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

Optimal. Leaf size=381 \[ -\frac{\left (5 a e^2+7 c d^2\right ) \left (c d^2-a e^2\right )^3 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{512 c^3 d^3 e^4}+\frac{\left (5 a e^2+7 c d^2\right ) \left (c d^2-a e^2\right ) \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{192 c^2 d^2 e^3}+\frac{\left (5 a e^2+7 c d^2\right ) \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{1024 c^{7/2} d^{7/2} e^{9/2}}+\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{6 c d e (d+e x)}-\frac{1}{60} \left (\frac{5 a}{c d}+\frac{7 d}{e^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} \]

[Out]

-((c*d^2 - a*e^2)^3*(7*c*d^2 + 5*a*e^2)*(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
])/(512*c^3*d^3*e^4) + ((c*d^2 - a*e^2)*(7*c*d^2 + 5*a*e^2)*(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))/(192*c^2*d^2*e^3) - (((5*a)/(c*d) + (7*d)/e^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2
)^(5/2))/60 + (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2)/(6*c*d*e*(d + e*x)) + ((c*d^2 - a*e^2)^5*(7*c*d^2
+ 5*a*e^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])])/(1024*c^(7/2)*d^(7/2)*e^(9/2))

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Rubi [A]  time = 0.386024, antiderivative size = 381, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.132, Rules used = {794, 664, 612, 621, 206} \[ -\frac{\left (5 a e^2+7 c d^2\right ) \left (c d^2-a e^2\right )^3 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{512 c^3 d^3 e^4}+\frac{\left (5 a e^2+7 c d^2\right ) \left (c d^2-a e^2\right ) \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{192 c^2 d^2 e^3}+\frac{\left (5 a e^2+7 c d^2\right ) \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{1024 c^{7/2} d^{7/2} e^{9/2}}+\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{6 c d e (d+e x)}-\frac{1}{60} \left (\frac{5 a}{c d}+\frac{7 d}{e^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} \]

Antiderivative was successfully verified.

[In]

Int[(x*(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*(7*c*d^2 + 5*a*e^2)*(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
])/(512*c^3*d^3*e^4) + ((c*d^2 - a*e^2)*(7*c*d^2 + 5*a*e^2)*(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))/(192*c^2*d^2*e^3) - (((5*a)/(c*d) + (7*d)/e^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2
)^(5/2))/60 + (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2)/(6*c*d*e*(d + e*x)) + ((c*d^2 - a*e^2)^5*(7*c*d^2
+ 5*a*e^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])])/(1024*c^(7/2)*d^(7/2)*e^(9/2))

Rule 794

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

Rule 664

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

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 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])

Rubi steps

\begin{align*} \int \frac{x \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 &=\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{6 c d e (d+e x)}+\frac{1}{12} \left (-\frac{7 d}{e}-\frac{5 a e}{c d}\right ) \int \frac{\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\\ &=-\frac{1}{60} \left (\frac{5 a}{c d}+\frac{7 d}{e^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}+\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{6 c d e (d+e x)}+\frac{\left (\left (\frac{7 d}{e}+\frac{5 a e}{c d}\right ) \left (c d^2-a e^2\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}{24 e}\\ &=\frac{\left (c d^2-a e^2\right ) \left (7 c d^2+5 a e^2\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}}{192 c^2 d^2 e^3}-\frac{1}{60} \left (\frac{5 a}{c d}+\frac{7 d}{e^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}+\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{6 c d e (d+e x)}-\frac{\left (\left (c d^2-a e^2\right )^3 \left (7 c d^2+5 a e^2\right )\right ) \int \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{128 c^2 d^2 e^3}\\ &=-\frac{\left (c d^2-a e^2\right )^3 \left (7 c d^2+5 a e^2\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}}{512 c^3 d^3 e^4}+\frac{\left (c d^2-a e^2\right ) \left (7 c d^2+5 a e^2\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}}{192 c^2 d^2 e^3}-\frac{1}{60} \left (\frac{5 a}{c d}+\frac{7 d}{e^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}+\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{6 c d e (d+e x)}+\frac{\left (\left (c d^2-a e^2\right )^5 \left (7 c d^2+5 a e^2\right )\right ) \int \frac{1}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{1024 c^3 d^3 e^4}\\ &=-\frac{\left (c d^2-a e^2\right )^3 \left (7 c d^2+5 a e^2\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}}{512 c^3 d^3 e^4}+\frac{\left (c d^2-a e^2\right ) \left (7 c d^2+5 a e^2\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}}{192 c^2 d^2 e^3}-\frac{1}{60} \left (\frac{5 a}{c d}+\frac{7 d}{e^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}+\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{6 c d e (d+e x)}+\frac{\left (\left (c d^2-a e^2\right )^5 \left (7 c d^2+5 a e^2\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 )}{512 c^3 d^3 e^4}\\ &=-\frac{\left (c d^2-a e^2\right )^3 \left (7 c d^2+5 a e^2\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}}{512 c^3 d^3 e^4}+\frac{\left (c d^2-a e^2\right ) \left (7 c d^2+5 a e^2\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}}{192 c^2 d^2 e^3}-\frac{1}{60} \left (\frac{5 a}{c d}+\frac{7 d}{e^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}+\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{6 c d e (d+e x)}+\frac{\left (c d^2-a e^2\right )^5 \left (7 c d^2+5 a e^2\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 )}{1024 c^{7/2} d^{7/2} e^{9/2}}\\ \end{align*}

