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

Optimal. Leaf size=297 \[ -\frac {2 \left (x \left (-3 a^3 e^6-a^2 c d^2 e^4-7 a c^2 d^4 e^2+3 c^3 d^6\right )+a d e \left (c d^2-3 a e^2\right ) \left (a e^2+3 c d^2\right )\right )}{3 c d e^2 \left (c d^2-a e^2\right )^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {\tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{c^{3/2} d^{3/2} e^{5/2}}-\frac {2 d x^2 \left (c d x \left (c d^2-a e^2\right )+a e \left (c d^2-a e^2\right )\right )}{3 e \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

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Rubi [A]  time = 0.29, antiderivative size = 297, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {849, 818, 777, 621, 206} \begin {gather*} -\frac {2 \left (x \left (-a^2 c d^2 e^4-3 a^3 e^6-7 a c^2 d^4 e^2+3 c^3 d^6\right )+a d e \left (c d^2-3 a e^2\right ) \left (a e^2+3 c d^2\right )\right )}{3 c d e^2 \left (c d^2-a e^2\right )^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {\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 )}{c^{3/2} d^{3/2} e^{5/2}}-\frac {2 d x^2 \left (c d x \left (c d^2-a e^2\right )+a e \left (c d^2-a e^2\right )\right )}{3 e \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*d*x^2*(a*e*(c*d^2 - a*e^2) + c*d*(c*d^2 - a*e^2)*x))/(3*e*(c*d^2 - a*e^2)^2*(a*d*e + (c*d^2 + a*e^2)*x + c
*d*e*x^2)^(3/2)) - (2*(a*d*e*(c*d^2 - 3*a*e^2)*(3*c*d^2 + a*e^2) + (3*c^3*d^6 - 7*a*c^2*d^4*e^2 - a^2*c*d^2*e^
4 - 3*a^3*e^6)*x))/(3*c*d*e^2*(c*d^2 - a*e^2)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + 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])]/(c^(3/2)*d^(3/2)
*e^(5/2))

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 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 777

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

Rule 818

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*(2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g
- c*(b*e*f + b*d*g + 2*a*e*g))*x))/(c*(p + 1)*(b^2 - 4*a*c)), x] - Dist[1/(c*(p + 1)*(b^2 - 4*a*c)), Int[(d +
e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Simp[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a
*e*(e*f*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*(m + p + 1) + 2*c^2*d*f*(m +
2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2*p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && Ne
Q[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] &&
RationalQ[a, b, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 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}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx &=\int \frac {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 )^{5/2}} \, dx\\ &=-\frac {2 d x^2 \left (a e \left (c d^2-a e^2\right )+c d \left (c d^2-a e^2\right ) x\right )}{3 e \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {2 \int \frac {x \left (2 a c d^2 e \left (c d^2-a e^2\right )+\frac {3}{2} c d \left (c d^2-a e^2\right )^2 x\right )}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{3 c d e \left (c d^2-a e^2\right )^2}\\ &=-\frac {2 d x^2 \left (a e \left (c d^2-a e^2\right )+c d \left (c d^2-a e^2\right ) x\right )}{3 e \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {2 \left (a d e \left (c d^2-3 a e^2\right ) \left (3 c d^2+a e^2\right )+\left (3 c^3 d^6-7 a c^2 d^4 e^2-a^2 c d^2 e^4-3 a^3 e^6\right ) x\right )}{3 c d e^2 \left (c d^2-a e^2\right )^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{c d e^2}\\ &=-\frac {2 d x^2 \left (a e \left (c d^2-a e^2\right )+c d \left (c d^2-a e^2\right ) x\right )}{3 e \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {2 \left (a d e \left (c d^2-3 a e^2\right ) \left (3 c d^2+a e^2\right )+\left (3 c^3 d^6-7 a c^2 d^4 e^2-a^2 c d^2 e^4-3 a^3 e^6\right ) x\right )}{3 c d e^2 \left (c d^2-a e^2\right )^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {2 \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 )}{c d e^2}\\ &=-\frac {2 d x^2 \left (a e \left (c d^2-a e^2\right )+c d \left (c d^2-a e^2\right ) x\right )}{3 e \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {2 \left (a d e \left (c d^2-3 a e^2\right ) \left (3 c d^2+a e^2\right )+\left (3 c^3 d^6-7 a c^2 d^4 e^2-a^2 c d^2 e^4-3 a^3 e^6\right ) x\right )}{3 c d e^2 \left (c d^2-a e^2\right )^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\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 )}{c^{3/2} d^{3/2} e^{5/2}}\\ \end {align*}

