3.20.32 \(\int (d+e x)^3 (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2} \, dx\) [1932]

Optimal. Leaf size=474 \[ \frac {55 \left (c d^2-a e^2\right )^7 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{32768 c^6 d^6 e^3}-\frac {55 \left (c d^2-a e^2\right )^5 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{12288 c^5 d^5 e^2}+\frac {11 \left (c d^2-a e^2\right )^3 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{768 c^4 d^4 e}+\frac {11 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{224 c^3 d^3}+\frac {11 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{144 c^2 d^2}+\frac {(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{9 c d}-\frac {55 \left (c d^2-a e^2\right )^9 \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{65536 c^{13/2} d^{13/2} e^{7/2}} \]

[Out]

-55/12288*(-a*e^2+c*d^2)^5*(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^5/d^5/e^2+11/768*
(-a*e^2+c*d^2)^3*(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)^(5/2)/c^4/d^4/e+11/224*(-a*e^2+c*d^
2)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c^3/d^3+11/144*(-a*e^2+c*d^2)*(e*x+d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*
e*x^2)^(7/2)/c^2/d^2+1/9*(e*x+d)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c/d-55/65536*(-a*e^2+c*d^2)^9*arcta
nh(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^(13/2)/d^(13
/2)/e^(7/2)+55/32768*(-a*e^2+c*d^2)^7*(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^6/d^6/
e^3

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Rubi [A]
time = 0.32, antiderivative size = 474, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {684, 654, 626, 635, 212} \begin {gather*} -\frac {55 \left (c d^2-a e^2\right )^9 \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 )}{65536 c^{13/2} d^{13/2} e^{7/2}}+\frac {55 \left (c d^2-a e^2\right )^7 \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}}{32768 c^6 d^6 e^3}-\frac {55 \left (c d^2-a e^2\right )^5 \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}}{12288 c^5 d^5 e^2}+\frac {11 \left (c d^2-a e^2\right )^3 \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 )^{5/2}}{768 c^4 d^4 e}+\frac {11 \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 )^{7/2}}{224 c^3 d^3}+\frac {11 (d+e x) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{144 c^2 d^2}+\frac {(d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{9 c d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(55*(c*d^2 - a*e^2)^7*(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])/(32768*c^6*d^6*
e^3) - (55*(c*d^2 - a*e^2)^5*(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))/(12288
*c^5*d^5*e^2) + (11*(c*d^2 - a*e^2)^3*(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)^(5/2
))/(768*c^4*d^4*e) + (11*(c*d^2 - a*e^2)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(224*c^3*d^3) + (11*
(c*d^2 - a*e^2)*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(144*c^2*d^2) + ((d + e*x)^2*(a*d*e +
(c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(9*c*d) - (55*(c*d^2 - a*e^2)^9*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])])/(65536*c^(13/2)*d^(13/2)*e^(7/2))

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 626

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 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 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 (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx &=\frac {(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{9 c d}+\frac {\left (11 \left (d^2-\frac {a e^2}{c}\right )\right ) \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx}{18 d}\\ &=\frac {11 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{144 c^2 d^2}+\frac {(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{9 c d}+\frac {\left (11 \left (d^2-\frac {a e^2}{c}\right )^2\right ) \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx}{32 d^2}\\ &=\frac {11 \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 )^{7/2}}{224 c^3 d^3}+\frac {11 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{144 c^2 d^2}+\frac {(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{9 c d}+\frac {\left (11 \left (d^2-\frac {a e^2}{c}\right )^3\right ) \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx}{64 d^3}\\ &=\frac {11 \left (c d^2-a e^2\right )^3 \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 )^{5/2}}{768 c^4 d^4 e}+\frac {11 \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 )^{7/2}}{224 c^3 d^3}+\frac {11 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{144 c^2 d^2}+\frac {(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{9 