Mathematica [A]  time = 3.61385, size = 506, normalized size = 1.33 \[ \frac{(a e+c d x) ((d+e x) (a e+c d x))^{5/2} \left (7-\frac{7 \sqrt{c d} \sqrt{c d^2-a e^2} \left (5 a e^2+7 c d^2\right ) \left (\frac{c d (d+e x)}{c d^2-a e^2}\right )^{3/2} \left (15 \sqrt{e} \sqrt{c d} \left (c d^2-a e^2\right )^{11/2} (a e+c d x) \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}-10 e^{3/2} \sqrt{c d} \left (c d^2-a e^2\right )^{9/2} (a e+c d x)^2 \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}+8 e^{5/2} \sqrt{c d} \left (c d^2-a e^2\right )^{7/2} (a e+c d x)^3 \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}+16 e^{7/2} \sqrt{c d} \left (c d^2-a e^2\right )^{3/2} (a e+c d x)^4 \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}} \left (c d (11 d+8 e x)-3 a e^2\right )-15 \sqrt{c} \sqrt{d} \left (c d^2-a e^2\right )^6 \sqrt{a e+c d x} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right )\right )}{1280 c^5 d^5 e^{7/2} (d+e x)^4 (a e+c d x)^4}\right )}{42 c d e} \]

Antiderivative was successfully verified.

[In]

Integrate[(x*(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)*((a*e + c*d*x)*(d + e*x))^(5/2)*(7 - (7*Sqrt[c*d]*Sqrt[c*d^2 - a*e^2]*(7*c*d^2 + 5*a*e^2)*((c*d
*(d + e*x))/(c*d^2 - a*e^2))^(3/2)*(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*Sqrt[(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])]))/(1280*c^5*d^5*e^(7/2)*(a*e + c*d*x)^4*(d + e*x)^4)))/(42
*c*d*e)

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Maple [B]  time = 0.06, size = 2411, normalized size = 6.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*d*a*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(3/2)*x+1/16*d^4/e^3*c*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x)
)^(3/2)-3/128*d^7/e^4*c^2*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)-1/16*e*a^2/c*(c*d*e*(d/e+x)^2+(a*e^2-c
*d^2)*(d/e+x))^(3/2)+5/48*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*a+1/12/d/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*
x^2)^(5/2)*a-75/1024*e^4*d/c*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^
(1/2))/(d*e*c)^(1/2)*a^4+9/64*d^4/e*a*c*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)*x-3/256/d*e^6*a^5/c^2*ln
((1/2*a*e^2-1/2*c*d^2+(d/e+x)*c*d*e)/(d*e*c)^(1/2)+(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2))/(d*e*c)^(1/2
)-15/256*d^7/e^2*a*c^2*ln((1/2*a*e^2-1/2*c*d^2+(d/e+x)*c*d*e)/(d*e*c)^(1/2)+(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/
e+x))^(1/2))/(d*e*c)^(1/2)+15/256*d*e^4*a^4/c*ln((1/2*a*e^2-1/2*c*d^2+(d/e+x)*c*d*e)/(d*e*c)^(1/2)+(c*d*e*(d/e
+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2))/(d*e*c)^(1/2)-5/64/e*d^4*c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a+15/
512/e^2*d^7*c^2*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(d*e*c
)^(1/2)*a+15/512*e^6/d/c^2*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1
/2))/(d*e*c)^(1/2)*a^5-5/1024*e^8/d^3/c^3*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2)*
x+c*d*e*x^2)^(1/2))/(d*e*c)^(1/2)*a^6-5/96*e^2/d/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*a^2+5/256*e^5/d^2
/c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^4-15/512*e^4/d/c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^
4+5/256*e^2*d/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^3-75/1024*d^5*c*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*
e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(d*e*c)^(1/2)*a^2+5/256/e^3*d^6*c^2*(a*d*e+(a*e^2+c*d^2)*x
+c*d*e*x^2)^(1/2)*x+5/512*e^6/d^3/c^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^5+15/128*e*d^2*(a*d*e+(a*e^2+c
*d^2)*x+c*d*e*x^2)^(1/2)*x*a^2-3/64*d*e^2*a^3/c*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)+1/8*d^3/e^2*c*(c
*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(3/2)*x-15/128*d^3*e^2*a^3*ln((1/2*a*e^2-1/2*c*d^2+(d/e+x)*c*d*e)/(d*e*c
)^(1/2)+(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2))/(d*e*c)^(1/2)+3/128/d*e^4*a^4/c^2*(c*d*e*(d/e+x)^2+(a*e
^2-c*d^2)*(d/e+x))^(1/2)+3/64*d^5/e^2*a*c*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)-9/64*d^2*e*a^2*(c*d*e*
(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)*x+3/64*e^3*a^3/c*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)*x+15/128
*d^5*a^2*c*ln((1/2*a*e^2-1/2*c*d^2+(d/e+x)*c*d*e)/(d*e*c)^(1/2)+(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2))
/(d*e*c)^(1/2)+3/256*d^9/e^4*c^3*ln((1/2*a*e^2-1/2*c*d^2+(d/e+x)*c*d*e)/(d*e*c)^(1/2)+(c*d*e*(d/e+x)^2+(a*e^2-
c*d^2)*(d/e+x))^(1/2))/(d*e*c)^(1/2)+25/256*e^2*d^3*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*d*e+(a*e
^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(d*e*c)^(1/2)*a^3-5/64*e^3/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^3-3/64*
d^6/e^3*c^2*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)*x-5/96/e^2*d^3*c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(
3/2)*x+5/192*e/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^2+5/192/e*d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/
2)*a-5/192/e^3*d^4*c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+5/512/e^4*d^7*c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^
2)^(1/2)+1/6/e*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*x+1/12/e^2*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)-1/
5*d/e^2*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(5/2)+5/256*d^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^2-5/
1024/e^4*d^9*c^3*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(d*e*
c)^(1/2)-5/192*e^3/d^2/c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^3-15/512/e^2*d^5*c*(a*d*e+(a*e^2+c*d^2)*x
+c*d*e*x^2)^(1/2)*a

