3.21.35 \(\int \frac {\sqrt {a d e+(c d^2+a e^2) x+c d e x^2}}{(d+e x)^{9/2}} \, dx\) [2035]

Optimal. Leaf size=264 \[ -\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 e (d+e x)^{7/2}}+\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 e \left (c d^2-a e^2\right ) (d+e x)^{5/2}}+\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 e \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}+\frac {c^3 d^3 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{8 e^{3/2} \left (c d^2-a e^2\right )^{5/2}} \]

[Out]

1/8*c^3*d^3*arctan(e^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e^2+c*d^2)^(1/2)/(e*x+d)^(1/2))/e^(3/2)
/(-a*e^2+c*d^2)^(5/2)-1/3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/e/(e*x+d)^(7/2)+1/12*c*d*(a*d*e+(a*e^2+c*d^2
)*x+c*d*e*x^2)^(1/2)/e/(-a*e^2+c*d^2)/(e*x+d)^(5/2)+1/8*c^2*d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/e/(-a*
e^2+c*d^2)^2/(e*x+d)^(3/2)

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Rubi [A]
time = 0.12, antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {676, 686, 674, 211} \begin {gather*} \frac {c^3 d^3 \text {ArcTan}\left (\frac {\sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d^2-a e^2}}\right )}{8 e^{3/2} \left (c d^2-a e^2\right )^{5/2}}+\frac {c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 e (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}+\frac {c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 e (d+e x)^{5/2} \left (c d^2-a e^2\right )}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 e (d+e x)^{7/2}} \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)^(9/2),x]

[Out]

-1/3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(e*(d + e*x)^(7/2)) + (c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c
*d*e*x^2])/(12*e*(c*d^2 - a*e^2)*(d + e*x)^(5/2)) + (c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(8*e
*(c*d^2 - a*e^2)^2*(d + e*x)^(3/2)) + (c^3*d^3*ArcTan[(Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(S
qrt[c*d^2 - a*e^2]*Sqrt[d + e*x])])/(8*e^(3/2)*(c*d^2 - a*e^2)^(5/2))

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 674

Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[2*e, Subst[Int[1/(
2*c*d - b*e + e^2*x^2), x], x, Sqrt[a + b*x + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^
2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 676

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

Rule 686

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*((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, p}, x] && NeQ[b^2 - 4*a*c, 0] && E
qQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, 0] && NeQ[m + p + 1, 0] && IntegerQ[2*p]

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)^{9/2}} \, dx &=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 e (d+e x)^{7/2}}+\frac {(c d) \int \frac {1}{(d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{6 e}\\ &=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 e (d+e x)^{7/2}}+\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 e \left (c d^2-a e^2\right ) (d+e x)^{5/2}}+\frac {\left (c^2 d^2\right ) \int \frac {1}{(d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{8 e \left (c d^2-a e^2\right )}\\ &=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 e (d+e x)^{7/2}}+\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 e \left (c d^2-a e^2\right ) (d+e x)^{5/2}}+\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 e \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}+\frac {\left (c^3 d^3\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{16 e \left (c d^2-a e^2\right )^2}\\ &=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 e (d+e x)^{7/2}}+\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 e \left (c d^2-a e^2\right ) (d+e x)^{5/2}}+\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 e \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}+\frac {\left (c^3 d^3\right ) \text {Subst}\left (\int \frac {1}{2 c d^2 e-e \left (c d^2+a e^2\right )+e^2 x^2} \, dx,x,\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}}\right )}{8 \left (c d^2-a e^2\right )^2}\\ &=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 e (d+e x)^{7/2}}+\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 e \left (c d^2-a e^2\right ) (d+e x)^{5/2}}+\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 e \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}+\frac {c^3 d^3 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{8 e^{3/2} \left (c d^2-a e^2\right )^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.60, size = 200, normalized size = 0.76 \begin {gather*} \frac {\sqrt {(a e+c d x) (d+e x)} \left (\sqrt {e} \sqrt {c d^2-a e^2} \sqrt {a e+c d x} \left (-8 a^2 e^4+2 a c d e^2 (7 d-e x)+c^2 d^2 \left (-3 d^2+8 d e x+3 e^2 x^2\right )\right )+3 c^3 d^3 (d+e x)^3 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2}}\right )\right )}{24 e^{3/2} \left (c d^2-a e^2\right )^{5/2} \sqrt {a e+c d x} (d+e x)^{7/2}} \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)^(9/2),x]

[Out]

