3.20.17 \(\int \frac {\sqrt {a d e+(c d^2+a e^2) x+c d e x^2}}{(d+e x)^6} \, dx\) [1917]

Optimal. Leaf size=231 \[ \frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{9 \left (c d^2-a e^2\right ) (d+e x)^6}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{21 \left (c d^2-a e^2\right )^2 (d+e x)^5}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{105 \left (c d^2-a e^2\right )^3 (d+e x)^4}+\frac {32 c^3 d^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{315 \left (c d^2-a e^2\right )^4 (d+e x)^3} \]

[Out]

2/9*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(-a*e^2+c*d^2)/(e*x+d)^6+4/21*c*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2
)^(3/2)/(-a*e^2+c*d^2)^2/(e*x+d)^5+16/105*c^2*d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(-a*e^2+c*d^2)^3/(e*
x+d)^4+32/315*c^3*d^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(-a*e^2+c*d^2)^4/(e*x+d)^3

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Rubi [A]
time = 0.08, antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {672, 664} \begin {gather*} \frac {32 c^3 d^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{315 (d+e x)^3 \left (c d^2-a e^2\right )^4}+\frac {16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{105 (d+e x)^4 \left (c d^2-a e^2\right )^3}+\frac {4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{21 (d+e x)^5 \left (c d^2-a e^2\right )^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{9 (d+e x)^6 \left (c d^2-a e^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(d + e*x)^6,x]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(9*(c*d^2 - a*e^2)*(d + e*x)^6) + (4*c*d*(a*d*e + (c*d^2 + a
*e^2)*x + c*d*e*x^2)^(3/2))/(21*(c*d^2 - a*e^2)^2*(d + e*x)^5) + (16*c^2*d^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*
e*x^2)^(3/2))/(105*(c*d^2 - a*e^2)^3*(d + e*x)^4) + (32*c^3*d^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))
/(315*(c*d^2 - a*e^2)^4*(d + e*x)^3)

Rule 664

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

Rule 672

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

Rubi steps

\begin {align*} \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^6} \, dx &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{9 \left (c d^2-a e^2\right ) (d+e x)^6}+\frac {(2 c d) \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^5} \, dx}{3 \left (c d^2-a e^2\right )}\\ &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{9 \left (c d^2-a e^2\right ) (d+e x)^6}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{21 \left (c d^2-a e^2\right )^2 (d+e x)^5}+\frac {\left (8 c^2 d^2\right ) \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^4} \, dx}{21 \left (c d^2-a e^2\right )^2}\\ &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{9 \left (c d^2-a e^2\right ) (d+e x)^6}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{21 \left (c d^2-a e^2\right )^2 (d+e x)^5}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{105 \left (c d^2-a e^2\right )^3 (d+e x)^4}+\frac {\left (16 c^3 d^3\right ) \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^3} \, dx}{105 \left (c d^2-a e^2\right )^3}\\ &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{9 \left (c d^2-a e^2\right ) (d+e x)^6}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{21 \left (c d^2-a e^2\right )^2 (d+e x)^5}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{105 \left (c d^2-a e^2\right )^3 (d+e x)^4}+\frac {32 c^3 d^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{315 \left (c d^2-a e^2\right )^4 (d+e x)^3}\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 138, normalized size = 0.60 \begin {gather*} \frac {2 ((a e+c d x) (d+e x))^{3/2} \left (-35 a^3 e^6+15 a^2 c d e^4 (9 d+2 e x)-3 a c^2 d^2 e^2 \left (63 d^2+36 d e x+8 e^2 x^2\right )+c^3 d^3 \left (105 d^3+126 d^2 e x+72 d e^2 x^2+16 e^3 x^3\right )\right )}{315 \left (c d^2-a e^2\right )^4 (d+e x)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(d + e*x)^6,x]

[Out]