c d}-\frac {\left (55 \left (c d^2-a e^2\right )^5\right ) \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx}{1536 c^4 d^4 e}\\ &=-\frac {55 \left (c d^2-a e^2\right )^5 \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}}{12288 c^5 d^5 e^2}+\frac {11 \left (c d^2-a e^2\right )^3 \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 )^{5/2}}{768 c^4 d^4 e}+\frac {11 \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 )^{7/2}}{224 c^3 d^3}+\frac {11 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{144 c^2 d^2}+\frac {(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{9 c d}+\frac {\left (55 \left (c d^2-a e^2\right )^7\right ) \int \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{8192 c^5 d^5 e^2}\\ &=\frac {55 \left (c d^2-a e^2\right )^7 \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}}{32768 c^6 d^6 e^3}-\frac {55 \left (c d^2-a e^2\right )^5 \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}}{12288 c^5 d^5 e^2}+\frac {11 \left (c d^2-a e^2\right )^3 \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 )^{5/2}}{768 c^4 d^4 e}+\frac {11 \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 )^{7/2}}{224 c^3 d^3}+\frac {11 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{144 c^2 d^2}+\frac {(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{9 c d}-\frac {\left (55 \left (c d^2-a e^2\right )^9\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{65536 c^6 d^6 e^3}\\ &=\frac {55 \left (c d^2-a e^2\right )^7 \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}}{32768 c^6 d^6 e^3}-\frac {55 \left (c d^2-a e^2\right )^5 \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}}{12288 c^5 d^5 e^2}+\frac {11 \left (c d^2-a e^2\right )^3 \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 )^{5/2}}{768 c^4 d^4 e}+\frac {11 \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 )^{7/2}}{224 c^3 d^3}+\frac {11 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{144 c^2 d^2}+\frac {(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{9 c d}-\frac {\left (55 \left (c d^2-a e^2\right )^9\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 )}{32768 c^6 d^6 e^3}\\ &=\frac {55 \left (c d^2-a e^2\right )^7 \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}}{32768 c^6 d^6 e^3}-\frac {55 \left (c d^2-a e^2\right )^5 \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}}{12288 c^5 d^5 e^2}+\frac {11 \left (c d^2-a e^2\right )^3 \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 )^{5/2}}{768 c^4 d^4 e}+\frac {11 \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 )^{7/2}}{224 c^3 d^3}+\frac {11 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{144 c^2 d^2}+\frac {(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{9 c d}-\frac {55 \left (c d^2-a e^2\right )^9 \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 )}{65536 c^{13/2} d^{13/2} e^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 2.07, size = 379, normalized size = 0.80 \begin {gather*} \frac {\left (c d^2-a e^2\right )^9 ((a e+c d x) (d+e x))^{5/2} \left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} (a e+c d x)^6 \left (-3465 e^8+\frac {30030 c d e^7 (d+e x)}{a e+c d x}-\frac {115038 c^2 d^2 e^6 (d+e x)^2}{(a e+c d x)^2}+\frac {255222 c^3 d^3 e^5 (d+e x)^3}{(a e+c d x)^3}-\frac {360448 c^4 d^4 e^4 (d+e x)^4}{(a e+c d x)^4}+\frac {334602 c^5 d^5 e^3 (d+e x)^5}{(a e+c d x)^5}+\frac {115038 c^6 d^6 e^2 (d+e x)^6}{(a e+c d x)^6}-\frac {30030 c^7 d^7 e (d+e x)^7}{(a e+c d x)^7}+\frac {3465 c^8 d^8 (d+e x)^8}{(a e+c d x)^8}\right )}{\left (c d^2-a e^2\right )^9 (d+e x)^2}-\frac {3465 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{(a e+c d x)^{5/2} (d+e x)^{5/2}}\right )}{2064384 c^{13/2} d^{13/2} e^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((c*d^2 - a*e^2)^9*((a*e + c*d*x)*(d + e*x))^(5/2)*((Sqrt[c]*Sqrt[d]*Sqrt[e]*(a*e + c*d*x)^6*(-3465*e^8 + (300
30*c*d*e^7*(d + e*x))/(a*e + c*d*x) - (115038*c^2*d^2*e^6*(d + e*x)^2)/(a*e + c*d*x)^2 + (255222*c^3*d^3*e^5*(
d + e*x)^3)/(a*e + c*d*x)^3 - (360448*c^4*d^4*e^4*(d + e*x)^4)/(a*e + c*d*x)^4 + (334602*c^5*d^5*e^3*(d + e*x)
^5)/(a*e + c*d*x)^5 + (115038*c^6*d^6*e^2*(d + e*x)^6)/(a*e + c*d*x)^6 - (30030*c^7*d^7*e*(d + e*x)^7)/(a*e +
c*d*x)^7 + (3465*c^8*d^8*(d + e*x)^8)/(a*e + c*d*x)^8))/((c*d^2 - a*e^2)^9*(d + e*x)^2) - (3465*ArcTanh[(Sqrt[
c]*Sqrt[d]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a*e + c*d*x])])/((a*e + c*d*x)^(5/2)*(d + e*x)^(5/2))))/(2064384*c^(13
/2)*d^(13/2)*e^(7/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2844\) vs. \(2(432)=864\).
