3.20.21 \(\int (d+e x) (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2} \, dx\) [1921]

Optimal. Leaf size=283 \[ -\frac {3 \left (c d^2-a e^2\right )^3 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c^3 d^3 e^2}+\frac {\left (c d^2-a e^2\right ) \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{16 c^2 d^2 e}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 c d}+\frac {3 \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{256 c^{7/2} d^{7/2} e^{5/2}} \]

[Out]

1/16*(-a*e^2+c*d^2)*(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^2/d^2/e+1/5*(a*d*e+(a*e^
2+c*d^2)*x+c*d*e*x^2)^(5/2)/c/d+3/256*(-a*e^2+c*d^2)^5*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))/c^(7/2)/d^(7/2)/e^(5/2)-3/128*(-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)^(1/2)/c^3/d^3/e^2

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Rubi [A]
time = 0.10, antiderivative size = 283, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {654, 626, 635, 212} \begin {gather*} \frac {3 \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 )}{256 c^{7/2} d^{7/2} e^{5/2}}-\frac {3 \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}}{128 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 )^{3/2}}{16 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 )^{5/2}}{5 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)^(3/2),x]

[Out]

(-3*(c*d^2 - a*e^2)^3*(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])/(128*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)^(3/2))/(16*c^2*d^2*e
) + (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(5*c*d) + (3*(c*d^2 - a*e^2)^5*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])])/(256*c^(7/2)*d^(7/2)*e^(5/2))

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]

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 )^{3/2} \, dx &=\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 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 )^{3/2} \, 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 )^{3/2}}{16 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 )^{5/2}}{5 c d}-\frac {\left (3 \left (c d^2-a e^2\right )^3\right ) \int \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{32 c^2 d^2 e}\\ &=-\frac {3 \left (c d^2-a e^2\right )^3 \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}}{128 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 )^{3/2}}{16 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 )^{5/2}}{5 c d}+\frac {\left (3 \left (c d^2-a e^2\right )^5\right ) \int \frac {1}{\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 {3 \left (c d^2-a e^2\right )^3 \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}}{128 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 )^{3/2}}{16 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 )^{5/2}}{5 c d}+\frac {\left (3 \left (c d^2-a e^2\right )^5\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 )}{128 c^3 d^3 e^2}\\ &=-\frac {3 \left (c d^2-a e^2\right )^3 \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}}{128 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 )^{3/2}}{16 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 )^{5/2}}{5 c d}+\frac {3 \left (c d^2-a e^2\right )^5 \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 )}{256 c^{7/2} d^{7/2} e^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.74, size = 268, normalized size = 0.95 \begin {gather*} \frac {\left (c d^2-a e^2\right )^5 ((a e+c d x) (d+e x))^{3/2} \left (-\frac {\sqrt {c} \sqrt {d} \sqrt {e} (a e+c d x)^3 \left (-15 e^4+\frac {70 c d e^3 (d+e x)}{a e+c d x}-\frac {128 c^2 d^2 e^2 (d+e x)^2}{(a e+c d x)^2}-\frac {70 c^3 d^3 e (d+e x)^3}{(a e+c d x)^3}+\frac {15 c^4 d^4 (d+e x)^4}{(a e+c d x)^4}\right )}{\left (c d^2-a e^2\right )^5 (d+e x)}+\frac {15 \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)^{3/2} (d+e x)^{3/2}}\right )}{640 c^{7/2} d^{7/2} e^{5/2}} \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)^(3/2),x]

[Out]