(2*((a*e + c*d*x)*(d + e*x))^(3/2)*(-35*a^3*e^6 + 15*a^2*c*d*e^4*(9*d + 2*e*x) - 3*a*c^2*d^2*e^2*(63*d^2 + 36*
d*e*x + 8*e^2*x^2) + c^3*d^3*(105*d^3 + 126*d^2*e*x + 72*d*e^2*x^2 + 16*e^3*x^3)))/(315*(c*d^2 - a*e^2)^4*(d +
e*x)^6)

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Maple [A]
time = 0.70, size = 293, normalized size = 1.27

method	result	size
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (-16 c^{3} d^{3} e^{3} x^{3}+24 a \,c^{2} d^{2} e^{4} x^{2}-72 c^{3} d^{4} e^{2} x^{2}-30 a^{2} c d \,e^{5} x +108 a \,c^{2} d^{3} e^{3} x -126 c^{3} d^{5} e x +35 e^{6} a^{3}-135 e^{4} d^{2} a^{2} c +189 d^{4} e^{2} c^{2} a -105 d^{6} c^{3}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{315 \left (e x +d \right )^{5} \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right )}\)	\(217\)
trager	\(-\frac {2 \left (-16 c^{4} d^{4} e^{3} x^{4}+8 a \,c^{3} d^{3} e^{4} x^{3}-72 c^{4} d^{5} e^{2} x^{3}-6 a^{2} c^{2} d^{2} e^{5} x^{2}+36 a \,c^{3} d^{4} e^{3} x^{2}-126 c^{4} d^{6} e \,x^{2}+5 d \,e^{6} c \,a^{3} x -27 d^{3} e^{4} a^{2} c^{2} x +63 d^{5} e^{2} c^{3} a x -105 d^{7} c^{4} x +35 e^{7} a^{4}-135 a^{3} c \,d^{2} e^{5}+189 a^{2} c^{2} d^{4} e^{3}-105 a \,c^{3} d^{6} e \right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{315 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \left (e x +d \right )^{5}}\)	\(271\)
default	\(\frac {-\frac {2 \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{9 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{6}}-\frac {2 c d e \left (-\frac {2 \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{7 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{5}}-\frac {4 c d e \left (-\frac {2 \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{5 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{4}}+\frac {4 c d e \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{15 \left (e^{2} a -c \,d^{2}\right )^{2} \left (x +\frac {d}{e}\right )^{3}}\right )}{7 \left (e^{2} a -c \,d^{2}\right )}\right )}{3 \left (e^{2} a -c \,d^{2}\right )}}{e^{6}}\)	\(293\)

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)^(1/2)/(e*x+d)^6,x,method=_RETURNVERBOSE)

[Out]

1/e^6*(-2/9/(a*e^2-c*d^2)/(x+d/e)^6*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(3/2)-2/3*c*d*e/(a*e^2-c*d^2)*(-2/
7/(a*e^2-c*d^2)/(x+d/e)^5*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(3/2)-4/7*c*d*e/(a*e^2-c*d^2)*(-2/5/(a*e^2-c
*d^2)/(x+d/e)^4*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(3/2)+4/15*c*d*e/(a*e^2-c*d^2)^2/(x+d/e)^3*(c*d*e*(x+d
/e)^2+(a*e^2-c*d^2)*(x+d/e))^(3/2))))