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Mathematica [C]  time = 4.64, size = 1443, normalized size = 4.86 \begin {gather*} \frac {a^3 e^3 (a e+c d x)^2 \left (\frac {c d (d+e x)}{c d^2-a e^2}\right )^{5/2} \left (\frac {56 \, _2F_1\left (\frac {3}{2},\frac {9}{2};\frac {11}{2};\frac {e (a e+c d x)}{a e^2-c d^2}\right ) (a e+c d x)^5}{a^3 e \left (c d^2-a e^2\right )^2}-\frac {280 \, _2F_1\left (\frac {3}{2},\frac {9}{2};\frac {11}{2};\frac {e (a e+c d x)}{a e^2-c d^2}\right ) (a e+c d x)^4}{a^2 \left (c d^2-a e^2\right )^2}+\frac {96 \, _4F_3\left (\frac {1}{2},2,2,\frac {7}{2};1,1,\frac {9}{2};\frac {e (a e+c d x)}{a e^2-c d^2}\right ) (a e+c d x)^4}{a^3 e^2 \left (a e^2-c d^2\right )}+\frac {294 \sin ^{-1}\left (\sqrt {\frac {e (a e+c d x)}{a e^2-c d^2}}\right ) (a e+c d x)^3}{a^3 e^3 \left (\frac {e (a e+c d x)}{a e^2-c d^2}\right )^{5/2}}+\frac {392 e \, _2F_1\left (\frac {3}{2},\frac {9}{2};\frac {11}{2};\frac {e (a e+c d x)}{a e^2-c d^2}\right ) (a e+c d x)^3}{a \left (c d^2-a e^2\right )^2}-\frac {288 \, _4F_3\left (\frac {1}{2},2,2,\frac {7}{2};1,1,\frac {9}{2};\frac {e (a e+c d x)}{a e^2-c d^2}\right ) (a e+c d x)^3}{a^2 \left (a e^3-c d^2 e\right )}-\frac {1575 \sin ^{-1}\left (\sqrt {\frac {e (a e+c d x)}{a e^2-c d^2}}\right ) (a e+c d x)^2}{a^2 e^2 \left (\frac {e (a e+c d x)}{a e^2-c d^2}\right )^{5/2}}-\frac {168 e^2 \, _2F_1\left (\frac {3}{2},\frac {9}{2};\frac {11}{2};\frac {e (a e+c d x)}{a e^2-c d^2}\right ) (a e+c d x)^2}{\left (c d^2-a e^2\right )^2}+\frac {288 \, _4F_3\left (\frac {1}{2},2,2,\frac {7}{2};1,1,\frac {9}{2};\frac {e (a e+c d x)}{a e^2-c d^2}\right ) (a e+c d x)^2}{a \left (a e^2-c d^2\right )}+\frac {196 c d (d+e x) (a e+c d x)^2}{a^3 e^4 \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}}}+\frac {3780 \sin ^{-1}\left (\sqrt {\frac {e (a e+c d x)}{a e^2-c d^2}}\right ) (a e+c d x)}{a e \left (\frac {e (a e+c d x)}{a e^2-c d^2}\right )^{5/2}}-\frac {96 e \, _4F_3\left (\frac {1}{2},2,2,\frac {7}{2};1,1,\frac {9}{2};\frac {e (a e+c d x)}{a e^2-c d^2}\right ) (a e+c d x)}{a e^2-c d^2}-\frac {294 \left (c d^2-a e^2\right )^2 \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}} (a e+c d x)}{a^3 e^5}-\frac {1050 c d (d+e x) (a e+c d x)}{a^2 e^3 \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}}}-\frac {1995 \sin ^{-1}\left (\sqrt {\frac {e (a e+c d x)}{a e^2-c d^2}}\right )}{\left (\frac {e (a e+c d x)}{a e^2-c d^2}\right )^{5/2}}-56 \left (\frac {c d x}{a e}+1\right )^3 \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}}+\frac {1575 \left (c d^2-a e^2\right )^2 \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}}}{a^2 e^4}+336 \left (\frac {c d x}{a e}+1\right )^2 \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}}-504 \left (\frac {c d x}{a e}+1\right ) \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}}+1568 \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}}+\frac {2520 c d (d+e x)}{a e^2 \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}}}-\frac {3780 \left (c d^2-a e^2\right )^2 \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}}}{a e^3 (a e+c d x)}-\frac {1330 c d (d+e x)}{e \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}} (a e+c d x)}+\frac {1995 \left (c d^2-a e^2\right )^2 \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}}}{e^2 (a e+c d x)^2}\right )}{252 c^4 d^4 ((a e+c d x) (d+e x))^{5/2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a^3*e^3*(a*e + c*d*x)^2*((c*d*(d + e*x))/(c*d^2 - a*e^2))^(5/2)*((2520*c*d*(d + e*x))/(a*e^2*Sqrt[(c*d*(d + e