time = 0.81, size = 2845, normalized size = 6.00

method	result	size
default	\(\text {Expression too large to display}\)	\(2845\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^3*(1/9*x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c/d/e-11/18*(a*e^2+c*d^2)/c/d/e*(1/8*x*(a*d*e+(a*e^2+c*d^
2)*x+c*d*e*x^2)^(7/2)/c/d/e-9/16*(a*e^2+c*d^2)/c/d/e*(1/7*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c/d/e-1/2*(a
*e^2+c*d^2)/c/d/e*(1/12*(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)^(5/2)/c/d/e+5/24*(4*a*c*d^2*
e^2-(a*e^2+c*d^2)^2)/c/d/e*(1/8*(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/d/e+3/16*(4*
a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c/d/e*(1/4*(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/d/e+
1/8*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/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)))))-1/8*a/c*(1/12*(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)^
(5/2)/c/d/e+5/24*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c/d/e*(1/8*(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/d/e+3/16*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c/d/e*(1/4*(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/d/e+1/8*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/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)))))-2/9*a/c*(1/7*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^
2)^(7/2)/c/d/e-1/2*(a*e^2+c*d^2)/c/d/e*(1/12*(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)^(5/2)/c
/d/e+5/24*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c/d/e*(1/8*(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/d/e+3/16*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c/d/e*(1/4*(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/d/e+1/8*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/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))))))+3*d*e^2*(1/8*x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(
7/2)/c/d/e-9/16*(a*e^2+c*d^2)/c/d/e*(1/7*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c/d/e-1/2*(a*e^2+c*d^2)/c/d/e
*(1/12*(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)^(5/2)/c/d/e+5/24*(4*a*c*d^2*e^2-(a*e^2+c*d^2)
^2)/c/d/e*(1/8*(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/d/e+3/16*(4*a*c*d^2*e^2-(a*e^
2+c*d^2)^2)/c/d/e*(1/4*(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/d/e+1/8*(4*a*c*d^2*e^
2-(a*e^2+c*d^2)^2)/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)))))-1/8*a/c*(1/12*(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)^(5/2)/c/d/e+5/24*
(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c/d/e*(1/8*(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/d
/e+3/16*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c/d/e*(1/4*(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/d/e+1/8*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/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)))))+3*d^2*e*(1/7*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c/d/e-1/
2*(a*e^2+c*d^2)/c/d/e*(1/12*(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)^(5/2)/c/d/e+5/24*(4*a*c*
d^2*e^2-(a*e^2+c*d^2)^2)/c/d/e*(1/8*(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/d/e+3/16
*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c/d/e*(1/4*(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/
d/e+1/8*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/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)))))+d^3*(1/12*(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)^
(5/2)/c/d/e+5/24*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c/d/e*(1/8*(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/d/e+3/16*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c/d/e*(1/4*(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/d/e+1/8*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/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))))

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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)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),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 [B] Leaf count of result is larger than twice the leaf count of optimal. 874 vs. \(2 (430) = 860\).
