3.20.35 \(\int (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2} \, dx\) [1935]

Optimal. Leaf size=305 \[ \frac {5 \left (c d^2-a e^2\right )^4 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 c^3 d^3 e^3}-\frac {5 \left (c d^2-a e^2\right )^2 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{192 c^2 d^2 e^2}+\frac {\left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{12 c d e}-\frac {5 \left (c d^2-a e^2\right )^6 \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{1024 c^{7/2} d^{7/2} e^{7/2}} \]

[Out]

-5/192*(-a*e^2+c*d^2)^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^2+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-5/1024*(-a*e^2+c*d^2)^6*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^(7/2)+5/512*
(-a*e^2+c*d^2)^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^3/d^3/e^3

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Rubi [A]
time = 0.09, antiderivative size = 305, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {626, 635, 212} \begin {gather*} -\frac {5 \left (c d^2-a e^2\right )^6 \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^{7/2}}+\frac {5 \left (c d^2-a e^2\right )^4 \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^3}-\frac {5 \left (c d^2-a e^2\right )^2 \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^2}+\frac {\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}}{12 c d e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(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)^4*(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^3
) - (5*(c*d^2 - a*e^2)^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^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))/(12*c*d*e) - (5*(c*d^2 -
a*e^2)^6*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^(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]

Rubi steps

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

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Mathematica [A]
time = 0.97, size = 295, normalized size = 0.97 \begin {gather*} \frac {\left (c d^2-a e^2\right )^6 ((a e+c d x) (d+e x))^{5/2} \left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} (a e+c d x)^3 \left (15 e^5-\frac {85 c d e^4 (d+e x)}{a e+c d x}+\frac {198 c^2 d^2 e^3 (d+e x)^2}{(a e+c d x)^2}+\frac {198 c^3 d^3 e^2 (d+e x)^3}{(a e+c d x)^3}-\frac {85 c^4 d^4 e (d+e x)^4}{(a e+c d x)^4}+\frac {15 c^5 d^5 (d+e x)^5}{(a e+c d x)^5}\right )}{\left (c d^2-a e^2\right )^6 (d+e x)^2}-\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)^{5/2} (d+e x)^{5/2}}\right )}{1536 c^{7/2} d^{7/2} e^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(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)^6*((a*e + c*d*x)*(d + e*x))^(5/2)*((Sqrt[c]*Sqrt[d]*Sqrt[e]*(a*e + c*d*x)^3*(15*e^5 - (85*c*d
*e^4*(d + e*x))/(a*e + c*d*x) + (198*c^2*d^2*e^3*(d + e*x)^2)/(a*e + c*d*x)^2 + (198*c^3*d^3*e^2*(d + e*x)^3)/
(a*e + c*d*x)^3 - (85*c^4*d^4*e*(d + e*x)^4)/(a*e + c*d*x)^4 + (15*c^5*d^5*(d + e*x)^5)/(a*e + c*d*x)^5))/((c*
d^2 - a*e^2)^6*(d + e*x)^2) - (15*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))))/(1536*c^(7/2)*d^(7/2)*e^(7/2))

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Maple [A]
time = 0.72, size = 340, normalized size = 1.11

