3.457 \(\int \frac {x^3 (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{d+e x} \, dx\)

Optimal. Leaf size=574 \[ -\frac {\left (-105 a^3 e^6-10 c d e x \left (-15 a^2 e^4-10 a c d^2 e^2+33 c^2 d^4\right )-95 a^2 c d^2 e^4-15 a c^2 d^4 e^2+231 c^3 d^6\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{4480 c^3 d^3 e^4}+\frac {3 \left (15 a^3 e^6+35 a^2 c d^2 e^4+45 a c^2 d^4 e^2+33 c^3 d^6\right ) \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{32768 c^{11/2} d^{11/2} e^{13/2}}-\frac {3 \left (15 a^3 e^6+35 a^2 c d^2 e^4+45 a c^2 d^4 e^2+33 c^3 d^6\right ) \left (c d^2-a e^2\right )^3 \left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{16384 c^5 d^5 e^6}+\frac {\left (15 a^3 e^6+35 a^2 c d^2 e^4+45 a c^2 d^4 e^2+33 c^3 d^6\right ) \left (c d^2-a e^2\right ) \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{2048 c^4 d^4 e^5}+\frac {1}{112} x^2 \left (\frac {5 a}{c d}-\frac {11 d}{e^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}+\frac {x^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{8 e} \]

[Out]

1/2048*(-a*e^2+c*d^2)*(15*a^3*e^6+35*a^2*c*d^2*e^4+45*a*c^2*d^4*e^2+33*c^3*d^6)*(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^4/d^4/e^5+1/112*(5*a/c/d-11*d/e^2)*x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(
5/2)+1/8*x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/e-1/4480*(231*c^3*d^6-15*a*c^2*d^4*e^2-95*a^2*c*d^2*e^4-1
05*a^3*e^6-10*c*d*e*(-15*a^2*e^4-10*a*c*d^2*e^2+33*c^2*d^4)*x)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c^3/d^3
/e^4+3/32768*(-a*e^2+c*d^2)^5*(15*a^3*e^6+35*a^2*c*d^2*e^4+45*a*c^2*d^4*e^2+33*c^3*d^6)*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^(11/2)/d^(11/2)/e^(13/2)-3/16
384*(-a*e^2+c*d^2)^3*(15*a^3*e^6+35*a^2*c*d^2*e^4+45*a*c^2*d^4*e^2+33*c^3*d^6)*(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^5/d^5/e^6

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Rubi [A]  time = 0.69, antiderivative size = 574, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {849, 832, 779, 612, 621, 206} \[ -\frac {3 \left (35 a^2 c d^2 e^4+15 a^3 e^6+45 a c^2 d^4 e^2+33 c^3 d^6\right ) \left (c d^2-a e^2\right )^3 \left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{16384 c^5 d^5 e^6}+\frac {\left (35 a^2 c d^2 e^4+15 a^3 e^6+45 a c^2 d^4 e^2+33 c^3 d^6\right ) \left (c d^2-a e^2\right ) \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{2048 c^4 d^4 e^5}-\frac {\left (-10 c d e x \left (-15 a^2 e^4-10 a c d^2 e^2+33 c^2 d^4\right )-95 a^2 c d^2 e^4-105 a^3 e^6-15 a c^2 d^4 e^2+231 c^3 d^6\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{4480 c^3 d^3 e^4}+\frac {3 \left (35 a^2 c d^2 e^4+15 a^3 e^6+45 a c^2 d^4 e^2+33 c^3 d^6\right ) \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{32768 c^{11/2} d^{11/2} e^{13/2}}+\frac {x^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{8 e}+\frac {1}{112} x^2 \left (\frac {5 a}{c d}-\frac {11 d}{e^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(d + e*x),x]

[Out]

