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

Optimal. Leaf size=356 \[ -\frac {5 \left (c d^2-a e^2\right )^7 \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2048 c^{9/2} d^{9/2} e^{7/2}}+\frac {5 \left (c d^2-a e^2\right )^5 \left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{1024 c^4 d^4 e^3}-\frac {5 \left (c d^2-a e^2\right )^3 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{384 c^3 d^3 e^2}+\frac {\left (c d^2-a e^2\right ) \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{24 c^2 d^2 e}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{7 c d} \]

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Rubi [A]  time = 0.24, antiderivative size = 356, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {640, 612, 621, 206} \begin {gather*} \frac {5 \left (c d^2-a e^2\right )^5 \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}}{1024 c^4 d^4 e^3}-\frac {5 \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 )^{3/2}}{384 c^3 d^3 e^2}+\frac {\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 )^{5/2}}{24 c^2 d^2 e}-\frac {5 \left (c d^2-a e^2\right )^7 \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 )}{2048 c^{9/2} d^{9/2} e^{7/2}}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{7 c d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(5*(c*d^2 - a*e^2)^5*(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])/(1024*c^4*d^4*e^
3) - (5*(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)^(3/2))/(384*c^3*
d^3*e^2) + ((c*d^2 - 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)^(5/2))/(24*c^2
*d^2*e) + (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2)/(7*c*d) - (5*(c*d^2 - a*e^2)^7*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])])/(2048*c^(9/2)*d^(9/2)*e
^(7/2))

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 640

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]

Rubi steps

\begin {align*} \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 &=\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{7 c d}+\frac {\left (d^2-\frac {a e^2}{c}\right ) \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx}{2 d}\\ &=\frac {\left (c d^2-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 )^{5/2}}{24 c^2 d^2 e}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{7 c d}-\frac {\left (5 \left (c d^2-a e^2\right )^3\right ) \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx}{48 c^2 d^2 e}\\ &=-\frac {5 \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 )^{3/2}}{384 c^3 d^3 e^2}+\frac {\left (c d^2-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 )^{5/2}}{24 c^2 d^2 e}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{7 c d}+\frac {\left (5 \left (c d^2-a e^2\right )^5\right ) \int \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{256 c^3 d^3 e^2}\\ &=\frac {5 \left (c d^2-a e^2\right )^5 \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}}{1024 c^4 d^4 e^3}-\frac {5 \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 )^{3/2}}{384 c^3 d^3 e^2}+\frac {\left (c d^2-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 )^{5/2}}{24 c^2 d^2 e}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{7 c d}-\frac {\left (5 \left (c d^2-a e^2\right )^7\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{2048 c^4 d^4 e^3}\\ &=\frac {5 \left (c d^2-a e^2\right )^5 \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}}{1024 c^4 d^4 e^3}-\frac {5 \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 )^{3/2}}{384 c^3 d^3 e^2}+\frac {\left (c d^2-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 )^{5/2}}{24 c^2 d^2 e}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{7 c d}-\frac {\left (5 \left (c d^2-a e^2\right )^7\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 )}{1024 c^4 d^4 e^3}\\ &=\frac {5 \left (c d^2-a e^2\right )^5 \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}}{1024 c^4 d^4 e^3}-\frac {5 \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 )^{3/2}}{384 c^3 d^3 e^2}+\frac {\left (c d^2-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 )^{5/2}}{24 c^2 d^2 e}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{7 c d}-\frac {5 \left (c d^2-a e^2\right )^7 \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 )}{2048 c^{9/2} d^{9/2} e^{7/2}}\\ \end {align*}