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(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

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Fricas [A]  time = 2.96248, size = 2264, normalized size = 5.94 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/30720*(15*(7*c^6*d^12 - 30*a*c^5*d^10*e^2 + 45*a^2*c^4*d^8*e^4 - 20*a^3*c^3*d^6*e^6 - 15*a^4*c^2*d^4*e^8 +
18*a^5*c*d^2*e^10 - 5*a^6*e^12)*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*sqr
t(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*(1280*c^6*d^6*e^6*x^5 - 105*c^6*d^11*e + 415*a*c^5*d^9*e^3 - 546*a^2*c^4*d^7*e^5 + 150*a^3*c^3*d^5*e^7 -
245*a^4*c^2*d^3*e^9 + 75*a^5*c*d*e^11 + 128*(13*c^6*d^7*e^5 + 25*a*c^5*d^5*e^7)*x^4 + 16*(3*c^6*d^8*e^4 + 278
*a*c^5*d^6*e^6 + 135*a^2*c^4*d^4*e^8)*x^3 - 8*(7*c^6*d^9*e^3 - 27*a*c^5*d^7*e^5 - 423*a^2*c^4*d^5*e^7 - 5*a^3*
c^3*d^3*e^9)*x^2 + 2*(35*c^6*d^10*e^2 - 136*a*c^5*d^8*e^4 + 174*a^2*c^4*d^6*e^6 + 80*a^3*c^3*d^4*e^8 - 25*a^4*
c^2*d^2*e^10)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^4*d^4*e^5), -1/15360*(15*(7*c^6*d^12 - 30*a*c
^5*d^10*e^2 + 45*a^2*c^4*d^8*e^4 - 20*a^3*c^3*d^6*e^6 - 15*a^4*c^2*d^4*e^8 + 18*a^5*c*d^2*e^10 - 5*a^6*e^12)*s
qrt(-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*(1280*c^6*d^6*e^6*x^5 - 105*c^6*d^11*e + 415*a*
c^5*d^9*e^3 - 546*a^2*c^4*d^7*e^5 + 150*a^3*c^3*d^5*e^7 - 245*a^4*c^2*d^3*e^9 + 75*a^5*c*d*e^11 + 128*(13*c^6*
d^7*e^5 + 25*a*c^5*d^5*e^7)*x^4 + 16*(3*c^6*d^8*e^4 + 278*a*c^5*d^6*e^6 + 135*a^2*c^4*d^4*e^8)*x^3 - 8*(7*c^6*
d^9*e^3 - 27*a*c^5*d^7*e^5 - 423*a^2*c^4*d^5*e^7 - 5*a^3*c^3*d^3*e^9)*x^2 + 2*(35*c^6*d^10*e^2 - 136*a*c^5*d^8
*e^4 + 174*a^2*c^4*d^6*e^6 + 80*a^3*c^3*d^4*e^8 - 25*a^4*c^2*d^2*e^10)*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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(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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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(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