(-3*(c*d^2 - a*e^2)^3*(33*c^3*d^6 + 45*a*c^2*d^4*e^2 + 35*a^2*c*d^2*e^4 + 15*a^3*e^6)*(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])/(16384*c^5*d^5*e^6) + ((c*d^2 - a*e^2)*(33*c^3*d^6 + 45*a*c^2
*d^4*e^2 + 35*a^2*c*d^2*e^4 + 15*a^3*e^6)*(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))/(2048*c^4*d^4*e^5) + (((5*a)/(c*d) - (11*d)/e^2)*x^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/112
+ (x^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(8*e) - ((231*c^3*d^6 - 15*a*c^2*d^4*e^2 - 95*a^2*c*d^2
*e^4 - 105*a^3*e^6 - 10*c*d*e*(33*c^2*d^4 - 10*a*c*d^2*e^2 - 15*a^2*e^4)*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e
*x^2)^(5/2))/(4480*c^3*d^3*e^4) + (3*(c*d^2 - a*e^2)^5*(33*c^3*d^6 + 45*a*c^2*d^4*e^2 + 35*a^2*c*d^2*e^4 + 15*
a^3*e^6)*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])])/(32768*c^(11/2)*d^(11/2)*e^(13/2))

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 849

Int[((x_)^(n_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_))/((d_) + (e_.)*(x_)), x_Symbol] :> Int[x^n*(a/d + (c*
x)/e)*(a + b*x + c*x^2)^(p - 1), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b
*d*e + a*e^2, 0] &&  !IntegerQ[p] && ( !IntegerQ[n] ||  !IntegerQ[2*p] || IGtQ[n, 2] || (GtQ[p, 0] && NeQ[n, 2
]))

