3.20.54 \(\int \frac {(d+e x)^5}{(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [1954]

Optimal. Leaf size=302 \[ -\frac {2 (d+e x)^4}{c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 e \left (c d^2-a e^2\right )^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 c^4 d^4}+\frac {35 e \left (c d^2-a e^2\right ) (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 c^3 d^3}+\frac {7 e (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^2 d^2}+\frac {35 \sqrt {e} \left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{16 c^{9/2} d^{9/2}} \]

[Out]

35/16*(-a*e^2+c*d^2)^3*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))*e^(1/2)/c^(9/2)/d^(9/2)-2*(e*x+d)^4/c/d/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+35/8*e*(-a*e^2+c
*d^2)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^4/d^4+35/12*e*(-a*e^2+c*d^2)*(e*x+d)*(a*d*e+(a*e^2+c*d^2)*x+
c*d*e*x^2)^(1/2)/c^3/d^3+7/3*e*(e*x+d)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^2/d^2

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Rubi [A]
time = 0.17, antiderivative size = 302, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {682, 684, 654, 635, 212} \begin {gather*} \frac {35 \sqrt {e} \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 )}{16 c^{9/2} d^{9/2}}+\frac {35 e \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 c^4 d^4}+\frac {35 e (d+e x) \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 c^3 d^3}+\frac {7 e (d+e x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c^2 d^2}-\frac {2 (d+e x)^4}{c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(d + e*x)^4)/(c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (35*e*(c*d^2 - a*e^2)^2*Sqrt[a*d*e + (c*d
^2 + a*e^2)*x + c*d*e*x^2])/(8*c^4*d^4) + (35*e*(c*d^2 - a*e^2)*(d + e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d
*e*x^2])/(12*c^3*d^3) + (7*e*(d + e*x)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(3*c^2*d^2) + (35*Sqrt[e
]*(c*d^2 - a*e^2)^3*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])])/(16*c^(9/2)*d^(9/2))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 635

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 654

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

Rule 682

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*
((a + b*x + c*x^2)^(p + 1)/(c*(p + 1))), x] - Dist[e^2*((m + p)/(c*(p + 1))), Int[(d + e*x)^(m - 2)*(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] &
& LtQ[p, -1] && GtQ[m, 1] && IntegerQ[2*p]

Rule 684

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

Rubi steps

\begin {align*} \int \frac {(d+e x)^5}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx &=-\frac {2 (d+e x)^4}{c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {(7 e) \int \frac {(d+e x)^3}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{c d}\\ &=-\frac {2 (d+e x)^4}{c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {7 e (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^2 d^2}+\frac {\left (35 e \left (c d^2-a e^2\right )\right ) \int \frac {(d+e x)^2}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{6 c^2 d^2}\\ &=-\frac {2 (d+e x)^4}{c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 e \left (c d^2-a e^2\right ) (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 c^3 d^3}+\frac {7 e (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^2 d^2}+\frac {\left (35 e \left (c d^2-a e^2\right )^2\right ) \int \frac {d+e x}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{8 c^3 d^3}\\ &=-\frac {2 (d+e x)^4}{c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 e \left (c d^2-a e^2\right )^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 c^4 d^4}+\frac {35 e \left (c d^2-a e^2\right ) (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 c^3 d^3}+\frac {7 e (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^2 d^2}+\frac {\left (35 e \left (c d^2-a e^2\right )^3\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{16 c^4 d^4}\\ &=-\frac {2 (d+e x)^4}{c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 e \left (c d^2-a e^2\right )^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 c^4 d^4}+\frac {35 e \left (c d^2-a e^2\right ) (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 c^3 d^3}+\frac {7 e (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^2 d^2}+\frac {\left (35 e \left (c d^2-a e^2\right )^3\right ) \text {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 )}{8 c^4 d^4}\\ &=-\frac {2 (d+e x)^4}{c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 e \left (c d^2-a e^2\right )^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 c^4 d^4}+\frac {35 e \left (c d^2-a e^2\right ) (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 c^3 d^3}+\frac {7 e (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^2 d^2}+\frac {35 \sqrt {e} \left (c d^2-a e^2\right )^3 \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 )}{16 c^{9/2} d^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 0.54, size = 235, normalized size = 0.78 \begin {gather*} \frac {\left (c d^2-a e^2\right )^3 \left (-\frac {\sqrt {c} \sqrt {d} (a e+c d x)^4 (d+e x)^2 \left (-105 e^3+\frac {280 c d e^2 (d+e x)}{a e+c d x}-\frac {231 c^2 d^2 e (d+e x)^2}{(a e+c d x)^2}+\frac {48 c^3 d^3 (d+e x)^3}{(a e+c d x)^3}\right )}{\left (c d^2-a e^2\right )^3}+105 \sqrt {e} (a e+c d x)^{3/2} (d+e x)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )\right )}{24 c^{9/2} d^{9/2} ((a e+c d x) (d+e x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((c*d^2 - a*e^2)^3*(-((Sqrt[c]*Sqrt[d]*(a*e + c*d*x)^4*(d + e*x)^2*(-105*e^3 + (280*c*d*e^2*(d + e*x))/(a*e +
c*d*x) - (231*c^2*d^2*e*(d + e*x)^2)/(a*e + c*d*x)^2 + (48*c^3*d^3*(d + e*x)^3)/(a*e + c*d*x)^3))/(c*d^2 - a*e
^2)^3) + 105*Sqrt[e]*(a*e + c*d*x)^(3/2)*(d + e*x)^(3/2)*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt
[a*e + c*d*x])]))/(24*c^(9/2)*d^(9/2)*((a*e + c*d*x)*(d + e*x))^(3/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(3116\) vs. \(2(270)=540\).
time = 0.72, size = 3117, normalized size = 10.32