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Mathematica [B]  time = 6.22, size = 1199, normalized size = 3.37 \begin {gather*} \frac {2 \left (c d^2-a e^2\right )^3 (a e+c d x) ((a e+c d x) (d+e x))^{5/2} \left (\frac {c d e (a e+c d x)}{\left (c d^2-a e^2\right ) \left (\frac {c^2 d^3}{c d^2-a e^2}-\frac {a c d e^2}{c d^2-a e^2}\right )}+1\right )^{9/2} \left (\frac {35 \left (c d^2-a e^2\right )^4 \left (\frac {16 c^3 d^3 e^3 (a e+c d x)^3}{15 \left (c d^2-a e^2\right )^3 \left (\frac {c^2 d^3}{c d^2-a e^2}-\frac {a c d e^2}{c d^2-a e^2}\right )^3}-\frac {4 c^2 d^2 e^2 (a e+c d x)^2}{3 \left (c d^2-a e^2\right )^2 \left (\frac {c^2 d^3}{c d^2-a e^2}-\frac {a c d e^2}{c d^2-a e^2}\right )^2}+\frac {2 c d e (a e+c d x)}{\left (c d^2-a e^2\right ) \left (\frac {c^2 d^3}{c d^2-a e^2}-\frac {a c d e^2}{c d^2-a e^2}\right )}-\frac {2 \sqrt {c} \sqrt {d} \sqrt {e} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2} \sqrt {\frac {c^2 d^3}{c d^2-a e^2}-\frac {a c d e^2}{c d^2-a e^2}}}\right ) \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2} \sqrt {\frac {c^2 d^3}{c d^2-a e^2}-\frac {a c d e^2}{c d^2-a e^2}} \sqrt {\frac {c d e (a e+c d x)}{\left (c d^2-a e^2\right ) \left (\frac {c^2 d^3}{c d^2-a e^2}-\frac {a c d e^2}{c d^2-a e^2}\right )}+1}}\right ) \left (\frac {c^2 d^3}{c d^2-a e^2}-\frac {a c d e^2}{c d^2-a e^2}\right )^4}{4096 c^4 d^4 e^4 (a e+c d x)^4 \left (\frac {c d e (a e+c d x)}{\left (c d^2-a e^2\right ) \left (\frac {c^2 d^3}{c d^2-a e^2}-\frac {a c d e^2}{c d^2-a e^2}\right )}+1\right )^4}+\frac {1}{2} \left (\frac {1}{\frac {c d e (a e+c d x)}{\left (c d^2-a e^2\right ) \left (\frac {c^2 d^3}{c d^2-a e^2}-\frac {a c d e^2}{c d^2-a e^2}\right )}+1}+\frac {7}{12 \left (\frac {c d e (a e+c d x)}{\left (c d^2-a e^2\right ) \left (\frac {c^2 d^3}{c d^2-a e^2}-\frac {a c d e^2}{c d^2-a e^2}\right )}+1\right )^2}+\frac {7}{24 \left (\frac {c d e (a e+c d x)}{\left (c d^2-a e^2\right ) \left (\frac {c^2 d^3}{c d^2-a e^2}-\frac {a c d e^2}{c d^2-a e^2}\right )}+1\right )^3}+\frac {7}{64 \left (\frac {c d e (a e+c d x)}{\left (c d^2-a e^2\right ) \left (\frac {c^2 d^3}{c d^2-a e^2}-\frac {a c d e^2}{c d^2-a e^2}\right )}+1\right )^4}\right )\right )}{7 c^4 d^4 \left (\frac {c d}{\frac {c^2 d^3}{c d^2-a e^2}-\frac {a c d e^2}{c d^2-a e^2}}\right )^{7/2} (d+e x)^2 \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(c*d^2 - a*e^2)^3*(a*e + c*d*x)*((a*e + c*d*x)*(d + e*x))^(5/2)*(1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)
*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^(9/2)*((7/(64*(1 + (c*d*e*(a*e + c*d*x))/((c*d^2
- a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^4) + 7/(24*(1 + (c*d*e*(a*e + c*d*x))/((c
*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^3) + 7/(12*(1 + (c*d*e*(a*e + c*d*x)
)/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^2) + (1 + (c*d*e*(a*e + c*d*x))
/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))))^(-1))/2 + (35*(c*d^2 - a*e^2)^4*
((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))^4*((2*c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^
3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))) - (4*c^2*d^2*e^2*(a*e + c*d*x)^2)/(3*(c*d^2 - a*e^2)^2*((c^
2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))^2) + (16*c^3*d^3*e^3*(a*e + c*d*x)^3)/(15*(c*d^2 - a*e^2
)^3*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))^3) - (2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x
]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d^2 - a*e^2]*Sqrt[(c^2*d^3)/(c*d^2 - a*e^2) - (a
*c*d*e^2)/(c*d^2 - a*e^2)])])/(Sqrt[c*d^2 - a*e^2]*Sqrt[(c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2
)]*Sqrt[1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2)))]
)))/(4096*c^4*d^4*e^4*(a*e + c*d*x)^4*(1 + (c*d*e*(a*e + c*d*x))/((c*d^2 - a*e^2)*((c^2*d^3)/(c*d^2 - a*e^2) -
(a*c*d*e^2)/(c*d^2 - a*e^2))))^4)))/(7*c^4*d^4*((c*d)/((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2
)))^(7/2)*(d + e*x)^2*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])