Rubi steps

\begin {align*} \int \frac {x^3 \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^3 (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^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{8 e}+\frac {\int x^2 \left (-3 a c d^2 e-\frac {1}{2} c d \left (11 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}{8 c d e}\\ &=\frac {1}{112} \left (\frac {5 a}{c d}-\frac {11 d}{e^2}\right ) x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}+\frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{8 e}+\frac {\int x \left (a c d^2 e \left (11 c d^2-5 a e^2\right )+\frac {3}{4} c d \left (33 c^2 d^4-10 a c d^2 e^2-15 a^2 e^4\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}{56 c^2 d^2 e^2}\\ &=\frac {1}{112} \left (\frac {5 a}{c d}-\frac {11 d}{e^2}\right ) x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}+\frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{8 e}-\frac {\left (231 c^3 d^6-15 a c^2 d^4 e^2-95 a^2 c d^2 e^4-105 a^3 e^6-10 c d e \left (33 c^2 d^4-10 a c d^2 e^2-15 a^2 e^4\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{4480 c^3 d^3 e^4}+\frac {\left (\left (c d^2-a e^2\right ) \left (33 c^3 d^6+45 a c^2 d^4 e^2+35 a^2 c d^2 e^4+15 a^3 e^6\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}{256 c^3 d^3 e^4}\\ &=\frac {\left (c d^2-a e^2\right ) \left (33 c^3 d^6+45 a c^2 d^4 e^2+35 a^2 c d^2 e^4+15 a^3 e^6\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}}{2048 c^4 d^4 e^5}+\frac {1}{112} \left (\frac {5 a}{c d}-\frac {11 d}{e^2}\right ) x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}+\frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{8 e}-\frac {\left (231 c^3 d^6-15 a c^2 d^4 e^2-95 a^2 c d^2 e^4-105 a^3 e^6-10 c d e \left (33 c^2 d^4-10 a c d^2 e^2-15 a^2 e^4\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{4480 c^3 d^3 e^4}-\frac {\left (3 \left (c d^2-a e^2\right )^3 \left (33 c^3 d^6+45 a c^2 d^4 e^2+35 a^2 c d^2 e^4+15 a^3 e^6\right )\right ) \int \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{4096 c^4 d^4 e^5}\\ &=-\frac {3 \left (c d^2-a e^2\right )^3 \left (33 c^3 d^6+45 a c^2 d^4 e^2+35 a^2 c d^2 e^4+15 a^3 e^6\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}}{16384 c^5 d^5 e^6}+\frac {\left (c d^2-a e^2\right ) \left (33 c^3 d^6+45 a c^2 d^4 e^2+35 a^2 c d^2 e^4+15 a^3 e^6\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}}{2048 c^4 d^4 e^5}+\frac {1}{112} \left (\frac {5 a}{c d}-\frac {11 d}{e^2}\right ) x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}+\frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{8 e}-\frac {\left (231 c^3 d^6-15 a c^2 d^4 e^2-95 a^2 c d^2 e^4-105 a^3 e^6-10 c d e \left (33 c^2 d^4-10 a c d^2 e^2-15 a^2 e^4\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{4480 c^3 d^3 e^4}+\frac {\left (3 \left (c d^2-a e^2\right )^5 \left (33 c^3 d^6+45 a c^2 d^4 e^2+35 a^2 c d^2 e^4+15 a^3 e^6\right )\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{32768 c^5 d^5 e^6}\\ &=-\frac {3 \left (c d^2-a e^2\right )^3 \left (33 c^3 d^6+45 a c^2 d^4 e^2+35 a^2 c d^2 e^4+15 a^3 e^6\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}}{16384 c^5 d^5 e^6}+\frac {\left (c d^2-a e^2\right ) \left (33 c^3 d^6+45 a c^2 d^4 e^2+35 a^2 c d^2 e^4+15 a^3 e^6\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}}{2048 c^4 d^4 e^5}+\frac {1}{112} \left (\frac {5 a}{c d}-\frac {11 d}{e^2}\right ) x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}+\frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{8 e}-\frac {\left (231 c^3 d^6-15 a c^2 d^4 e^2-95 a^2 c d^2 e^4-105 a^3 e^6-10 c d e \left (33 c^2 d^4-10 a c d^2 e^2-15 a^2 e^4\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{4480 c^3 d^3 e^4}+\frac {\left (3 \left (c d^2-a e^2\right )^5 \left (33 c^3 d^6+45 a c^2 d^4 e^2+35 a^2 c d^2 e^4+15 a^3 e^6\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 )}{16384 c^5 d^5 e^6}\\ &=-\frac {3 \left (c d^2-a e^2\right )^3 \left (33 c^3 d^6+45 a c^2 d^4 e^2+35 a^2 c d^2 e^4+15 a^3 e^6\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}}{16384 c^5 d^5 e^6}+\frac {\left (c d^2-a e^2\right ) \left (33 c^3 d^6+45 a c^2 d^4 e^2+35 a^2 c d^2 e^4+15 a^3 e^6\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}}{2048 c^4 d^4 e^5}+\frac {1}{112} \left (\frac {5 a}{c d}-\frac {11 d}{e^2}\right ) x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}+\frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{8 e}-\frac {\left (231 c^3 d^6-15 a c^2 d^4 e^2-95 a^2 c d^2 e^4-105 a^3 e^6-10 c d e \left (33 c^2 d^4-10 a c d^2 e^2-15 a^2 e^4\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{4480 c^3 d^3 e^4}+\frac {3 \left (c d^2-a e^2\right )^5 \left (33 c^3 d^6+45 a c^2 d^4 e^2+35 a^2 c d^2 e^4+15 a^3 e^6\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 )}{32768 c^{11/2} d^{11/2} e^{13/2}}\\ \end {align*}

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Mathematica [A]  time = 3.61, size = 681, normalized size = 1.19 \[ \frac {\sqrt {(d+e x) (a e+c d x)} \left (\frac {105 \sqrt {c} \sqrt {d} \left (15 a^3 e^6+35 a^2 c d^2 e^4+45 a c^2 d^4 e^2+33 c^3 d^6\right ) \left (c d^2-a e^2\right )^{9/2} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d} \sqrt {c d^2-a e^2}}\right )}{\sqrt {c d} \sqrt {a e+c d x} \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}}}+\frac {\sqrt {e} \left (1575 a^8 e^{15}-525 a^7 c d e^{13} (7 d-e x)+35 a^6 c^2 d^2 e^{11} \left (29 d^2-37 d e x-6 e^2 x^2\right )+5 a^5 c^3 d^3 e^9 \left (185 d^3+93 d^2 e x+100 d e^2 x^2+24 e^3 x^3\right )+5 a^4 c^4 d^4 e^7 \left (265 d^4+65 d^3 e x-30 d^2 e^2 x^2-56 d e^3 x^3-16 e^4 x^4\right )+a^3 c^5 d^5 e^5 \left (-11193 d^5+8359 d^4 e x-6088 d^3 e^2 x^2+5040 d^2 e^3 x^3+139200 d e^4 x^4+104320 e^5 x^5\right )+a^2 c^6 d^6 e^3 \left (11445 d^6-18669 d^5 e x+12962 d^4 e^2 x^2-10544 d^3 e^3 x^3+9120 d^2 e^4 x^4+350080 d e^5 x^5+272640 e^6 x^6\right )+a c^7 d^7 e \left (-3465 d^7+13755 d^6 e x-9324 d^5 e^2 x^2+7512 d^4 e^3 x^3-6464 d^3 e^4 x^4+5760 d^2 e^5 x^5+299520 d e^6 x^6+240640 e^7 x^7\right )+c^8 d^8 x \left (-3465 d^7+2310 d^6 e x-1848 d^5 e^2 x^2+1584 d^4 e^3 x^3-1408 d^3 e^4 x^4+1280 d^2 e^5 x^5+87040 d e^6 x^6+71680 e^7 x^7\right )\right )}{a e+c d x}\right )}{573440 c^5 d^5 e^{13/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(d + e*x),x]