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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)^(1/2)/(e*x+d)^6,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. 580 vs. \(2 (219) = 438\).
time = 30.59, size = 580, normalized size = 2.51 \begin {gather*} \frac {2 \, {\left (105 \, c^{4} d^{7} x - 5 \, a^{3} c d x e^{6} - 35 \, a^{4} e^{7} + 3 \, {\left (2 \, a^{2} c^{2} d^{2} x^{2} + 45 \, a^{3} c d^{2}\right )} e^{5} - {\left (8 \, a c^{3} d^{3} x^{3} - 27 \, a^{2} c^{2} d^{3} x\right )} e^{4} + {\left (16 \, c^{4} d^{4} x^{4} - 36 \, a c^{3} d^{4} x^{2} - 189 \, a^{2} c^{2} d^{4}\right )} e^{3} + 9 \, {\left (8 \, c^{4} d^{5} x^{3} - 7 \, a c^{3} d^{5} x\right )} e^{2} + 21 \, {\left (6 \, c^{4} d^{6} x^{2} + 5 \, a c^{3} d^{6}\right )} e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}}{315 \, {\left (5 \, c^{4} d^{12} x e + c^{4} d^{13} + a^{4} x^{5} e^{13} + 5 \, a^{4} d x^{4} e^{12} - 2 \, {\left (2 \, a^{3} c d^{2} x^{5} - 5 \, a^{4} d^{2} x^{3}\right )} e^{11} - 10 \, {\left (2 \, a^{3} c d^{3} x^{4} - a^{4} d^{3} x^{2}\right )} e^{10} + {\left (6 \, a^{2} c^{2} d^{4} x^{5} - 40 \, a^{3} c d^{4} x^{3} + 5 \, a^{4} d^{4} x\right )} e^{9} + {\left (30 \, a^{2} c^{2} d^{5} x^{4} - 40 \, a^{3} c d^{5} x^{2} + a^{4} d^{5}\right )} e^{8} - 4 \, {\left (a c^{3} d^{6} x^{5} - 15 \, a^{2} c^{2} d^{6} x^{3} + 5 \, a^{3} c d^{6} x\right )} e^{7} - 4 \, {\left (5 \, a c^{3} d^{7} x^{4} - 15 \, a^{2} c^{2} d^{7} x^{2} + a^{3} c d^{7}\right )} e^{6} + {\left (c^{4} d^{8} x^{5} - 40 \, a c^{3} d^{8} x^{3} + 30 \, a^{2} c^{2} d^{8} x\right )} e^{5} + {\left (5 \, c^{4} d^{9} x^{4} - 40 \, a c^{3} d^{9} x^{2} + 6 \, a^{2} c^{2} d^{9}\right )} e^{4} + 10 \, {\left (c^{4} d^{10} x^{3} - 2 \, a c^{3} d^{10} x\right )} e^{3} + 2 \, {\left (5 \, c^{4} d^{11} x^{2} - 2 \, a c^{3} d^{11}\right )} e^{2}\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)^(1/2)/(e*x+d)^6,x, algorithm="fricas")

[Out]

2/315*(105*c^4*d^7*x - 5*a^3*c*d*x*e^6 - 35*a^4*e^7 + 3*(2*a^2*c^2*d^2*x^2 + 45*a^3*c*d^2)*e^5 - (8*a*c^3*d^3*
x^3 - 27*a^2*c^2*d^3*x)*e^4 + (16*c^4*d^4*x^4 - 36*a*c^3*d^4*x^2 - 189*a^2*c^2*d^4)*e^3 + 9*(8*c^4*d^5*x^3 - 7
*a*c^3*d^5*x)*e^2 + 21*(6*c^4*d^6*x^2 + 5*a*c^3*d^6)*e)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)/(5*c^4*d^1
2*x*e + c^4*d^13 + a^4*x^5*e^13 + 5*a^4*d*x^4*e^12 - 2*(2*a^3*c*d^2*x^5 - 5*a^4*d^2*x^3)*e^11 - 10*(2*a^3*c*d^
3*x^4 - a^4*d^3*x^2)*e^10 + (6*a^2*c^2*d^4*x^5 - 40*a^3*c*d^4*x^3 + 5*a^4*d^4*x)*e^9 + (30*a^2*c^2*d^5*x^4 - 4
0*a^3*c*d^5*x^2 + a^4*d^5)*e^8 - 4*(a*c^3*d^6*x^5 - 15*a^2*c^2*d^6*x^3 + 5*a^3*c*d^6*x)*e^7 - 4*(5*a*c^3*d^7*x
^4 - 15*a^2*c^2*d^7*x^2 + a^3*c*d^7)*e^6 + (c^4*d^8*x^5 - 40*a*c^3*d^8*x^3 + 30*a^2*c^2*d^8*x)*e^5 + (5*c^4*d^
9*x^4 - 40*a*c^3*d^9*x^2 + 6*a^2*c^2*d^9)*e^4 + 10*(c^4*d^10*x^3 - 2*a*c^3*d^10*x)*e^3 + 2*(5*c^4*d^11*x^2 - 2
*a*c^3*d^11)*e^2)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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)**(1/2)/(e*x+d)**6,x)