((c*d^2 - a*e^2)^5*((a*e + c*d*x)*(d + e*x))^(3/2)*(-((Sqrt[c]*Sqrt[d]*Sqrt[e]*(a*e + c*d*x)^3*(-15*e^4 + (70*
c*d*e^3*(d + e*x))/(a*e + c*d*x) - (128*c^2*d^2*e^2*(d + e*x)^2)/(a*e + c*d*x)^2 - (70*c^3*d^3*e*(d + e*x)^3)/
(a*e + c*d*x)^3 + (15*c^4*d^4*(d + e*x)^4)/(a*e + c*d*x)^4))/((c*d^2 - a*e^2)^5*(d + e*x))) + (15*ArcTanh[(Sqr
t[c]*Sqrt[d]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a*e + c*d*x])])/((a*e + c*d*x)^(3/2)*(d + e*x)^(3/2))))/(640*c^(7/2)
*d^(7/2)*e^(5/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(558\) vs. \(2(253)=506\).
time = 0.73, size = 559, normalized size = 1.98

method	result	size
default	\(e \left (\frac {\left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {5}{2}}}{5 c d e}-\frac {\left (e^{2} a +c \,d^{2}\right ) \left (\frac {\left (2 c d e x +e^{2} a +c \,d^{2}\right ) \left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}{8 c d e}+\frac {3 \left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right ) \left (\frac {\left (2 c d e x +e^{2} a +c \,d^{2}\right ) \sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}{4 c d e}+\frac {\left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right ) \ln \left (\frac {\frac {1}{2} e^{2} a +\frac {1}{2} c \,d^{2}+c d e x}{\sqrt {c d e}}+\sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}\right )}{8 c d e \sqrt {c d e}}\right )}{16 c d e}\right )}{2 c d e}\right )+d \left (\frac {\left (2 c d e x +e^{2} a +c \,d^{2}\right ) \left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}{8 c d e}+\frac {3 \left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right ) \left (\frac {\left (2 c d e x +e^{2} a +c \,d^{2}\right ) \sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}{4 c d e}+\frac {\left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right ) \ln \left (\frac {\frac {1}{2} e^{2} a +\frac {1}{2} c \,d^{2}+c d e x}{\sqrt {c d e}}+\sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}\right )}{8 c d e \sqrt {c d e}}\right )}{16 c d e}\right )\)	\(559\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e*(1/5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c/d/e-1/2*(a*e^2+c*d^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*(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)*(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 = 3.30, size = 815, normalized size = 2.88 \begin {gather*} \left [\frac {{\left (15 \, {\left (c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}\right )} \sqrt {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 \, \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 {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 (10 \, c^{5} d^{8} x e^{2} - 15 \, c^{5} d^{9} e - 10 \, a^{3} c^{2} d^{2} x e^{8} + 15 \, a^{4} c d e^{9} + 2 \, {\left (4 \, a^{2} c^{3} d^{3} x^{2} - 35 \, a^{3} c^{2} d^{3}\right )} e^{7} + 2 \, {\left (88 \, a c^{4} d^{4} x^{3} + 23 \, a^{2} c^{3} d^{4} x\right )} e^{6} + 128 \, {\left (c^{5} d^{5} x^{4} + 4 \, a c^{4} d^{5} x^{2} + a^{2} c^{3} d^{5}\right )} e^{5} + 2 \, {\left (168 \, c^{5} d^{6} x^{3} + 233 \, a c^{4} d^{6} x\right )} e^{4} + 2 \, {\left (124 \, c^{5} d^{7} x^{2} + 35 \, a c^{4} d^{7}\right )} e^{3}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}\right )} e^{\left (-3\right )}}{2560 \, c^{4} d^{4}}, -\frac {{\left (15 \, {\left (c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}\right )} \sqrt {-c d e} \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 {-c d e}}{2 \, {\left (c^{2} d^{3} x e + a c d x e^{3} + {\left (c^{2} d^{2} x^{2} + a c d^{2}\right )} e^{2}\right )}}\right ) - 2 \, {\left (10 \, c^{5} d^{8} x e^{2} - 15 \, c^{5} d^{9} e - 10 \, a^{3} c^{2} d^{2} x e^{8} + 15 \, a^{4} c d e^{9} + 2 \, {\left (4 \, a^{2} c^{3} d^{3} x^{2} - 35 \, a^{3} c^{2} d^{3}\right )} e^{7} + 2 \, {\left (88 \, a c^{4} d^{4} x^{3} + 23 \, a^{2} c^{3} d^{4} x\right )} e^{6} + 128 \, {\left (c^{5} d^{5} x^{4} + 4 \, a c^{4} d^{5} x^{2} + a^{2} c^{3} d^{5}\right )} e^{5} + 2 \, {\left (168 \, c^{5} d^{6} x^{3} + 233 \, a c^{4} d^{6} x\right )} e^{4} + 2 \, {\left (124 \, c^{5} d^{7} x^{2} + 35 \, a c^{4} d^{7}\right )} e^{3}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}\right )} e^{\left (-3\right )}}{1280 \, c^{4} d^{4}}\right ] \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)^(3/2),x, algorithm="fricas")