time = 3.18, size = 1763, normalized size = 3.72 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/8257536*(3465*(c^9*d^18 - 9*a*c^8*d^16*e^2 + 36*a^2*c^7*d^14*e^4 - 84*a^3*c^6*d^12*e^6 + 126*a^4*c^5*d^10*e
^8 - 126*a^5*c^4*d^8*e^10 + 84*a^6*c^3*d^6*e^12 - 36*a^7*c^2*d^4*e^14 + 9*a^8*c*d^2*e^16 - a^9*e^18)*sqrt(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*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(c*d)*e^(1/2) + 2*(4*c^2*d^2*x^2 + 3*a*c*d^2)*e^2) - 4*(2310*c^9*d^16*x*e^2 -
3465*c^9*d^17*e - 2310*a^7*c^2*d^2*x*e^16 + 3465*a^8*c*d*e^17 + 462*(4*a^6*c^3*d^3*x^2 - 65*a^7*c^2*d^3)*e^15
- 66*(24*a^5*c^4*d^4*x^3 - 301*a^6*c^3*d^4*x)*e^14 + 22*(64*a^4*c^5*d^5*x^4 - 720*a^5*c^4*d^5*x^2 + 5229*a^6*
c^3*d^5)*e^13 - 2*(640*a^3*c^6*d^6*x^5 - 6776*a^4*c^5*d^6*x^3 + 37719*a^5*c^4*d^6*x)*e^12 - 2*(158208*a^2*c^7*
d^7*x^6 + 6016*a^3*c^6*d^7*x^4 - 29964*a^4*c^5*d^7*x^2 + 127611*a^5*c^4*d^7)*e^11 - 2*(265216*a*c^8*d^8*x^7 +
947328*a^2*c^7*d^8*x^5 + 25584*a^3*c^6*d^8*x^3 - 82841*a^4*c^5*d^8*x)*e^10 - 32768*(7*c^9*d^9*x^8 + 94*a*c^8*d
^9*x^6 + 144*a^2*c^7*d^9*x^4 + 4*a^3*c^6*d^9*x^2 - 11*a^4*c^5*d^9)*e^9 - 2*(652288*c^9*d^10*x^7 + 3672960*a*c^
8*d^10*x^5 + 3120144*a^2*c^7*d^10*x^3 + 115609*a^3*c^6*d^10*x)*e^8 - 2*(1512960*c^9*d^11*x^6 + 4548736*a*c^8*d
^11*x^4 + 2290956*a^2*c^7*d^11*x^2 + 167301*a^3*c^6*d^11)*e^7 - 2*(1801600*c^9*d^12*x^5 + 2988664*a*c^8*d^12*x
^3 + 847017*a^2*c^7*d^12*x)*e^6 - 2*(1114816*c^9*d^13*x^4 + 876816*a*c^8*d^13*x^2 + 57519*a^2*c^7*d^13)*e^5 -
258*(2280*c^9*d^14*x^3 + 77*a*c^8*d^14*x)*e^4 - 462*(4*c^9*d^15*x^2 - 65*a*c^8*d^15)*e^3)*sqrt(c*d^2*x + a*x*e
^2 + (c*d*x^2 + a*d)*e))*e^(-4)/(c^7*d^7), 1/4128768*(3465*(c^9*d^18 - 9*a*c^8*d^16*e^2 + 36*a^2*c^7*d^14*e^4
- 84*a^3*c^6*d^12*e^6 + 126*a^4*c^5*d^10*e^8 - 126*a^5*c^4*d^8*e^10 + 84*a^6*c^3*d^6*e^12 - 36*a^7*c^2*d^4*e^1
4 + 9*a^8*c*d^2*e^16 - a^9*e^18)*sqrt(-c*d*e)*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(-c*d*e)/(c^2*d^3*x*e + a*c*d*x*e^3 + (c^2*d^2*x^2 + a*c*d^2)*e^2)) - 2*(2310*c^9*d^16*
x*e^2 - 3465*c^9*d^17*e - 2310*a^7*c^2*d^2*x*e^16 + 3465*a^8*c*d*e^17 + 462*(4*a^6*c^3*d^3*x^2 - 65*a^7*c^2*d^
3)*e^15 - 66*(24*a^5*c^4*d^4*x^3 - 301*a^6*c^3*d^4*x)*e^14 + 22*(64*a^4*c^5*d^5*x^4 - 720*a^5*c^4*d^5*x^2 + 52
29*a^6*c^3*d^5)*e^13 - 2*(640*a^3*c^6*d^6*x^5 - 6776*a^4*c^5*d^6*x^3 + 37719*a^5*c^4*d^6*x)*e^12 - 2*(158208*a
^2*c^7*d^7*x^6 + 6016*a^3*c^6*d^7*x^4 - 29964*a^4*c^5*d^7*x^2 + 127611*a^5*c^4*d^7)*e^11 - 2*(265216*a*c^8*d^8
*x^7 + 947328*a^2*c^7*d^8*x^5 + 25584*a^3*c^6*d^8*x^3 - 82841*a^4*c^5*d^8*x)*e^10 - 32768*(7*c^9*d^9*x^8 + 94*
a*c^8*d^9*x^6 + 144*a^2*c^7*d^9*x^4 + 4*a^3*c^6*d^9*x^2 - 11*a^4*c^5*d^9)*e^9 - 2*(652288*c^9*d^10*x^7 + 36729
60*a*c^8*d^10*x^5 + 3120144*a^2*c^7*d^10*x^3 + 115609*a^3*c^6*d^10*x)*e^8 - 2*(1512960*c^9*d^11*x^6 + 4548736*
a*c^8*d^11*x^4 + 2290956*a^2*c^7*d^11*x^2 + 167301*a^3*c^6*d^11)*e^7 - 2*(1801600*c^9*d^12*x^5 + 2988664*a*c^8
*d^12*x^3 + 847017*a^2*c^7*d^12*x)*e^6 - 2*(1114816*c^9*d^13*x^4 + 876816*a*c^8*d^13*x^2 + 57519*a^2*c^7*d^13)
*e^5 - 258*(2280*c^9*d^14*x^3 + 77*a*c^8*d^14*x)*e^4 - 462*(4*c^9*d^15*x^2 - 65*a*c^8*d^15)*e^3)*sqrt(c*d^2*x
+ a*x*e^2 + (c*d*x^2 + a*d)*e))*e^(-4)/(c^7*d^7)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 875 vs. \(2 (430) = 860\).