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IntegrateAlgebraic [F]  time = 180.12, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

$Aborted

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fricas [A]  time = 0.53, size = 1270, normalized size = 3.57

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/86016*(105*(c^7*d^14 - 7*a*c^6*d^12*e^2 + 21*a^2*c^5*d^10*e^4 - 35*a^3*c^4*d^8*e^6 + 35*a^4*c^3*d^6*e^8 - 2
1*a^5*c^2*d^4*e^10 + 7*a^6*c*d^2*e^12 - a^7*e^14)*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*(3072*c^7*d^7*e^7*x^6 + 105*c^7*d^13*e - 700*a*c^6*d^11*e^3 + 1981*a^2*c^5*d^9*e^5 + 30
72*a^3*c^4*d^7*e^7 - 1981*a^4*c^3*d^5*e^9 + 700*a^5*c^2*d^3*e^11 - 105*a^6*c*d*e^13 + 256*(43*c^7*d^8*e^6 + 29
*a*c^6*d^6*e^8)*x^5 + 128*(107*c^7*d^9*e^5 + 216*a*c^6*d^7*e^7 + 37*a^2*c^5*d^5*e^9)*x^4 + 16*(381*c^7*d^10*e^
4 + 2281*a*c^6*d^8*e^6 + 1175*a^2*c^5*d^6*e^8 + 3*a^3*c^4*d^4*e^10)*x^3 + 8*(7*c^7*d^11*e^3 + 2258*a*c^6*d^9*e
^5 + 3456*a^2*c^5*d^7*e^7 + 46*a^3*c^4*d^5*e^9 - 7*a^4*c^3*d^3*e^11)*x^2 - 2*(35*c^7*d^12*e^2 - 231*a*c^6*d^10
*e^4 - 8570*a^2*c^5*d^8*e^6 - 646*a^3*c^4*d^6*e^8 + 231*a^4*c^3*d^4*e^10 - 35*a^5*c^2*d^2*e^12)*x)*sqrt(c*d*e*
x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^5*d^5*e^4), 1/43008*(105*(c^7*d^14 - 7*a*c^6*d^12*e^2 + 21*a^2*c^5*d^10*e
^4 - 35*a^3*c^4*d^8*e^6 + 35*a^4*c^3*d^6*e^8 - 21*a^5*c^2*d^4*e^10 + 7*a^6*c*d^2*e^12 - a^7*e^14)*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*(3072*c^7*d^7*e^7*x^6 + 105*c^7*d^13*e - 700*a*c^6*d^11*e^
3 + 1981*a^2*c^5*d^9*e^5 + 3072*a^3*c^4*d^7*e^7 - 1981*a^4*c^3*d^5*e^9 + 700*a^5*c^2*d^3*e^11 - 105*a^6*c*d*e^
13 + 256*(43*c^7*d^8*e^6 + 29*a*c^6*d^6*e^8)*x^5 + 128*(107*c^7*d^9*e^5 + 216*a*c^6*d^7*e^7 + 37*a^2*c^5*d^5*e
^9)*x^4 + 16*(381*c^7*d^10*e^4 + 2281*a*c^6*d^8*e^6 + 1175*a^2*c^5*d^6*e^8 + 3*a^3*c^4*d^4*e^10)*x^3 + 8*(7*c^
7*d^11*e^3 + 2258*a*c^6*d^9*e^5 + 3456*a^2*c^5*d^7*e^7 + 46*a^3*c^4*d^5*e^9 - 7*a^4*c^3*d^3*e^11)*x^2 - 2*(35*
c^7*d^12*e^2 - 231*a*c^6*d^10*e^4 - 8570*a^2*c^5*d^8*e^6 - 646*a^3*c^4*d^6*e^8 + 231*a^4*c^3*d^4*e^10 - 35*a^5
*c^2*d^2*e^12)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^5*d^5*e^4)]