[Out]

[1/2560*(15*(c^5*d^10 - 5*a*c^4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 - 10*a^3*c^2*d^4*e^6 + 5*a^4*c*d^2*e^8 - a^5*e^10
)*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*(10*c^5*d^8
*x*e^2 - 15*c^5*d^9*e - 10*a^3*c^2*d^2*x*e^8 + 15*a^4*c*d*e^9 + 2*(4*a^2*c^3*d^3*x^2 - 35*a^3*c^2*d^3)*e^7 + 2
*(88*a*c^4*d^4*x^3 + 23*a^2*c^3*d^4*x)*e^6 + 128*(c^5*d^5*x^4 + 4*a*c^4*d^5*x^2 + a^2*c^3*d^5)*e^5 + 2*(168*c^
5*d^6*x^3 + 233*a*c^4*d^6*x)*e^4 + 2*(124*c^5*d^7*x^2 + 35*a*c^4*d^7)*e^3)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 +
a*d)*e))*e^(-3)/(c^4*d^4), -1/1280*(15*(c^5*d^10 - 5*a*c^4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 - 10*a^3*c^2*d^4*e^6
+ 5*a^4*c*d^2*e^8 - a^5*e^10)*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*(10*c^5*d^8*x*e^2
- 15*c^5*d^9*e - 10*a^3*c^2*d^2*x*e^8 + 15*a^4*c*d*e^9 + 2*(4*a^2*c^3*d^3*x^2 - 35*a^3*c^2*d^3)*e^7 + 2*(88*a*
c^4*d^4*x^3 + 23*a^2*c^3*d^4*x)*e^6 + 128*(c^5*d^5*x^4 + 4*a*c^4*d^5*x^2 + a^2*c^3*d^5)*e^5 + 2*(168*c^5*d^6*x
^3 + 233*a*c^4*d^6*x)*e^4 + 2*(124*c^5*d^7*x^2 + 35*a*c^4*d^7)*e^3)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e
))*e^(-3)/(c^4*d^4)]

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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 {3}{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)**(3/2),x)

[Out]

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

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Giac [A]
time = 0.67, size = 388, normalized size = 1.37 \begin {gather*} \frac {1}{640} \, \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 \, c d x e^{2} + \frac {{\left (21 \, c^{5} d^{6} e^{5} + 11 \, a c^{4} d^{4} e^{7}\right )} e^{\left (-4\right )}}{c^{4} d^{4}}\right )} x + \frac {{\left (31 \, c^{5} d^{7} e^{4} + 64 \, a c^{4} d^{5} e^{6} + a^{2} c^{3} d^{3} e^{8}\right )} e^{\left (-4\right )}}{c^{4} d^{4}}\right )} x + \frac {{\left (5 \, c^{5} d^{8} e^{3} + 233 \, a c^{4} d^{6} e^{5} + 23 \, a^{2} c^{3} d^{4} e^{7} - 5 \, a^{3} c^{2} d^{2} e^{9}\right )} e^{\left (-4\right )}}{c^{4} d^{4}}\right )} x - \frac {{\left (15 \, c^{5} d^{9} e^{2} - 70 \, a c^{4} d^{7} e^{4} - 128 \, a^{2} c^{3} d^{5} e^{6} + 70 \, a^{3} c^{2} d^{3} e^{8} - 15 \, a^{4} c d e^{10}\right )} e^{\left (-4\right )}}{c^{4} d^{4}}\right )} - \frac {3 \, {\left (c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}\right )} e^{\left (-\frac {5}{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 )}{256 \, \sqrt {c d} c^{3} d^{3}} \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)^(3/2),x, algorithm="giac")