method	result	size
default	\(\text {Expression too large to display}\)	\(3117\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^5*(1/3*x^4/c/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-7/6*(a*e^2+c*d^2)/c/d/e*(1/2*x^3/c/d/e/(a*d*e+(a*e^
2+c*d^2)*x+c*d*e*x^2)^(1/2)-5/4*(a*e^2+c*d^2)/c/d/e*(x^2/c/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-3/2*(a*
e^2+c*d^2)/c/d/e*(-x/c/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-1/2*(a*e^2+c*d^2)/c/d/e*(-1/c/d/e/(a*d*e+(a
*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-(a*e^2+c*d^2)/c/d/e*(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)/(a*
d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))+1/c/d/e*ln((1/2*e^2*a+1/2*c*d^2+c*d*e*x)/(c*d*e)^(1/2)+(a*d*e+(a*e^2+c*d
^2)*x+c*d*e*x^2)^(1/2))/(c*d*e)^(1/2))-2*a/c*(-1/c/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-(a*e^2+c*d^2)/c
/d/e*(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)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)))-3/2*a/c
*(-x/c/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-1/2*(a*e^2+c*d^2)/c/d/e*(-1/c/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*
d*e*x^2)^(1/2)-(a*e^2+c*d^2)/c/d/e*(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)/(a*d*e+(a*e^2+c*d^2
)*x+c*d*e*x^2)^(1/2))+1/c/d/e*ln((1/2*e^2*a+1/2*c*d^2+c*d*e*x)/(c*d*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)
^(1/2))/(c*d*e)^(1/2)))-4/3*a/c*(x^2/c/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-3/2*(a*e^2+c*d^2)/c/d/e*(-x
/c/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-1/2*(a*e^2+c*d^2)/c/d/e*(-1/c/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*
x^2)^(1/2)-(a*e^2+c*d^2)/c/d/e*(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)/(a*d*e+(a*e^2+c*d^2)*x+
c*d*e*x^2)^(1/2))+1/c/d/e*ln((1/2*e^2*a+1/2*c*d^2+c*d*e*x)/(c*d*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/
2))/(c*d*e)^(1/2))-2*a/c*(-1/c/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-(a*e^2+c*d^2)/c/d/e*(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)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))))+5*d*e^4*(1/2*x^3/c/d/e/(a*
d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-5/4*(a*e^2+c*d^2)/c/d/e*(x^2/c/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2
)-3/2*(a*e^2+c*d^2)/c/d/e*(-x/c/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-1/2*(a*e^2+c*d^2)/c/d/e*(-1/c/d/e/
(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-(a*e^2+c*d^2)/c/d/e*(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)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))+1/c/d/e*ln((1/2*e^2*a+1/2*c*d^2+c*d*e*x)/(c*d*e)^(1/2)+(a*d*e+(
a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(c*d*e)^(1/2))-2*a/c*(-1/c/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-(a*e^2
+c*d^2)/c/d/e*(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)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))
)-3/2*a/c*(-x/c/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-1/2*(a*e^2+c*d^2)/c/d/e*(-1/c/d/e/(a*d*e+(a*e^2+c*
d^2)*x+c*d*e*x^2)^(1/2)-(a*e^2+c*d^2)/c/d/e*(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)/(a*d*e+(a*
e^2+c*d^2)*x+c*d*e*x^2)^(1/2))+1/c/d/e*ln((1/2*e^2*a+1/2*c*d^2+c*d*e*x)/(c*d*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c
*d*e*x^2)^(1/2))/(c*d*e)^(1/2)))+10*d^2*e^3*(x^2/c/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-3/2*(a*e^2+c*d^
2)/c/d/e*(-x/c/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-1/2*(a*e^2+c*d^2)/c/d/e*(-1/c/d/e/(a*d*e+(a*e^2+c*d
^2)*x+c*d*e*x^2)^(1/2)-(a*e^2+c*d^2)/c/d/e*(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)/(a*d*e+(a*e
^2+c*d^2)*x+c*d*e*x^2)^(1/2))+1/c/d/e*ln((1/2*e^2*a+1/2*c*d^2+c*d*e*x)/(c*d*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*
d*e*x^2)^(1/2))/(c*d*e)^(1/2))-2*a/c*(-1/c/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-(a*e^2+c*d^2)/c/d/e*(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)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)))+10*d^3*e^2*(-x/
c/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-1/2*(a*e^2+c*d^2)/c/d/e*(-1/c/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x
^2)^(1/2)-(a*e^2+c*d^2)/c/d/e*(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)/(a*d*e+(a*e^2+c*d^2)*x+c
*d*e*x^2)^(1/2))+1/c/d/e*ln((1/2*e^2*a+1/2*c*d^2+c*d*e*x)/(c*d*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2
))/(c*d*e)^(1/2))+5*d^4*e*(-1/c/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-(a*e^2+c*d^2)/c/d/e*(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)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))+2*d^5*(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)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(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((e*x+d)^5/(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(c*d^2-%e^2*a>0)', see `assume?
` for more d