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giac [A]  time = 0.60, size = 610, normalized size = 1.71 \begin {gather*} \frac {1}{21504} \, \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 (12 \, c^{2} d^{2} x e^{3} + \frac {{\left (43 \, c^{8} d^{9} e^{8} + 29 \, a c^{7} d^{7} e^{10}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} x + \frac {{\left (107 \, c^{8} d^{10} e^{7} + 216 \, a c^{7} d^{8} e^{9} + 37 \, a^{2} c^{6} d^{6} e^{11}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} x + \frac {{\left (381 \, c^{8} d^{11} e^{6} + 2281 \, a c^{7} d^{9} e^{8} + 1175 \, a^{2} c^{6} d^{7} e^{10} + 3 \, a^{3} c^{5} d^{5} e^{12}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} x + \frac {{\left (7 \, c^{8} d^{12} e^{5} + 2258 \, a c^{7} d^{10} e^{7} + 3456 \, a^{2} c^{6} d^{8} e^{9} + 46 \, a^{3} c^{5} d^{6} e^{11} - 7 \, a^{4} c^{4} d^{4} e^{13}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} x - \frac {{\left (35 \, c^{8} d^{13} e^{4} - 231 \, a c^{7} d^{11} e^{6} - 8570 \, a^{2} c^{6} d^{9} e^{8} - 646 \, a^{3} c^{5} d^{7} e^{10} + 231 \, a^{4} c^{4} d^{5} e^{12} - 35 \, a^{5} c^{3} d^{3} e^{14}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} x + \frac {{\left (105 \, c^{8} d^{14} e^{3} - 700 \, a c^{7} d^{12} e^{5} + 1981 \, a^{2} c^{6} d^{10} e^{7} + 3072 \, a^{3} c^{5} d^{8} e^{9} - 1981 \, a^{4} c^{4} d^{6} e^{11} + 700 \, a^{5} c^{3} d^{4} e^{13} - 105 \, a^{6} c^{2} d^{2} e^{15}\right )} e^{\left (-6\right )}}{c^{6} d^{6}}\right )} + \frac {5 \, {\left (c^{7} d^{14} - 7 \, a c^{6} d^{12} e^{2} + 21 \, a^{2} c^{5} d^{10} e^{4} - 35 \, a^{3} c^{4} d^{8} e^{6} + 35 \, a^{4} c^{3} d^{6} e^{8} - 21 \, a^{5} c^{2} d^{4} e^{10} + 7 \, a^{6} c d^{2} e^{12} - a^{7} e^{14}\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 )}{2048 \, \sqrt {c d} c^{4} d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/21504*sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*(2*(4*(2*(8*(2*(12*c^2*d^2*x*e^3 + (43*c^8*d^9*e^8 + 29*a*
c^7*d^7*e^10)*e^(-6)/(c^6*d^6))*x + (107*c^8*d^10*e^7 + 216*a*c^7*d^8*e^9 + 37*a^2*c^6*d^6*e^11)*e^(-6)/(c^6*d
^6))*x + (381*c^8*d^11*e^6 + 2281*a*c^7*d^9*e^8 + 1175*a^2*c^6*d^7*e^10 + 3*a^3*c^5*d^5*e^12)*e^(-6)/(c^6*d^6)
)*x + (7*c^8*d^12*e^5 + 2258*a*c^7*d^10*e^7 + 3456*a^2*c^6*d^8*e^9 + 46*a^3*c^5*d^6*e^11 - 7*a^4*c^4*d^4*e^13)
*e^(-6)/(c^6*d^6))*x - (35*c^8*d^13*e^4 - 231*a*c^7*d^11*e^6 - 8570*a^2*c^6*d^9*e^8 - 646*a^3*c^5*d^7*e^10 + 2
31*a^4*c^4*d^5*e^12 - 35*a^5*c^3*d^3*e^14)*e^(-6)/(c^6*d^6))*x + (105*c^8*d^14*e^3 - 700*a*c^7*d^12*e^5 + 1981
*a^2*c^6*d^10*e^7 + 3072*a^3*c^5*d^8*e^9 - 1981*a^4*c^4*d^6*e^11 + 700*a^5*c^3*d^4*e^13 - 105*a^6*c^2*d^2*e^15
)*e^(-6)/(c^6*d^6)) + 5/2048*(c^7*d^14 - 7*a*c^6*d^12*e^2 + 21*a^2*c^5*d^10*e^4 - 35*a^3*c^4*d^8*e^6 + 35*a^4*
c^3*d^6*e^8 - 21*a^5*c^2*d^4*e^10 + 7*a^6*c*d^2*e^12 - a^7*e^14)*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^4*d^4)