(Sqrt[(a*e + c*d*x)*(d + e*x)]*(Sqrt[e]*Sqrt[c*d^2 - a*e^2]*Sqrt[a*e + c*d*x]*(-8*a^2*e^4 + 2*a*c*d*e^2*(7*d -
e*x) + c^2*d^2*(-3*d^2 + 8*d*e*x + 3*e^2*x^2)) + 3*c^3*d^3*(d + e*x)^3*ArcTan[(Sqrt[e]*Sqrt[a*e + c*d*x])/Sqr
t[c*d^2 - a*e^2]]))/(24*e^(3/2)*(c*d^2 - a*e^2)^(5/2)*Sqrt[a*e + c*d*x]*(d + e*x)^(7/2))

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Maple [A]
time = 0.77, size = 447, normalized size = 1.69

method	result	size
default	\(-\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (3 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{3} d^{3} e^{3} x^{3}+9 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{3} d^{4} e^{2} x^{2}+9 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{3} d^{5} e x +3 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{3} d^{6}-3 c^{2} d^{2} e^{2} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}+2 a c d \,e^{3} x \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}-8 c^{2} d^{3} e x \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}+8 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, \sqrt {c d x +a e}\, a^{2} e^{4}-14 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, \sqrt {c d x +a e}\, a c \,d^{2} e^{2}+3 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, \sqrt {c d x +a e}\, c^{2} d^{4}\right )}{24 \left (e x +d \right )^{\frac {7}{2}} \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, e \left (e^{2} a -c \,d^{2}\right )^{2} \sqrt {c d x +a e}}\)	\(447\)

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)^(9/2),x,method=_RETURNVERBOSE)

[Out]

-1/24*((c*d*x+a*e)*(e*x+d))^(1/2)*(3*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*c^3*d^3*e^3*x^3+9*ar
ctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*c^3*d^4*e^2*x^2+9*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2
)*e)^(1/2))*c^3*d^5*e*x+3*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*c^3*d^6-3*c^2*d^2*e^2*x^2*(c*d*
x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+2*a*c*d*e^3*x*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)-8*c^2*d^3*e*x*(c*
d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+8*((a*e^2-c*d^2)*e)^(1/2)*(c*d*x+a*e)^(1/2)*a^2*e^4-14*((a*e^2-c*d^2)*e
)^(1/2)*(c*d*x+a*e)^(1/2)*a*c*d^2*e^2+3*((a*e^2-c*d^2)*e)^(1/2)*(c*d*x+a*e)^(1/2)*c^2*d^4)/(e*x+d)^(7/2)/((a*e
^2-c*d^2)*e)^(1/2)/e/(a*e^2-c*d^2)^2/(c*d*x+a*e)^(1/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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)^(9/2),x, algorithm="maxima")

[Out]