*x))/(c*d^2 - a*e^2)]) - (1330*c*d*(d + e*x))/(e*(a*e + c*d*x)*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]) - (1050*
c*d*(a*e + c*d*x)*(d + e*x))/(a^2*e^3*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]) + (196*c*d*(a*e + c*d*x)^2*(d + e
*x))/(a^3*e^4*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]) + 1568*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)] + (1575*(c*d
^2 - a*e^2)^2*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])/(a^2*e^4) + (1995*(c*d^2 - a*e^2)^2*Sqrt[(c*d*(d + e*x))/
(c*d^2 - a*e^2)])/(e^2*(a*e + c*d*x)^2) - (3780*(c*d^2 - a*e^2)^2*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])/(a*e^
3*(a*e + c*d*x)) - (294*(c*d^2 - a*e^2)^2*(a*e + c*d*x)*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])/(a^3*e^5) - 504
*(1 + (c*d*x)/(a*e))*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)] + 336*(1 + (c*d*x)/(a*e))^2*Sqrt[(c*d*(d + e*x))/(c
*d^2 - a*e^2)] - 56*(1 + (c*d*x)/(a*e))^3*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)] - (1995*ArcSin[Sqrt[(e*(a*e +
c*d*x))/(-(c*d^2) + a*e^2)]])/((e*(a*e + c*d*x))/(-(c*d^2) + a*e^2))^(5/2) + (3780*(a*e + c*d*x)*ArcSin[Sqrt[(
e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)]])/(a*e*((e*(a*e + c*d*x))/(-(c*d^2) + a*e^2))^(5/2)) - (1575*(a*e + c*d*x
)^2*ArcSin[Sqrt[(e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)]])/(a^2*e^2*((e*(a*e + c*d*x))/(-(c*d^2) + a*e^2))^(5/2))
+ (294*(a*e + c*d*x)^3*ArcSin[Sqrt[(e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)]])/(a^3*e^3*((e*(a*e + c*d*x))/(-(c*d
^2) + a*e^2))^(5/2)) - (168*e^2*(a*e + c*d*x)^2*Hypergeometric2F1[3/2, 9/2, 11/2, (e*(a*e + c*d*x))/(-(c*d^2)
+ a*e^2)])/(c*d^2 - a*e^2)^2 + (392*e*(a*e + c*d*x)^3*Hypergeometric2F1[3/2, 9/2, 11/2, (e*(a*e + c*d*x))/(-(c
*d^2) + a*e^2)])/(a*(c*d^2 - a*e^2)^2) - (280*(a*e + c*d*x)^4*Hypergeometric2F1[3/2, 9/2, 11/2, (e*(a*e + c*d*
x))/(-(c*d^2) + a*e^2)])/(a^2*(c*d^2 - a*e^2)^2) + (56*(a*e + c*d*x)^5*Hypergeometric2F1[3/2, 9/2, 11/2, (e*(a
*e + c*d*x))/(-(c*d^2) + a*e^2)])/(a^3*e*(c*d^2 - a*e^2)^2) - (96*e*(a*e + c*d*x)*HypergeometricPFQ[{1/2, 2, 2
, 7/2}, {1, 1, 9/2}, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(-(c*d^2) + a*e^2) + (288*(a*e + c*d*x)^2*Hypergeo
metricPFQ[{1/2, 2, 2, 7/2}, {1, 1, 9/2}, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(a*(-(c*d^2) + a*e^2)) - (288*
(a*e + c*d*x)^3*HypergeometricPFQ[{1/2, 2, 2, 7/2}, {1, 1, 9/2}, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(a^2*(
-(c*d^2*e) + a*e^3)) + (96*(a*e + c*d*x)^4*HypergeometricPFQ[{1/2, 2, 2, 7/2}, {1, 1, 9/2}, (e*(a*e + c*d*x))/
(-(c*d^2) + a*e^2)])/(a^3*e^2*(-(c*d^2) + a*e^2))))/(252*c^4*d^4*((a*e + c*d*x)*(d + e*x))^(5/2))