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maple [B]  time = 0.05, size = 1533, normalized size = 4.31

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12/c/d*e^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*x*a-105/2048*e*c*d^6*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^2-35/2048/c^3/d^2*e^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^6+1/7*(c*d*e*x^2+a*d*e+(a
*e^2+c*d^2)*x)^(7/2)/c/d-25/1024/c^2*e^5*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a^4+25/1024*e*d^4*(c*d*e*x^2+
a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a^2+5/1024*c^2*d^8/e^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)-5/384*c*d^5/e^2*(c
*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)+5/192*d^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*a+1/24*d^2/e*(c*d*e*x^
2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)+1/12*d*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*x-25/512*c*d^5*(c*d*e*x^2+a*d*e+
(a*e^2+c*d^2)*x)^(1/2)*x*a-5/192/c^2/d*e^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*a^3+5/512*c^2*d^7/e^2*(c*d*
e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x+5/64*e*d^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*x*a+25/256*d^3*e^2*(c*
d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x*a^2+105/2048/c^2*e^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^5-5/256*c*d^6/e*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*
a-5/192*c*d^4/e*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*x-1/24/c^2/d^2*e^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(
5/2)*a^2+5/256/c^3/d^2*e^7*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a^5+175/2048*d^4*e^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^3-5/64/c*e^3*(c*d*e*x^2+a*d*
e+(a*e^2+c*d^2)*x)^(3/2)*x*a^2-5/2048*c^3*d^10/e^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)+5/384/c^3/d^3*e^6*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*a^4-5/102
4/c^4/d^4*e^9*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a^6+25/512/c^2/d*e^6*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(
1/2)*x*a^4-25/256/c*d*e^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x*a^3-175/2048/c*d^2*e^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^4+35/2048*c^2*d^8/e*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+5/192/c^2/d
^2*e^5*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*x*a^3-5/512/c^3/d^3*e^8*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)
*x*a^5+5/2048/c^4/d^4*e^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)*a^7

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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)*(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(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 \left (d+e\,x\right )\,{\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)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2),x)

[Out]

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

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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 )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(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), x)

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