integrate(sqrt(c*d*x^2*e + a*d*e + (c*d^2 + a*e^2)*x)/(x*e + d)^(9/2), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 540 vs. \(2 (231) = 462\).
time = 3.28, size = 1099, normalized size = 4.16 \begin {gather*} \left [-\frac {3 \, {\left (c^{3} d^{3} x^{4} e^{4} + 4 \, c^{3} d^{4} x^{3} e^{3} + 6 \, c^{3} d^{5} x^{2} e^{2} + 4 \, c^{3} d^{6} x e + c^{3} d^{7}\right )} \sqrt {-c d^{2} e + a e^{3}} \log \left (\frac {c d^{3} - 2 \, a x e^{3} - {\left (c d x^{2} + 2 \, a d\right )} e^{2} + 2 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {-c d^{2} e + a e^{3}} \sqrt {x e + d}}{x^{2} e^{2} + 2 \, d x e + d^{2}}\right ) - 2 \, {\left (8 \, c^{3} d^{5} x e^{2} - 3 \, c^{3} d^{6} e - 10 \, a c^{2} d^{3} x e^{4} + 2 \, a^{2} c d x e^{6} + 8 \, a^{3} e^{7} - {\left (3 \, a c^{2} d^{2} x^{2} + 22 \, a^{2} c d^{2}\right )} e^{5} + {\left (3 \, c^{3} d^{4} x^{2} + 17 \, a c^{2} d^{4}\right )} e^{3}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{48 \, {\left (4 \, c^{3} d^{9} x e^{3} + c^{3} d^{10} e^{2} - a^{3} x^{4} e^{12} - 4 \, a^{3} d x^{3} e^{11} + 3 \, {\left (a^{2} c d^{2} x^{4} - 2 \, a^{3} d^{2} x^{2}\right )} e^{10} + 4 \, {\left (3 \, a^{2} c d^{3} x^{3} - a^{3} d^{3} x\right )} e^{9} - {\left (3 \, a c^{2} d^{4} x^{4} - 18 \, a^{2} c d^{4} x^{2} + a^{3} d^{4}\right )} e^{8} - 12 \, {\left (a c^{2} d^{5} x^{3} - a^{2} c d^{5} x\right )} e^{7} + {\left (c^{3} d^{6} x^{4} - 18 \, a c^{2} d^{6} x^{2} + 3 \, a^{2} c d^{6}\right )} e^{6} + 4 \, {\left (c^{3} d^{7} x^{3} - 3 \, a c^{2} d^{7} x\right )} e^{5} + 3 \, {\left (2 \, c^{3} d^{8} x^{2} - a c^{2} d^{8}\right )} e^{4}\right )}}, -\frac {3 \, {\left (c^{3} d^{3} x^{4} e^{4} + 4 \, c^{3} d^{4} x^{3} e^{3} + 6 \, c^{3} d^{5} x^{2} e^{2} + 4 \, c^{3} d^{6} x e + c^{3} d^{7}\right )} \sqrt {c d^{2} e - a e^{3}} \arctan \left (\frac {\sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {c d^{2} e - a e^{3}} \sqrt {x e + d}}{c d^{2} x e + a x e^{3} + {\left (c d x^{2} + a d\right )} e^{2}}\right ) - {\left (8 \, c^{3} d^{5} x e^{2} - 3 \, c^{3} d^{6} e - 10 \, a c^{2} d^{3} x e^{4} + 2 \, a^{2} c d x e^{6} + 8 \, a^{3} e^{7} - {\left (3 \, a c^{2} d^{2} x^{2} + 22 \, a^{2} c d^{2}\right )} e^{5} + {\left (3 \, c^{3} d^{4} x^{2} + 17 \, a c^{2} d^{4}\right )} e^{3}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{24 \, {\left (4 \, c^{3} d^{9} x e^{3} + c^{3} d^{10} e^{2} - a^{3} x^{4} e^{12} - 4 \, a^{3} d x^{3} e^{11} + 3 \, {\left (a^{2} c d^{2} x^{4} - 2 \, a^{3} d^{2} x^{2}\right )} e^{10} + 4 \, {\left (3 \, a^{2} c d^{3} x^{3} - a^{3} d^{3} x\right )} e^{9} - {\left (3 \, a c^{2} d^{4} x^{4} - 18 \, a^{2} c d^{4} x^{2} + a^{3} d^{4}\right )} e^{8} - 12 \, {\left (a c^{2} d^{5} x^{3} - a^{2} c d^{5} x\right )} e^{7} + {\left (c^{3} d^{6} x^{4} - 18 \, a c^{2} d^{6} x^{2} + 3 \, a^{2} c d^{6}\right )} e^{6} + 4 \, {\left (c^{3} d^{7} x^{3} - 3 \, a c^{2} d^{7} x\right )} e^{5} + 3 \, {\left (2 \, c^{3} d^{8} x^{2} - a c^{2} d^{8}\right )} e^{4}\right )}}\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)^(9/2),x, algorithm="fricas")

[Out]