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IntegrateAlgebraic [B]  time = 10.11, size = 10635, normalized size = 35.81 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Result too large to show

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fricas [B]  time = 3.24, size = 1466, normalized size = 4.94

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/6*(3*(a*c^3*d^8*e - 3*a^2*c^2*d^6*e^3 + 3*a^3*c*d^4*e^5 - a^4*d^2*e^7 + (c^4*d^7*e^2 - 3*a*c^3*d^5*e^4 + 3*
a^2*c^2*d^3*e^6 - a^3*c*d*e^8)*x^3 + (2*c^4*d^8*e - 5*a*c^3*d^6*e^3 + 3*a^2*c^2*d^4*e^5 + a^3*c*d^2*e^7 - a^4*
e^9)*x^2 + (c^4*d^9 - a*c^3*d^7*e^2 - 3*a^2*c^2*d^5*e^4 + 5*a^3*c*d^3*e^6 - 2*a^4*d*e^8)*x)*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*(3*a*c^3*d^7*e^2 - 8*a^2*c^2*d^5*e^4 - 3*a^3*
c*d^3*e^6 + (4*c^4*d^7*e^2 - 9*a*c^3*d^5*e^4 - 3*a^3*c*d*e^8)*x^2 + (3*c^4*d^8*e - 4*a*c^3*d^6*e^3 - 9*a^2*c^2
*d^4*e^5 - 6*a^3*c*d^2*e^7)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(a*c^5*d^10*e^4 - 3*a^2*c^4*d^8*e^
6 + 3*a^3*c^3*d^6*e^8 - a^4*c^2*d^4*e^10 + (c^6*d^9*e^5 - 3*a*c^5*d^7*e^7 + 3*a^2*c^4*d^5*e^9 - a^3*c^3*d^3*e^
11)*x^3 + (2*c^6*d^10*e^4 - 5*a*c^5*d^8*e^6 + 3*a^2*c^4*d^6*e^8 + a^3*c^3*d^4*e^10 - a^4*c^2*d^2*e^12)*x^2 + (
c^6*d^11*e^3 - a*c^5*d^9*e^5 - 3*a^2*c^4*d^7*e^7 + 5*a^3*c^3*d^5*e^9 - 2*a^4*c^2*d^3*e^11)*x), -1/3*(3*(a*c^3*
d^8*e - 3*a^2*c^2*d^6*e^3 + 3*a^3*c*d^4*e^5 - a^4*d^2*e^7 + (c^4*d^7*e^2 - 3*a*c^3*d^5*e^4 + 3*a^2*c^2*d^3*e^6
- a^3*c*d*e^8)*x^3 + (2*c^4*d^8*e - 5*a*c^3*d^6*e^3 + 3*a^2*c^2*d^4*e^5 + a^3*c*d^2*e^7 - a^4*e^9)*x^2 + (c^4
*d^9 - a*c^3*d^7*e^2 - 3*a^2*c^2*d^5*e^4 + 5*a^3*c*d^3*e^6 - 2*a^4*d*e^8)*x)*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*(3*a*c^3*d^7*e^2 - 8*a^2*c^2*d^5*e^4 - 3*a^3*c*d^3*e^6 + (4*c^4*d^7*e^2 - 9*a*c
^3*d^5*e^4 - 3*a^3*c*d*e^8)*x^2 + (3*c^4*d^8*e - 4*a*c^3*d^6*e^3 - 9*a^2*c^2*d^4*e^5 - 6*a^3*c*d^2*e^7)*x)*sqr
t(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(a*c^5*d^10*e^4 - 3*a^2*c^4*d^8*e^6 + 3*a^3*c^3*d^6*e^8 - a^4*c^2*d^
4*e^10 + (c^6*d^9*e^5 - 3*a*c^5*d^7*e^7 + 3*a^2*c^4*d^5*e^9 - a^3*c^3*d^3*e^11)*x^3 + (2*c^6*d^10*e^4 - 5*a*c^
5*d^8*e^6 + 3*a^2*c^4*d^6*e^8 + a^3*c^3*d^4*e^10 - a^4*c^2*d^2*e^12)*x^2 + (c^6*d^11*e^3 - a*c^5*d^9*e^5 - 3*a
^2*c^4*d^7*e^7 + 5*a^3*c^3*d^5*e^9 - 2*a^4*c^2*d^3*e^11)*x)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unab
le to transpose Error: Bad Argument Value