method	result	size
default	\(\frac {\left (2 c d e x +e^{2} a +c \,d^{2}\right ) \left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {5}{2}}}{12 c d e}+\frac {5 \left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right ) \left (\frac {\left (2 c d e x +e^{2} a +c \,d^{2}\right ) \left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}{8 c d e}+\frac {3 \left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right ) \left (\frac {\left (2 c d e x +e^{2} a +c \,d^{2}\right ) \sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}{4 c d e}+\frac {\left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right ) \ln \left (\frac {\frac {1}{2} e^{2} a +\frac {1}{2} c \,d^{2}+c d e x}{\sqrt {c d e}}+\sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}\right )}{8 c d e \sqrt {c d e}}\right )}{16 c d e}\right )}{24 c d e}\)	\(340\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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((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 [A]
time = 2.56, size = 1011, normalized size = 3.31 \begin {gather*} \left [\frac {{\left (15 \, {\left (c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, 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} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}\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^{6} d^{10} x e^{2} - 15 \, c^{6} d^{11} e + 10 \, a^{4} c^{2} d^{2} x e^{10} - 15 \, a^{5} c d e^{11} - {\left (8 \, a^{3} c^{3} d^{3} x^{2} - 85 \, a^{4} c^{2} d^{3}\right )} e^{9} - 8 \, {\left (54 \, a^{2} c^{4} d^{4} x^{3} + 7 \, a^{3} c^{3} d^{4} x\right )} e^{8} - 2 \, {\left (320 \, a c^{5} d^{5} x^{4} + 636 \, a^{2} c^{4} d^{5} x^{2} + 99 \, a^{3} c^{3} d^{5}\right )} e^{7} - 4 \, {\left (64 \, c^{6} d^{6} x^{5} + 424 \, a c^{5} d^{6} x^{3} + 297 \, a^{2} c^{4} d^{6} x\right )} e^{6} - 2 \, {\left (320 \, c^{6} d^{7} x^{4} + 636 \, a c^{5} d^{7} x^{2} + 99 \, a^{2} c^{4} d^{7}\right )} e^{5} - 8 \, {\left (54 \, c^{6} d^{8} x^{3} + 7 \, a c^{5} d^{8} x\right )} e^{4} - {\left (8 \, c^{6} d^{9} x^{2} - 85 \, a c^{5} d^{9}\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 (-4\right )}}{6144 \, c^{4} d^{4}}, \frac {{\left (15 \, {\left (c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, 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} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}\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^{6} d^{10} x e^{2} - 15 \, c^{6} d^{11} e + 10 \, a^{4} c^{2} d^{2} x e^{10} - 15 \, a^{5} c d e^{11} - {\left (8 \, a^{3} c^{3} d^{3} x^{2} - 85 \, a^{4} c^{2} d^{3}\right )} e^{9} - 8 \, {\left (54 \, a^{2} c^{4} d^{4} x^{3} + 7 \, a^{3} c^{3} d^{4} x\right )} e^{8} - 2 \, {\left (320 \, a c^{5} d^{5} x^{4} + 636 \, a^{2} c^{4} d^{5} x^{2} + 99 \, a^{3} c^{3} d^{5}\right )} e^{7} - 4 \, {\left (64 \, c^{6} d^{6} x^{5} + 424 \, a c^{5} d^{6} x^{3} + 297 \, a^{2} c^{4} d^{6} x\right )} e^{6} - 2 \, {\left (320 \, c^{6} d^{7} x^{4} + 636 \, a c^{5} d^{7} x^{2} + 99 \, a^{2} c^{4} d^{7}\right )} e^{5} - 8 \, {\left (54 \, c^{6} d^{8} x^{3} + 7 \, a c^{5} d^{8} x\right )} e^{4} - {\left (8 \, c^{6} d^{9} x^{2} - 85 \, a c^{5} d^{9}\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 (-4\right )}}{3072 \, c^{4} d^{4}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/6144*(15*(c^6*d^12 - 6*a*c^5*d^10*e^2 + 15*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 - 6*a^
5*c*d^2*e^10 + a^6*e^12)*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^6*d^10*x*e^2 - 15*c^6*d^11*e + 10*a^4*c^2*d^2*x*e^10 - 15*a^5*c*d*e^11 - (8*a^3*c^3*d^3*x^2
- 85*a^4*c^2*d^3)*e^9 - 8*(54*a^2*c^4*d^4*x^3 + 7*a^3*c^3*d^4*x)*e^8 - 2*(320*a*c^5*d^5*x^4 + 636*a^2*c^4*d^5*
x^2 + 99*a^3*c^3*d^5)*e^7 - 4*(64*c^6*d^6*x^5 + 424*a*c^5*d^6*x^3 + 297*a^2*c^4*d^6*x)*e^6 - 2*(320*c^6*d^7*x^
4 + 636*a*c^5*d^7*x^2 + 99*a^2*c^4*d^7)*e^5 - 8*(54*c^6*d^8*x^3 + 7*a*c^5*d^8*x)*e^4 - (8*c^6*d^9*x^2 - 85*a*c