time = 0.77, size = 875, normalized size = 1.85 \begin {gather*} \frac {1}{2064384} \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (4 \, {\left (14 \, {\left (16 \, c^{2} d^{2} x e^{5} + \frac {{\left (91 \, c^{10} d^{11} e^{12} + 37 \, a c^{9} d^{9} e^{14}\right )} e^{\left (-8\right )}}{c^{8} d^{8}}\right )} x + \frac {{\left (2955 \, c^{10} d^{12} e^{11} + 3008 \, a c^{9} d^{10} e^{13} + 309 \, a^{2} c^{8} d^{8} e^{15}\right )} e^{\left (-8\right )}}{c^{8} d^{8}}\right )} x + \frac {{\left (14075 \, c^{10} d^{13} e^{10} + 28695 \, a c^{9} d^{11} e^{12} + 7401 \, a^{2} c^{8} d^{9} e^{14} + 5 \, a^{3} c^{7} d^{7} e^{16}\right )} e^{\left (-8\right )}}{c^{8} d^{8}}\right )} x + \frac {{\left (17419 \, c^{10} d^{14} e^{9} + 71074 \, a c^{9} d^{12} e^{11} + 36864 \, a^{2} c^{8} d^{10} e^{13} + 94 \, a^{3} c^{7} d^{8} e^{15} - 11 \, a^{4} c^{6} d^{6} e^{17}\right )} e^{\left (-8\right )}}{c^{8} d^{8}}\right )} x + \frac {{\left (36765 \, c^{10} d^{15} e^{8} + 373583 \, a c^{9} d^{13} e^{10} + 390018 \, a^{2} c^{8} d^{11} e^{12} + 3198 \, a^{3} c^{7} d^{9} e^{14} - 847 \, a^{4} c^{6} d^{7} e^{16} + 99 \, a^{5} c^{5} d^{5} e^{18}\right )} e^{\left (-8\right )}}{c^{8} d^{8}}\right )} x + \frac {{\left (231 \, c^{10} d^{16} e^{7} + 219204 \, a c^{9} d^{14} e^{9} + 572739 \, a^{2} c^{8} d^{12} e^{11} + 16384 \, a^{3} c^{7} d^{10} e^{13} - 7491 \, a^{4} c^{6} d^{8} e^{15} + 1980 \, a^{5} c^{5} d^{6} e^{17} - 231 \, a^{6} c^{4} d^{4} e^{19}\right )} e^{\left (-8\right )}}{c^{8} d^{8}}\right )} x - \frac {{\left (1155 \, c^{10} d^{17} e^{6} - 9933 \, a c^{9} d^{15} e^{8} - 847017 \, a^{2} c^{8} d^{13} e^{10} - 115609 \, a^{3} c^{7} d^{11} e^{12} + 82841 \, a^{4} c^{6} d^{9} e^{14} - 37719 \, a^{5} c^{5} d^{7} e^{16} + 9933 \, a^{6} c^{4} d^{5} e^{18} - 1155 \, a^{7} c^{3} d^{3} e^{20}\right )} e^{\left (-8\right )}}{c^{8} d^{8}}\right )} x + \frac {{\left (3465 \, c^{10} d^{18} e^{5} - 30030 \, a c^{9} d^{16} e^{7} + 115038 \, a^{2} c^{8} d^{14} e^{9} + 334602 \, a^{3} c^{7} d^{12} e^{11} - 360448 \, a^{4} c^{6} d^{10} e^{13} + 255222 \, a^{5} c^{5} d^{8} e^{15} - 115038 \, a^{6} c^{4} d^{6} e^{17} + 30030 \, a^{7} c^{3} d^{4} e^{19} - 3465 \, a^{8} c^{2} d^{2} e^{21}\right )} e^{\left (-8\right )}}{c^{8} d^{8}}\right )} + \frac {55 \, {\left (c^{9} d^{18} - 9 \, a