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maple [B]  time = 0.01, size = 977, normalized size = 3.29 \begin {gather*} -\frac {16 c^{2} d^{5} x}{3 \left (a \,e^{2}-c \,d^{2}\right )^{3} \sqrt {\left (x +\frac {d}{e}\right )^{2} c d e +\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\, e^{2}}-\frac {8 a c \,d^{4}}{3 \left (a \,e^{2}-c \,d^{2}\right )^{3} \sqrt {\left (x +\frac {d}{e}\right )^{2} c d e +\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\, e}-\frac {8 c^{2} d^{6}}{3 \left (a \,e^{2}-c \,d^{2}\right )^{3} \sqrt {\left (x +\frac {d}{e}\right )^{2} c d e +\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\, e^{3}}+\frac {a^{2} e^{2} x}{\left (-a^{2} e^{4}+2 a c \,d^{2} e^{2}-c^{2} d^{4}\right ) \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, c d}+\frac {4 a d x}{\left (-a^{2} e^{4}+2 a c \,d^{2} e^{2}-c^{2} d^{4}\right ) \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}}+\frac {3 c \,d^{3} x}{\left (-a^{2} e^{4}+2 a c \,d^{2} e^{2}-c^{2} d^{4}\right ) \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, e^{2}}+\frac {a^{3} e^{3}}{2 \left (-a^{2} e^{4}+2 a c \,d^{2} e^{2}-c^{2} d^{4}\right ) \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, c^{2} d^{2}}+\frac {5 a^{2} e}{2 \left (-a^{2} e^{4}+2 a c \,d^{2} e^{2}-c^{2} d^{4}\right ) \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, c}+\frac {7 a \,d^{2}}{2 \left (-a^{2} e^{4}+2 a c \,d^{2} e^{2}-c^{2} d^{4}\right ) \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, e}+\frac {3 c \,d^{4}}{2 \left (-a^{2} e^{4}+2 a c \,d^{2} e^{2}-c^{2} d^{4}\right ) \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, e^{3}}+\frac {2 d^{3}}{3 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) \sqrt {\left (x +\frac {d}{e}\right )^{2} c d e +\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\, e^{4}}+\frac {2 \left (2 c d e x +a \,e^{2}+c \,d^{2}\right ) d^{2}}{\left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, e^{3}}-\frac {x}{\sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, c d \,e^{2}}+\frac {\ln \left (\frac {c d e x +\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}}{\sqrt {c d e}}+\sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\right )}{\sqrt {c d e}\, c d \,e^{2}}+\frac {a}{2 \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, c^{2} d^{2} e}+\frac {3}{2 \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, c \,e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/e^2*x/c/d/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)+1/2/e/c^2/d^2/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a+3
/2/e^3/c/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)+e^2/c/d/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(c*d*e*x^2+a*d*e+(a*
e^2+c*d^2)*x)^(1/2)*x*a^2+4*d/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x*a+3/e
^2*c*d^3/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x+1/2*e^3/c^2/d^2/(-a^2*e^4+
2*a*c*d^2*e^2-c^2*d^4)/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a^3+5/2*e/c/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(c
*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a^2+7/2/e*d^2/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(c*d*e*x^2+a*d*e+(a*e^2+c
*d^2)*x)^(1/2)*a+3/2/e^3*c*d^4/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)+1/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)+2*d^
2/e^3*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)+2/3*d^3/
e^4/(a*e^2-c*d^2)/(x+d/e)/((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2)-16/3*d^5/e^2*c^2/(a*e^2-c*d^2)^3/((x+d
/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2)*x-8/3*d^4/e*c/(a*e^2-c*d^2)^3/((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))
^(1/2)*a-8/3*d^6/e^3*c^2/(a*e^2-c*d^2)^3/((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),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 \begin {gather*} \int \frac {x^3}{\left (d+e\,x\right )\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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