^5*d^9)*e^3)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e))*e^(-4)/(c^4*d^4), 1/3072*(15*(c^6*d^12 - 6*a*c^5*d^1
0*e^2 + 15*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 - 6*a^5*c*d^2*e^10 + a^6*e^12)*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^6*d^10*x*e^2 - 15*c^6*d^11*e + 10*a^4*c^2*d^2*x*e^10
- 15*a^5*c*d*e^11 - (8*a^3*c^3*d^3*x^2 - 85*a^4*c^2*d^3)*e^9 - 8*(54*a^2*c^4*d^4*x^3 + 7*a^3*c^3*d^4*x)*e^8 -
2*(320*a*c^5*d^5*x^4 + 636*a^2*c^4*d^5*x^2 + 99*a^3*c^3*d^5)*e^7 - 4*(64*c^6*d^6*x^5 + 424*a*c^5*d^6*x^3 + 29
7*a^2*c^4*d^6*x)*e^6 - 2*(320*c^6*d^7*x^4 + 636*a*c^5*d^7*x^2 + 99*a^2*c^4*d^7)*e^5 - 8*(54*c^6*d^8*x^3 + 7*a*
c^5*d^8*x)*e^4 - (8*c^6*d^9*x^2 - 85*a*c^5*d^9)*e^3)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e))*e^(-4)/(c^4*
d^4)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.13, size = 493, normalized size = 1.62 \begin {gather*} \frac {1}{1536} \, \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 \, c^{2} d^{2} x e^{2} + \frac {5 \, {\left (c^{7} d^{8} e^{6} + a c^{6} d^{6} e^{8}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x + \frac {{\left (27 \, c^{7} d^{9} e^{5} + 106 \, a c^{6} d^{7} e^{7} + 27 \, a^{2} c^{5} d^{5} e^{9}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x + \frac {{\left (c^{7} d^{10} e^{4} + 159 \, a c^{6} d^{8} e^{6} + 159 \, a^{2} c^{5} d^{6} e^{8} + a^{3} c^{4} d^{4} e^{10}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x - \frac {{\left (5 \, c^{7} d^{11} e^{3} - 28 \, a c^{6} d^{9} e^{5} - 594 \, a^{2} c^{5} d^{7} e^{7} - 28 \, a^{3} c^{4} d^{5} e^{9} + 5 \, a^{4} c^{3} d^{3} e^{11}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x + \frac {{\left (15 \, c^{7} d^{12} e^{2} - 85 \, a c^{6} d^{10} e^{4} + 198 \, a^{2} c^{5} d^{8} e^{6} + 198 \, a^{3} c^{4} d^{6} e^{8} - 85 \, a^{4} c^{3} d^{4} e^{10} + 15 \, a^{5} c^{2} d^{2} e^{12}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} + \frac {5 \, {\left (c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, 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} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}\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 )}{1024 \, \sqrt {c d} c^{3} d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1536*sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*(2*(4*(2*(8*(2*c^2*d^2*x*e^2 + 5*(c^7*d^8*e^6 + a*c^6*d^6*e
^8)*e^(-5)/(c^5*d^5))*x + (27*c^7*d^9*e^5 + 106*a*c^6*d^7*e^7 + 27*a^2*c^5*d^5*e^9)*e^(-5)/(c^5*d^5))*x + (c^7
*d^10*e^4 + 159*a*c^6*d^8*e^6 + 159*a^2*c^5*d^6*e^8 + a^3*c^4*d^4*e^10)*e^(-5)/(c^5*d^5))*x - (5*c^7*d^11*e^3
- 28*a*c^6*d^9*e^5 - 594*a^2*c^5*d^7*e^7 - 28*a^3*c^4*d^5*e^9 + 5*a^4*c^3*d^3*e^11)*e^(-5)/(c^5*d^5))*x + (15*
c^7*d^12*e^2 - 85*a*c^6*d^10*e^4 + 198*a^2*c^5*d^8*e^6 + 198*a^3*c^4*d^6*e^8 - 85*a^4*c^3*d^4*e^10 + 15*a^5*c^
2*d^2*e^12)*e^(-5)/(c^5*d^5)) + 5/1024*(c^6*d^12 - 6*a*c^5*d^10*e^2 + 15*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 - 6*a^5*c*d^2*e^10 + a^6*e^12)*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^3*d^3)

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Mupad [B]
time = 0.79, size = 319, normalized size = 1.05 \begin {gather*} \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 )}^{5/2}}{6\,c\,d\,e}-\frac {\left (\frac {5\,{\left (c\,d^2+a\,e^2\right )}^2}{4}-5\,a\,c\,d^2\,e^2\right )\,\left (\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\,d\,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\,d\,e}\right )}{6\,c\,d\,e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(((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)^(5/2))/(6*c*d*e) - (((5*(a*e^2 + c*
d^2)^2)/4 - 5*a*c*d^2*e^2)*((((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*d*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*d*e)))/(6*c*d*e)

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