[Out]

1/640*sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*(2*(4*(2*(8*c*d*x*e^2 + (21*c^5*d^6*e^5 + 11*a*c^4*d^4*e^7)*
e^(-4)/(c^4*d^4))*x + (31*c^5*d^7*e^4 + 64*a*c^4*d^5*e^6 + a^2*c^3*d^3*e^8)*e^(-4)/(c^4*d^4))*x + (5*c^5*d^8*e
^3 + 233*a*c^4*d^6*e^5 + 23*a^2*c^3*d^4*e^7 - 5*a^3*c^2*d^2*e^9)*e^(-4)/(c^4*d^4))*x - (15*c^5*d^9*e^2 - 70*a*
c^4*d^7*e^4 - 128*a^2*c^3*d^5*e^6 + 70*a^3*c^2*d^3*e^8 - 15*a^4*c*d*e^10)*e^(-4)/(c^4*d^4)) - 3/256*(c^5*d^10
- 5*a*c^4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 - 10*a^3*c^2*d^4*e^6 + 5*a^4*c*d^2*e^8 - a^5*e^10)*e^(-5/2)*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^3*d^3)

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Mupad [B]
time = 1.43, size = 646, normalized size = 2.28 \begin {gather*} \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{5\,c\,d}+\frac {\left (\frac {c\,d^2}{2}+c\,x\,d\,e+\frac {a\,e^2}{2}\right )\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}}{4\,c\,e}-\frac {\left (\frac {3\,{\left (c\,d^2+a\,e^2\right )}^2}{4}-3\,a\,c\,d^2\,e^2\right )\,\left (\left (\frac {x}{2}+\frac {c\,d^2+a\,e^2}{4\,c\,d\,e}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}-\frac {\ln \left (2\,\sqrt {\left (a\,e+c\,d\,x\right )\,\left (d+e\,x\right )}\,\sqrt {c\,d\,e}+a\,e^2+c\,d^2+2\,c\,d\,e\,x\right )\,\left (\frac {{\left (c\,d^2+a\,e^2\right )}^2}{4}-a\,c\,d^2\,e^2\right )}{2\,{\left (c\,d\,e\right )}^{3/2}}\right )}{4\,c\,e}-\frac {\left (c\,d^2+a\,e^2\right )\,\left (\frac {x\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}}{4}-\frac {3\,a\,d\,e\,\left (\ln \left (2\,\sqrt {\left (a\,e+c\,d\,x\right )\,\left (d+e\,x\right )}\,\sqrt {c\,d\,e}+a\,e^2+c\,d^2+2\,c\,d\,e\,x\right )\,\left (\frac {{\left (c\,d^2+a\,e^2\right )}^2}{8\,{\left (c\,d\,e\right )}^{3/2}}-\frac {a\,d\,e}{2\,\sqrt {c\,d\,e}}\right )-\frac {\left (c\,d^2+2\,c\,x\,d\,e+a\,e^2\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{4\,c\,d\,e}\right )}{4}+\frac {\left (c\,d^2+a\,e^2\right )\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}}{8\,c\,d\,e}+\frac {3\,{\left (c\,d^2+a\,e^2\right )}^2\,\left (\ln \left (2\,\sqrt {\left (a\,e+c\,d\,x\right )\,\left (d+e\,x\right )}\,\sqrt {c\,d\,e}+a\,e^2+c\,d^2+2\,c\,d\,e\,x\right )\,\left (\frac {{\left (c\,d^2+a\,e^2\right )}^2}{8\,{\left (c\,d\,e\right )}^{3/2}}-\frac {a\,d\,e}{2\,\sqrt {c\,d\,e}}\right )-\frac {\left (c\,d^2+2\,c\,x\,d\,e+a\,e^2\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{4\,c\,d\,e}\right )}{16\,c\,d\,e}\right )}{2\,c\,d} \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)^(3/2),x)

[Out]

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

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