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Fricas [A]
time = 3.40, size = 740, normalized size = 2.45 \begin {gather*} \left [\frac {105 \, {\left (c^{4} d^{7} x - 3 \, a c^{3} d^{5} x e^{2} + a c^{3} d^{6} e + 3 \, a^{2} c^{2} d^{3} x e^{4} - 3 \, a^{2} c^{2} d^{4} e^{3} - a^{3} c d x e^{6} + 3 \, a^{3} c d^{2} e^{5} - a^{4} e^{7}\right )} \sqrt {\frac {1}{c d}} e^{\frac {1}{2}} \log \left (8 \, c^{2} d^{3} x e + c^{2} d^{4} + 8 \, a c d x e^{3} + a^{2} e^{4} + 4 \, {\left (2 \, c^{2} d^{2} x e + c^{2} d^{3} + a c d e^{2}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {\frac {1}{c d}} e^{\frac {1}{2}} + 2 \, {\left (4 \, c^{2} d^{2} x^{2} + 3 \, a c d^{2}\right )} e^{2}\right ) + 4 \, {\left (87 \, c^{3} d^{5} x e - 48 \, c^{3} d^{6} + 35 \, a^{2} c d x e^{5} + 105 \, a^{3} e^{6} - 14 \, {\left (a c^{2} d^{2} x^{2} + 20 \, a^{2} c d^{2}\right )} e^{4} + 2 \, {\left (4 \, c^{3} d^{3} x^{3} - 49 \, a c^{2} d^{3} x\right )} e^{3} + {\left (38 \, c^{3} d^{4} x^{2} + 231 \, a c^{2} d^{4}\right )} e^{2}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}}{96 \, {\left (c^{5} d^{5} x + a c^{4} d^{4} e\right )}}, -\frac {105 \, {\left (c^{4} d^{7} x - 3 \, a c^{3} d^{5} x e^{2} + a c^{3} d^{6} e + 3 \, a^{2} c^{2} d^{3} x e^{4} - 3 \, a^{2} c^{2} d^{4} e^{3} - a^{3} c d x e^{6} + 3 \, a^{3} c d^{2} e^{5} - a^{4} e^{7}\right )} \sqrt {-\frac {e}{c d}} \arctan \left (\frac {\sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (2 \, c d x e + c d^{2} + a e^{2}\right )} \sqrt {-\frac {e}{c d}}}{2 \, {\left (c d^{2} x e + a x e^{3} + {\left (c d x^{2} + a d\right )} e^{2}\right )}}\right ) - 2 \, {\left (87 \, c^{3} d^{5} x e - 48 \, c^{3} d^{6} + 35 \, a^{2} c d x e^{5} + 105 \, a^{3} e^{6} - 14 \, {\left (a c^{2} d^{2} x^{2} + 20 \, a^{2} c d^{2}\right )} e^{4} + 2 \, {\left (4 \, c^{3} d^{3} x^{3} - 49 \, a c^{2} d^{3} x\right )} e^{3} + {\left (38 \, c^{3} d^{4} x^{2} + 231 \, a c^{2} d^{4}\right )} e^{2}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}}{48 \, {\left (c^{5} d^{5} x + a c^{4} d^{4} e\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/96*(105*(c^4*d^7*x - 3*a*c^3*d^5*x*e^2 + a*c^3*d^6*e + 3*a^2*c^2*d^3*x*e^4 - 3*a^2*c^2*d^4*e^3 - a^3*c*d*x*
e^6 + 3*a^3*c*d^2*e^5 - a^4*e^7)*sqrt(1/(c*d))*e^(1/2)*log(8*c^2*d^3*x*e + c^2*d^4 + 8*a*c*d*x*e^3 + a^2*e^4 +
4*(2*c^2*d^2*x*e + c^2*d^3 + a*c*d*e^2)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(1/(c*d))*e^(1/2) + 2
*(4*c^2*d^2*x^2 + 3*a*c*d^2)*e^2) + 4*(87*c^3*d^5*x*e - 48*c^3*d^6 + 35*a^2*c*d*x*e^5 + 105*a^3*e^6 - 14*(a*c^
2*d^2*x^2 + 20*a^2*c*d^2)*e^4 + 2*(4*c^3*d^3*x^3 - 49*a*c^2*d^3*x)*e^3 + (38*c^3*d^4*x^2 + 231*a*c^2*d^4)*e^2)
*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e))/(c^5*d^5*x + a*c^4*d^4*e), -1/48*(105*(c^4*d^7*x - 3*a*c^3*d^5*x