[-1/48*(3*(c^3*d^3*x^4*e^4 + 4*c^3*d^4*x^3*e^3 + 6*c^3*d^5*x^2*e^2 + 4*c^3*d^6*x*e + c^3*d^7)*sqrt(-c*d^2*e +
a*e^3)*log((c*d^3 - 2*a*x*e^3 - (c*d*x^2 + 2*a*d)*e^2 + 2*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(-c*
d^2*e + a*e^3)*sqrt(x*e + d))/(x^2*e^2 + 2*d*x*e + d^2)) - 2*(8*c^3*d^5*x*e^2 - 3*c^3*d^6*e - 10*a*c^2*d^3*x*e
^4 + 2*a^2*c*d*x*e^6 + 8*a^3*e^7 - (3*a*c^2*d^2*x^2 + 22*a^2*c*d^2)*e^5 + (3*c^3*d^4*x^2 + 17*a*c^2*d^4)*e^3)*
sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(x*e + d))/(4*c^3*d^9*x*e^3 + c^3*d^10*e^2 - a^3*x^4*e^12 - 4*
a^3*d*x^3*e^11 + 3*(a^2*c*d^2*x^4 - 2*a^3*d^2*x^2)*e^10 + 4*(3*a^2*c*d^3*x^3 - a^3*d^3*x)*e^9 - (3*a*c^2*d^4*x
^4 - 18*a^2*c*d^4*x^2 + a^3*d^4)*e^8 - 12*(a*c^2*d^5*x^3 - a^2*c*d^5*x)*e^7 + (c^3*d^6*x^4 - 18*a*c^2*d^6*x^2
+ 3*a^2*c*d^6)*e^6 + 4*(c^3*d^7*x^3 - 3*a*c^2*d^7*x)*e^5 + 3*(2*c^3*d^8*x^2 - a*c^2*d^8)*e^4), -1/24*(3*(c^3*d
^3*x^4*e^4 + 4*c^3*d^4*x^3*e^3 + 6*c^3*d^5*x^2*e^2 + 4*c^3*d^6*x*e + c^3*d^7)*sqrt(c*d^2*e - a*e^3)*arctan(sqr
t(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(c*d^2*e - a*e^3)*sqrt(x*e + d)/(c*d^2*x*e + a*x*e^3 + (c*d*x^2 +
a*d)*e^2)) - (8*c^3*d^5*x*e^2 - 3*c^3*d^6*e - 10*a*c^2*d^3*x*e^4 + 2*a^2*c*d*x*e^6 + 8*a^3*e^7 - (3*a*c^2*d^2
*x^2 + 22*a^2*c*d^2)*e^5 + (3*c^3*d^4*x^2 + 17*a*c^2*d^4)*e^3)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqr
t(x*e + d))/(4*c^3*d^9*x*e^3 + c^3*d^10*e^2 - a^3*x^4*e^12 - 4*a^3*d*x^3*e^11 + 3*(a^2*c*d^2*x^4 - 2*a^3*d^2*x
^2)*e^10 + 4*(3*a^2*c*d^3*x^3 - a^3*d^3*x)*e^9 - (3*a*c^2*d^4*x^4 - 18*a^2*c*d^4*x^2 + a^3*d^4)*e^8 - 12*(a*c^
2*d^5*x^3 - a^2*c*d^5*x)*e^7 + (c^3*d^6*x^4 - 18*a*c^2*d^6*x^2 + 3*a^2*c*d^6)*e^6 + 4*(c^3*d^7*x^3 - 3*a*c^2*d
^7*x)*e^5 + 3*(2*c^3*d^8*x^2 - a*c^2*d^8)*e^4)]

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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)**(9/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3877 deep

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Giac [A]
time = 1.66, size = 369, normalized size = 1.40 \begin {gather*} \frac {{\left (\frac {3 \, c^{4} d^{4} \arctan \left (\frac {\sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}}}{\sqrt {c d^{2} e - a e^{3}}}\right ) e}{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {c d^{2} e - a e^{3}}} - \frac {{\left (3 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} c^{6} d^{8} e^{3} - 6 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a c^{5} d^{6} e^{5} - 8 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{5} d^{6} e^{2} + 3 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a^{2} c^{4} d^{4} e^{7} + 8 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a c^{4} d^{4} e^{4} - 3 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} c^{4} d^{4} e\right )} e^{\left (-3\right )}}{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} {\left (x e + d\right )}^{3} c^{3} d^{3}}\right )} e^{\left (-2\right )}}{24 \, c d} \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)^(9/2),x, algorithm="giac")

[Out]

1/24*(3*c^4*d^4*arctan(sqrt((x*e + d)*c*d*e - c*d^2*e + a*e^3)/sqrt(c*d^2*e - a*e^3))*e/((c^2*d^4 - 2*a*c*d^2*
e^2 + a^2*e^4)*sqrt(c*d^2*e - a*e^3)) - (3*sqrt((x*e + d)*c*d*e - c*d^2*e + a*e^3)*c^6*d^8*e^3 - 6*sqrt((x*e +
d)*c*d*e - c*d^2*e + a*e^3)*a*c^5*d^6*e^5 - 8*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*c^5*d^6*e^2 + 3*sqrt(
(x*e + d)*c*d*e - c*d^2*e + a*e^3)*a^2*c^4*d^4*e^7 + 8*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a*c^4*d^4*e^4
- 3*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*c^4*d^4*e)*e^(-3)/((c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*(x*e + d
)^3*c^3*d^3))*e^(-2)/(c*d)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^{9/2}} \,d x \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)^(9/2),x)

[Out]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)/(d + e*x)^(9/2), x)

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