c^{8} d^{16} e^{2} + 36 \, a^{2} c^{7} d^{14} e^{4} - 84 \, a^{3} c^{6} d^{12} e^{6} + 126 \, a^{4} c^{5} d^{10} e^{8} - 126 \, a^{5} c^{4} d^{8} e^{10} + 84 \, a^{6} c^{3} d^{6} e^{12} - 36 \, a^{7} c^{2} d^{4} e^{14} + 9 \, a^{8} c d^{2} e^{16} - a^{9} e^{18}\right )} e^{\left (-\frac {7}{2}\right )} \log \left ({\left | -c d^{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 )} \sqrt {c d} e^{\frac {1}{2}} - a e^{2} \right |}\right )}{65536 \, \sqrt {c d} c^{6} d^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2064384*sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*(2*(4*(2*(8*(2*(4*(14*(16*c^2*d^2*x*e^5 + (91*c^10*d^11*
e^12 + 37*a*c^9*d^9*e^14)*e^(-8)/(c^8*d^8))*x + (2955*c^10*d^12*e^11 + 3008*a*c^9*d^10*e^13 + 309*a^2*c^8*d^8*
e^15)*e^(-8)/(c^8*d^8))*x + (14075*c^10*d^13*e^10 + 28695*a*c^9*d^11*e^12 + 7401*a^2*c^8*d^9*e^14 + 5*a^3*c^7*
d^7*e^16)*e^(-8)/(c^8*d^8))*x + (17419*c^10*d^14*e^9 + 71074*a*c^9*d^12*e^11 + 36864*a^2*c^8*d^10*e^13 + 94*a^
3*c^7*d^8*e^15 - 11*a^4*c^6*d^6*e^17)*e^(-8)/(c^8*d^8))*x + (36765*c^10*d^15*e^8 + 373583*a*c^9*d^13*e^10 + 39
0018*a^2*c^8*d^11*e^12 + 3198*a^3*c^7*d^9*e^14 - 847*a^4*c^6*d^7*e^16 + 99*a^5*c^5*d^5*e^18)*e^(-8)/(c^8*d^8))
*x + (231*c^10*d^16*e^7 + 219204*a*c^9*d^14*e^9 + 572739*a^2*c^8*d^12*e^11 + 16384*a^3*c^7*d^10*e^13 - 7491*a^
4*c^6*d^8*e^15 + 1980*a^5*c^5*d^6*e^17 - 231*a^6*c^4*d^4*e^19)*e^(-8)/(c^8*d^8))*x - (1155*c^10*d^17*e^6 - 993
3*a*c^9*d^15*e^8 - 847017*a^2*c^8*d^13*e^10 - 115609*a^3*c^7*d^11*e^12 + 82841*a^4*c^6*d^9*e^14 - 37719*a^5*c^
5*d^7*e^16 + 9933*a^6*c^4*d^5*e^18 - 1155*a^7*c^3*d^3*e^20)*e^(-8)/(c^8*d^8))*x + (3465*c^10*d^18*e^5 - 30030*
a*c^9*d^16*e^7 + 115038*a^2*c^8*d^14*e^9 + 334602*a^3*c^7*d^12*e^11 - 360448*a^4*c^6*d^10*e^13 + 255222*a^5*c^
5*d^8*e^15 - 115038*a^6*c^4*d^6*e^17 + 30030*a^7*c^3*d^4*e^19 - 3465*a^8*c^2*d^2*e^21)*e^(-8)/(c^8*d^8)) + 55/
65536*(c^9*d^18 - 9*a*c^8*d^16*e^2 + 36*a^2*c^7*d^14*e^4 - 84*a^3*c^6*d^12*e^6 + 126*a^4*c^5*d^10*e^8 - 126*a^
5*c^4*d^8*e^10 + 84*a^6*c^3*d^6*e^12 - 36*a^7*c^2*d^4*e^14 + 9*a^8*c*d^2*e^16 - a^9*e^18)*e^(-7/2)*log(abs(-c*
d^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))*sqrt(c*d)*e^(1/2) - a*e^2))/(sqrt(
c*d)*c^6*d^6)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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