[Out]

(Sqrt[(a*e + c*d*x)*(d + e*x)]*((Sqrt[e]*(1575*a^8*e^15 - 525*a^7*c*d*e^13*(7*d - e*x) + 35*a^6*c^2*d^2*e^11*(
29*d^2 - 37*d*e*x - 6*e^2*x^2) + 5*a^5*c^3*d^3*e^9*(185*d^3 + 93*d^2*e*x + 100*d*e^2*x^2 + 24*e^3*x^3) + 5*a^4
*c^4*d^4*e^7*(265*d^4 + 65*d^3*e*x - 30*d^2*e^2*x^2 - 56*d*e^3*x^3 - 16*e^4*x^4) + a^3*c^5*d^5*e^5*(-11193*d^5
+ 8359*d^4*e*x - 6088*d^3*e^2*x^2 + 5040*d^2*e^3*x^3 + 139200*d*e^4*x^4 + 104320*e^5*x^5) + a^2*c^6*d^6*e^3*(
11445*d^6 - 18669*d^5*e*x + 12962*d^4*e^2*x^2 - 10544*d^3*e^3*x^3 + 9120*d^2*e^4*x^4 + 350080*d*e^5*x^5 + 2726
40*e^6*x^6) + c^8*d^8*x*(-3465*d^7 + 2310*d^6*e*x - 1848*d^5*e^2*x^2 + 1584*d^4*e^3*x^3 - 1408*d^3*e^4*x^4 + 1
280*d^2*e^5*x^5 + 87040*d*e^6*x^6 + 71680*e^7*x^7) + a*c^7*d^7*e*(-3465*d^7 + 13755*d^6*e*x - 9324*d^5*e^2*x^2
+ 7512*d^4*e^3*x^3 - 6464*d^3*e^4*x^4 + 5760*d^2*e^5*x^5 + 299520*d*e^6*x^6 + 240640*e^7*x^7)))/(a*e + c*d*x)
+ (105*Sqrt[c]*Sqrt[d]*(c*d^2 - a*e^2)^(9/2)*(33*c^3*d^6 + 45*a*c^2*d^4*e^2 + 35*a^2*c*d^2*e^4 + 15*a^3*e^6)*
ArcSinh[(Sqrt[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[a*e + c*
d*x]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])))/(573440*c^5*d^5*e^(13/2))

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fricas [A]  time = 1.23, size = 1524, normalized size = 2.66 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d),x, algorithm="fricas")

[Out]