[Out]

Timed out

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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)^(1/2)/(e*x+d)^6,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Unable to divide, perhaps due to rounding error%%%{%%%{1,[0,0,5]%%%},[10]%%%}+%%%{%%{[%%%{-10,[0,1,4]%%%},0
]:[1,0,%%%{

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Mupad [B]
time = 2.58, size = 1192, normalized size = 5.16 \begin {gather*} \frac {\left (\frac {4\,c^2\,d^3}{9\,\left (a\,e^2-c\,d^2\right )\,\left (7\,a\,e^3-7\,c\,d^2\,e\right )}-\frac {4\,a\,c\,d\,e^2}{9\,\left (a\,e^2-c\,d^2\right )\,\left (7\,a\,e^3-7\,c\,d^2\,e\right )}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^4}-\frac {\left (\frac {2\,a\,e^2}{9\,a\,e^3-9\,c\,d^2\,e}-\frac {2\,c\,d^2}{9\,a\,e^3-9\,c\,d^2\,e}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^5}+\frac {\left (\frac {4\,c^3\,d^4+4\,a\,c^2\,d^2\,e^2}{63\,{\left (a\,e^2-c\,d^2\right )}^2\,\left (5\,a\,e^3-5\,c\,d^2\,e\right )}-\frac {8\,c^3\,d^4}{63\,{\left (a\,e^2-c\,d^2\right )}^2\,\left (5\,a\,e^3-5\,c\,d^2\,e\right )}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^3}+\frac {\left (\frac {8\,c^4\,d^5+8\,a\,c^3\,d^3\,e^2}{315\,{\left (a\,e^2-c\,d^2\right )}^3\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}-\frac {16\,c^4\,d^5}{315\,{\left (a\,e^2-c\,d^2\right )}^3\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^2}+\frac {\left (\frac {16\,c^5\,d^6+16\,a\,c^4\,d^4\,e^2}{945\,e\,{\left (a\,e^2-c\,d^2\right )}^5}-\frac {32\,c^5\,d^6}{945\,e\,{\left (a\,e^2-c\,d^2\right )}^5}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{d+e\,x}+\frac {\left (\frac {2\,c^2\,d^3+2\,a\,c\,d\,e^2}{9\,\left (a\,e^2-c\,d^2\right )\,\left (7\,a\,e^3-7\,c\,d^2\,e\right )}-\frac {4\,c^2\,d^3}{9\,\left (a\,e^2-c\,d^2\right )\,\left (7\,a\,e^3-7\,c\,d^2\,e\right )}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^4}+\frac {\left (\frac {8\,c^3\,d^4}{63\,{\left (a\,e^2-c\,d^2\right )}^2\,\left (5\,a\,e^3-5\,c\,d^2\,e\right )}-\frac {8\,a\,c^2\,d^2\,e^2}{63\,{\left (a\,e^2-c\,d^2\right )}^2\,\left (5\,a\,e^3-5\,c\,d^2\,e\right )}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^3}+\frac {\left (\frac {16\,c^4\,d^5}{315\,{\left (a\,e^2-c\,d^2\right )}^3\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}-\frac {16\,a\,c^3\,d^3\,e^2}{315\,{\left (a\,e^2-c\,d^2\right )}^3\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^2}+\frac {\left (\frac {32\,c^5\,d^6}{945\,e\,{\left (a\,e^2-c\,d^2\right )}^5}-\frac {32\,a\,c^4\,d^4\,e}{945\,{\left (a\,e^2-c\,d^2\right )}^5}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{d+e\,x}-\frac {8\,c^3\,d^3\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{63\,{\left (a\,e^2-c\,d^2\right )}^2\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )\,{\left (d+e\,x\right )}^2}+\frac {16\,c^2\,d^2\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{63\,\left (a\,e^2-c\,d^2\right )\,\left (5\,a\,e^3-5\,c\,d^2\,e\right )\,{\left (d+e\,x\right )}^3}+\frac {16\,c^4\,d^4\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{135\,e\,{\left (a\,e^2-c\,d^2\right )}^4\,\left (d+e\,x\right )} \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)^(1/2)/(d + e*x)^6,x)

