c^{5} d^{5} e^{5} x^{4} - 105 \, c^{5} d^{9} e + 190 \, a c^{4} d^{7} e^{3} - 36 \, a^{2} c^{3} d^{5} e^{5} - 30 \, a^{3} c^{2} d^{3} e^{7} + 45 \, a^{4} c d e^{9} + 48 \, {\left (c^{5} d^{6} e^{4} + 11 \, a c^{4} d^{4} e^{6}\right )} x^{3} - 8 \, {\left (7 \, c^{5} d^{7} e^{3} - 12 \, a c^{4} d^{5} e^{5} - 3 \, a^{2} c^{3} d^{3} e^{7}\right )} x^{2} + 2 \, {\left (35 \, c^{5} d^{8} e^{2} - 61 \, a c^{4} d^{6} e^{4} + 9 \, a^{2} c^{3} d^{4} e^{6} - 15 \, a^{3} c^{2} d^{2} e^{8}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{7680 \, c^{4} d^{4} e^{5}}, -\frac {15 \, {\left (7 \, c^{5} d^{10} - 15 \, a c^{4} d^{8} e^{2} + 6 \, a^{2} c^{3} d^{6} e^{4} + 2 \, a^{3} c^{2} d^{4} e^{6} + 3 \, a^{4} c d^{2} e^{8} - 3 \, a^{5} e^{10}\right )} \sqrt {-c d e} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {-c d e}}{2 \, {\left (c^{2} d^{2} e^{2} x^{2} + a c d^{2} e^{2} + {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )}}\right ) - 2 \, {\left (384 \, c^{5} d^{5} e^{5} x^{4} - 105 \, c^{5} d^{9} e + 190 \, a c^{4} d^{7} e^{3} - 36 \, a^{2} c^{3} d^{5} e^{5} - 30 \, a^{3} c^{2} d^{3} e^{7} + 45 \, a^{4} c d e^{9} + 48 \, {\left (c^{5} d^{6} e^{4} + 11 \, a c^{4} d^{4} e^{6}\right )} x^{3} - 8 \, {\left (7 \, c^{5} d^{7} e^{3} - 12 \, a c^{4} d^{5} e^{5} - 3 \, a^{2} c^{3} d^{3} e^{7}\right )} x^{2} + 2 \, {\left (35 \, c^{5} d^{8} e^{2} - 61 \, a c^{4} d^{6} e^{4} + 9 \, a^{2} c^{3} d^{4} e^{6} - 15 \, a^{3} c^{2} d^{2} e^{8}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{3840 \, c^{4} d^{4} e^{5}}\right ] \]

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)^(3/2)/(e*x+d),x, algorithm="fricas")

[Out]

[-1/7680*(15*(7*c^5*d^10 - 15*a*c^4*d^8*e^2 + 6*a^2*c^3*d^6*e^4 + 2*a^3*c^2*d^4*e^6 + 3*a^4*c*d^2*e^8 - 3*a^5*
e^10)*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*(384*c^5*d^5*e^5*x^4
- 105*c^5*d^9*e + 190*a*c^4*d^7*e^3 - 36*a^2*c^3*d^5*e^5 - 30*a^3*c^2*d^3*e^7 + 45*a^4*c*d*e^9 + 48*(c^5*d^6*e
^4 + 11*a*c^4*d^4*e^6)*x^3 - 8*(7*c^5*d^7*e^3 - 12*a*c^4*d^5*e^5 - 3*a^2*c^3*d^3*e^7)*x^2 + 2*(35*c^5*d^8*e^2
- 61*a*c^4*d^6*e^4 + 9*a^2*c^3*d^4*e^6 - 15*a^3*c^2*d^2*e^8)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(
c^4*d^4*e^5), -1/3840*(15*(7*c^5*d^10 - 15*a*c^4*d^8*e^2 + 6*a^2*c^3*d^6*e^4 + 2*a^3*c^2*d^4*e^6 + 3*a^4*c*d^2
*e^8 - 3*a^5*e^10)*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*(384*c^5*d^5*e^5*x^4 - 105*
c^5*d^9*e + 190*a*c^4*d^7*e^3 - 36*a^2*c^3*d^5*e^5 - 30*a^3*c^2*d^3*e^7 + 45*a^4*c*d*e^9 + 48*(c^5*d^6*e^4 + 1
1*a*c^4*d^4*e^6)*x^3 - 8*(7*c^5*d^7*e^3 - 12*a*c^4*d^5*e^5 - 3*a^2*c^3*d^3*e^7)*x^2 + 2*(35*c^5*d^8*e^2 - 61*a
*c^4*d^6*e^4 + 9*a^2*c^3*d^4*e^6 - 15*a^3*c^2*d^2*e^8)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^4*d^
4*e^5)]

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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)^(3/2)/(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.Evaluation time: 0.42Error: Bad Argument Type

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maple [B]  time = 0.02, size = 1560, normalized size = 4.43 \[ \text {result too large to display} \]

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)^(3/2)/(e*x+d),x)

[Out]

-21/128*d^3*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^2-3/32/c*d*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a^2+9/128/e^4*c*d^5*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)
^(1/2)-1/4/e/c*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*a-3/64/e^2*d^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*
a+1/5/e^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)/c/d-3/8/e^2*d*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*x+1/8*
d*a^2/c*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2)+3/16*d^3*a^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^5/e^4*c*((x+d/e)^2*c*d*e+(a*e^2-c
*d^2)*(x+d/e))^(1/2)-1/16*d*e^2*a^3/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)-3/16*d^5/e^2*a*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)+3/64*e^3/c^2/d^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)
*x)^(1/2)*x*a^3-3/256*e^6/c^3/d^3*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+3/256*e^4/c^2/d*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+9/128*e^2/c*d*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+33/256/e^2*c*d^5*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+1/16*d^7/e^4*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)-1/16*e/c^2/d^2*(c*d*e*x^2+
a*d*e+(a*e^2+c*d^2)*x)^(3/2)*a^2+9/64/e^3*c*d^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x+3/128*e^4/c^3/d^3*(c
*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a^4-1/8/c/d*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*x*a+3/64*e/c*(c*d*e*
x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x*a^2+1/3*d^2/e^3*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(3/2)-3/16/e^3*d^2*
(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)+1/4*d^2/e*a*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2)*x-1/4*d^4/e^
3*c*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2)*x+3/64*e^2/c^2/d*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a^3
-15/64/e*d^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x*a-9/256/e^4*c^2*d^7*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)

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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)^(3/2)/(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 \[ \int \frac {x^2\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}}{d+e\,x} \,d x \]

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)^(3/2))/(d + e*x),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}}}{d + e x}\, dx \]

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)**(3/2)/(e*x+d),x)

[Out]

Integral(x**2*((d + e*x)*(a*e + c*d*x))**(3/2)/(d + e*x), x)

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