[1/2293760*(105*(33*c^8*d^16 - 120*a*c^7*d^14*e^2 + 140*a^2*c^6*d^12*e^4 - 40*a^3*c^5*d^10*e^6 - 10*a^4*c^4*d^
8*e^8 - 8*a^5*c^3*d^6*e^10 - 20*a^6*c^2*d^4*e^12 + 40*a^7*c*d^2*e^14 - 15*a^8*e^16)*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*(71680*c^8*d^8*e^8*x^7 - 3465*c^8*d^15*e + 11445*a*c^
7*d^13*e^3 - 11193*a^2*c^6*d^11*e^5 + 1325*a^3*c^5*d^9*e^7 + 925*a^4*c^4*d^7*e^9 + 1015*a^5*c^3*d^5*e^11 - 367
5*a^6*c^2*d^3*e^13 + 1575*a^7*c*d*e^15 + 5120*(17*c^8*d^9*e^7 + 33*a*c^7*d^7*e^9)*x^6 + 1280*(c^8*d^10*e^6 + 1
66*a*c^7*d^8*e^8 + 81*a^2*c^6*d^6*e^10)*x^5 - 128*(11*c^8*d^11*e^5 - 35*a*c^7*d^9*e^7 - 1075*a^2*c^6*d^7*e^9 -
5*a^3*c^5*d^5*e^11)*x^4 + 16*(99*c^8*d^12*e^4 - 316*a*c^7*d^10*e^6 + 290*a^2*c^6*d^8*e^8 + 100*a^3*c^5*d^6*e^
10 - 45*a^4*c^4*d^4*e^12)*x^3 - 8*(231*c^8*d^13*e^3 - 741*a*c^7*d^11*e^5 + 686*a^2*c^6*d^9*e^7 - 50*a^3*c^5*d^
7*e^9 + 235*a^4*c^4*d^5*e^11 - 105*a^5*c^3*d^3*e^13)*x^2 + 2*(1155*c^8*d^14*e^2 - 3738*a*c^7*d^12*e^4 + 3517*a
^2*c^6*d^10*e^6 - 300*a^3*c^5*d^8*e^8 - 275*a^4*c^4*d^6*e^10 + 1190*a^5*c^3*d^4*e^12 - 525*a^6*c^2*d^2*e^14)*x
)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^6*d^6*e^7), -1/1146880*(105*(33*c^8*d^16 - 120*a*c^7*d^14*e^
2 + 140*a^2*c^6*d^12*e^4 - 40*a^3*c^5*d^10*e^6 - 10*a^4*c^4*d^8*e^8 - 8*a^5*c^3*d^6*e^10 - 20*a^6*c^2*d^4*e^12
+ 40*a^7*c*d^2*e^14 - 15*a^8*e^16)*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*(71680*c^8
*d^8*e^8*x^7 - 3465*c^8*d^15*e + 11445*a*c^7*d^13*e^3 - 11193*a^2*c^6*d^11*e^5 + 1325*a^3*c^5*d^9*e^7 + 925*a^
4*c^4*d^7*e^9 + 1015*a^5*c^3*d^5*e^11 - 3675*a^6*c^2*d^3*e^13 + 1575*a^7*c*d*e^15 + 5120*(17*c^8*d^9*e^7 + 33*
a*c^7*d^7*e^9)*x^6 + 1280*(c^8*d^10*e^6 + 166*a*c^7*d^8*e^8 + 81*a^2*c^6*d^6*e^10)*x^5 - 128*(11*c^8*d^11*e^5
- 35*a*c^7*d^9*e^7 - 1075*a^2*c^6*d^7*e^9 - 5*a^3*c^5*d^5*e^11)*x^4 + 16*(99*c^8*d^12*e^4 - 316*a*c^7*d^10*e^6
+ 290*a^2*c^6*d^8*e^8 + 100*a^3*c^5*d^6*e^10 - 45*a^4*c^4*d^4*e^12)*x^3 - 8*(231*c^8*d^13*e^3 - 741*a*c^7*d^1
1*e^5 + 686*a^2*c^6*d^9*e^7 - 50*a^3*c^5*d^7*e^9 + 235*a^4*c^4*d^5*e^11 - 105*a^5*c^3*d^3*e^13)*x^2 + 2*(1155*
c^8*d^14*e^2 - 3738*a*c^7*d^12*e^4 + 3517*a^2*c^6*d^10*e^6 - 300*a^3*c^5*d^8*e^8 - 275*a^4*c^4*d^6*e^10 + 1190
*a^5*c^3*d^4*e^12 - 525*a^6*c^2*d^2*e^14)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^6*d^6*e^7)]