*e^2 + a*c^3*d^6*e + 3*a^2*c^2*d^3*x*e^4 - 3*a^2*c^2*d^4*e^3 - a^3*c*d*x*e^6 + 3*a^3*c*d^2*e^5 - a^4*e^7)*sqrt
(-e/(c*d))*arctan(1/2*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*(2*c*d*x*e + c*d^2 + a*e^2)*sqrt(-e/(c*d))/(
c*d^2*x*e + a*x*e^3 + (c*d*x^2 + a*d)*e^2)) - 2*(87*c^3*d^5*x*e - 48*c^3*d^6 + 35*a^2*c*d*x*e^5 + 105*a^3*e^6
- 14*(a*c^2*d^2*x^2 + 20*a^2*c*d^2)*e^4 + 2*(4*c^3*d^3*x^3 - 49*a*c^2*d^3*x)*e^3 + (38*c^3*d^4*x^2 + 231*a*c^2
*d^4)*e^2)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e))/(c^5*d^5*x + a*c^4*d^4*e)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d + e x\right )^{5}}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.40, size = 372, normalized size = 1.23 \begin {gather*} \frac {1}{24} \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} {\left (2 \, x {\left (\frac {4 \, x e^{3}}{c^{2} d^{2}} + \frac {{\left (19 \, c^{11} d^{12} e^{4} - 11 \, a c^{10} d^{10} e^{6}\right )} e^{\left (-2\right )}}{c^{13} d^{13}}\right )} + \frac {{\left (87 \, c^{11} d^{13} e^{3} - 136 \, a c^{10} d^{11} e^{5} + 57 \, a^{2} c^{9} d^{9} e^{7}\right )} e^{\left (-2\right )}}{c^{13} d^{13}}\right )} - \frac {35 \, {\left (\sqrt {c d} c^{3} d^{6} e^{\frac {3}{2}} - 3 \, \sqrt {c d} a c^{2} d^{4} e^{\frac {7}{2}} + 3 \, \sqrt {c d} a^{2} c d^{2} e^{\frac {11}{2}} - \sqrt {c d} a^{3} e^{\frac {15}{2}}\right )} e^{\left (-1\right )} \log \left ({\left | -\sqrt {c d} c d^{2} e^{\frac {1}{2}} - 2 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} c d e - \sqrt {c d} a e^{\frac {5}{2}} \right |}\right )}{16 \, c^{5} d^{5}} + \frac {2 \, {\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )}}{{\left ({\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} c d + \sqrt {c d} a e^{\frac {3}{2}}\right )} c^{4} d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*(2*x*(4*x*e^3/(c^2*d^2) + (19*c^11*d^12*e^4 - 11*a*c^10*d^10*
e^6)*e^(-2)/(c^13*d^13)) + (87*c^11*d^13*e^3 - 136*a*c^10*d^11*e^5 + 57*a^2*c^9*d^9*e^7)*e^(-2)/(c^13*d^13)) -
35/16*(sqrt(c*d)*c^3*d^6*e^(3/2) - 3*sqrt(c*d)*a*c^2*d^4*e^(7/2) + 3*sqrt(c*d)*a^2*c*d^2*e^(11/2) - sqrt(c*d)
*a^3*e^(15/2))*e^(-1)*log(abs(-sqrt(c*d)*c*d^2*e^(1/2) - 2*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a
*x*e^2 + a*d*e))*c*d*e - sqrt(c*d)*a*e^(5/2)))/(c^5*d^5) + 2*(c^4*d^8 - 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 -
4*a^3*c*d^2*e^6 + a^4*e^8)/(((sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))*c*d + sqrt(c*
d)*a*e^(3/2))*c^4*d^4)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^5}{{\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((d + e*x)^5/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2),x)

[Out]

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

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