[Out]

(((4*c^2*d^3)/(9*(a*e^2 - c*d^2)*(7*a*e^3 - 7*c*d^2*e)) - (4*a*c*d*e^2)/(9*(a*e^2 - c*d^2)*(7*a*e^3 - 7*c*d^2*
e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^4 - (((2*a*e^2)/(9*a*e^3 - 9*c*d^2*e) - (2*c*d^2
)/(9*a*e^3 - 9*c*d^2*e))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^5 + (((4*c^3*d^4 + 4*a*c^2*d
^2*e^2)/(63*(a*e^2 - c*d^2)^2*(5*a*e^3 - 5*c*d^2*e)) - (8*c^3*d^4)/(63*(a*e^2 - c*d^2)^2*(5*a*e^3 - 5*c*d^2*e)
))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^3 + (((8*c^4*d^5 + 8*a*c^3*d^3*e^2)/(315*(a*e^2 -
c*d^2)^3*(3*a*e^3 - 3*c*d^2*e)) - (16*c^4*d^5)/(315*(a*e^2 - c*d^2)^3*(3*a*e^3 - 3*c*d^2*e)))*(x*(a*e^2 + c*d^
2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^2 + (((16*c^5*d^6 + 16*a*c^4*d^4*e^2)/(945*e*(a*e^2 - c*d^2)^5) - (32
*c^5*d^6)/(945*e*(a*e^2 - c*d^2)^5))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x) + (((2*c^2*d^3 +
2*a*c*d*e^2)/(9*(a*e^2 - c*d^2)*(7*a*e^3 - 7*c*d^2*e)) - (4*c^2*d^3)/(9*(a*e^2 - c*d^2)*(7*a*e^3 - 7*c*d^2*e)
))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^4 + (((8*c^3*d^4)/(63*(a*e^2 - c*d^2)^2*(5*a*e^3 -
5*c*d^2*e)) - (8*a*c^2*d^2*e^2)/(63*(a*e^2 - c*d^2)^2*(5*a*e^3 - 5*c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*
d*e*x^2)^(1/2))/(d + e*x)^3 + (((16*c^4*d^5)/(315*(a*e^2 - c*d^2)^3*(3*a*e^3 - 3*c*d^2*e)) - (16*a*c^3*d^3*e^2
)/(315*(a*e^2 - c*d^2)^3*(3*a*e^3 - 3*c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^2 +
(((32*c^5*d^6)/(945*e*(a*e^2 - c*d^2)^5) - (32*a*c^4*d^4*e)/(945*(a*e^2 - c*d^2)^5))*(x*(a*e^2 + c*d^2) + a*d*
e + c*d*e*x^2)^(1/2))/(d + e*x) - (8*c^3*d^3*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(63*(a*e^2 - c*d^2
)^2*(3*a*e^3 - 3*c*d^2*e)*(d + e*x)^2) + (16*c^2*d^2*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(63*(a*e^2
- c*d^2)*(5*a*e^3 - 5*c*d^2*e)*(d + e*x)^3) + (16*c^4*d^4*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(135
*e*(a*e^2 - c*d^2)^4*(d + e*x))

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