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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^3*(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.Warning, replacing 0 by ` u`, a substitution variable sho
uld perhaps be purged.Warning, replacing 0 by ` u`, a substitution variable should perhaps be purged.Evaluatio
n time: 0.5Error: Bad Argument Type

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maple [B]  time = 0.03, size = 3178, normalized size = 5.54 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)/(e*x+d),x)

[Out]

-975/16384*e^2/c*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^4+195/4096/e^4*c^2*d^9*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+3/64*e/c^2/d^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*x*a^2+9/64*d^6/e^3*a*c*
((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2)*x-3/256*d*e^4*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)+15/128*d^7/e^2*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^9/e^
4*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)+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)*x+15/256*d^3*e^2*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)-5/256*e^
2/c^2*a^3/d*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*x-15/1024*e^4/c^3/d^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3
/2)*x*a^4+45/8192*e^7/c^4/d^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x*a^6-855/8192/e^2*c*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)*a^2-495/4096/e^3*c*d^6*
(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x*a-15/4096*e^5/c^3/d^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x*a^5-
105/2048*e/c*d^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x*a^3-45/32768*e^10/c^5/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^8+15/4096*e^8/c^4/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^7-15/8192*e^6/c
^3/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^6
+45/4096*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^5-3/64*d^3*a^3/c*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2)+1/16*d^6/e^5*c*((x+d/e)^2*c*d*e+(a
*e^2-c*d^2)*(x+d/e))^(3/2)-3/128*d^9/e^6*c^2*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2)-15/128*d^5*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)-
95/2048/e^5*c*d^6*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)+285/16384/e^6*c^2*d^9*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)
*x)^(1/2)+45/2048/e^3*d^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*a+165/16384/e^2*d^5*(c*d*e*x^2+a*d*e+(a*e^2+
c*d^2)*x)^(1/2)*a^2+19/64/e^3*d^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*x-15/1024*e/c^2*(c*d*e*x^2+a*d*e+(a*
e^2+c*d^2)*x)^(3/2)*a^3+465/4096*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^3+735/16384/c*d^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a^3+13/128/c^2/d*(c*d
*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*a^2-1/5*d^3/e^4*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(5/2)+19/128/e^4*d
^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)-25/112/e^3/c*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(7/2)+65/1024/e/c*d^
2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*a^2+285/8192/e^5*c^2*d^8*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x-1
/8*d^3/e^2*a*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(3/2)*x+1/8*d^5/e^4*c*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d
/e))^(3/2)*x-1/16*d^2/e*a^2/c*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(3/2)-9/64*d^4/e*a^2*((x+d/e)^2*c*d*e+(a
*e^2-c*d^2)*(x+d/e))^(1/2)*x+3/128*d*e^2*a^4/c^2*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2)+3/64*d^7/e^4*a*
c*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2)-3/64*d^8/e^5*c^2*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2)
*x+3/256*d^11/e^6*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)+45/16384*e^8/c^5/d^5*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a^7+15/16384*e^6/c^4/d^
3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a^6-75/16384*e^4/c^3/d*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a^5-4
65/16384*e^2/c^2*d*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a^4-285/32768/e^6*c^3*d^11*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)+3/128*e^2/c^3/d^3*(c*d*e*x^2+a*d
*e+(a*e^2+c*d^2)*x)^(5/2)*a^3+29/128/e^2/c*d*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*a-5/512/c*d*(c*d*e*x^2+a*
d*e+(a*e^2+c*d^2)*x)^(3/2)*x*a^2+35/256/e^2*a*d^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*x+5/32/e/c*(c*d*e*x^
2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*x*a-45/8192*e^3/c^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x*a^4+1155/8192/e*d
^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x*a^2+1/8/e^2*x*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(7/2)/c/d-705/163
84/e^4*c*d^7*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a-9/112/e/c^2/d^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(7/2)
*a-95/1024/e^4*c*d^5*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*x-15/2048*e^5/c^4/d^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d
^2)*x)^(3/2)*a^5-35/2048*e^3/c^3/d^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*a^4

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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^3*(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 \[ \int \frac {x^3\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{d+e\,x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2))/(d + e*x),x)

[Out]

int((x^3*(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**3*(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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