3.21.25 \(\int \frac {\sqrt {d+e x}}{(a d e+(c d^2+a e^2) x+c d e x^2)^3} \, dx\) [2025]

Optimal. Leaf size=208 \[ \frac {35 e^2}{12 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2}}-\frac {1}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2 (d+e x)^{3/2}}+\frac {7 e}{4 \left (c d^2-a e^2\right )^2 (a e+c d x) (d+e x)^{3/2}}+\frac {35 c d e^2}{4 \left (c d^2-a e^2\right )^4 \sqrt {d+e x}}-\frac {35 c^{3/2} d^{3/2} e^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{4 \left (c d^2-a e^2\right )^{9/2}} \]

[Out]

35/12*e^2/(-a*e^2+c*d^2)^3/(e*x+d)^(3/2)-1/2/(-a*e^2+c*d^2)/(c*d*x+a*e)^2/(e*x+d)^(3/2)+7/4*e/(-a*e^2+c*d^2)^2
/(c*d*x+a*e)/(e*x+d)^(3/2)-35/4*c^(3/2)*d^(3/2)*e^2*arctanh(c^(1/2)*d^(1/2)*(e*x+d)^(1/2)/(-a*e^2+c*d^2)^(1/2)
)/(-a*e^2+c*d^2)^(9/2)+35/4*c*d*e^2/(-a*e^2+c*d^2)^4/(e*x+d)^(1/2)

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Rubi [A]
time = 0.09, antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {640, 44, 53, 65, 214} \begin {gather*} -\frac {35 c^{3/2} d^{3/2} e^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{4 \left (c d^2-a e^2\right )^{9/2}}+\frac {35 c d e^2}{4 \sqrt {d+e x} \left (c d^2-a e^2\right )^4}+\frac {35 e^2}{12 (d+e x)^{3/2} \left (c d^2-a e^2\right )^3}+\frac {7 e}{4 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2 (a e+c d x)}-\frac {1}{2 (d+e x)^{3/2} \left (c d^2-a e^2\right ) (a e+c d x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[d + e*x]/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]

[Out]

(35*e^2)/(12*(c*d^2 - a*e^2)^3*(d + e*x)^(3/2)) - 1/(2*(c*d^2 - a*e^2)*(a*e + c*d*x)^2*(d + e*x)^(3/2)) + (7*e
)/(4*(c*d^2 - a*e^2)^2*(a*e + c*d*x)*(d + e*x)^(3/2)) + (35*c*d*e^2)/(4*(c*d^2 - a*e^2)^4*Sqrt[d + e*x]) - (35
*c^(3/2)*d^(3/2)*e^2*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[c*d^2 - a*e^2]])/(4*(c*d^2 - a*e^2)^(9/2))

Rule 44

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

Rule 53

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

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

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

Rule 640

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

Rubi steps

\begin {align*} \int \frac {\sqrt {d+e x}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx &=\int \frac {1}{(a e+c d x)^3 (d+e x)^{5/2}} \, dx\\ &=-\frac {1}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2 (d+e x)^{3/2}}-\frac {(7 e) \int \frac {1}{(a e+c d x)^2 (d+e x)^{5/2}} \, dx}{4 \left (c d^2-a e^2\right )}\\ &=-\frac {1}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2 (d+e x)^{3/2}}+\frac {7 e}{4 \left (c d^2-a e^2\right )^2 (a e+c d x) (d+e x)^{3/2}}+\frac {\left (35 e^2\right ) \int \frac {1}{(a e+c d x) (d+e x)^{5/2}} \, dx}{8 \left (c d^2-a e^2\right )^2}\\ &=\frac {35 e^2}{12 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2}}-\frac {1}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2 (d+e x)^{3/2}}+\frac {7 e}{4 \left (c d^2-a e^2\right )^2 (a e+c d x) (d+e x)^{3/2}}+\frac {\left (35 c d e^2\right ) \int \frac {1}{(a e+c d x) (d+e x)^{3/2}} \, dx}{8 \left (c d^2-a e^2\right )^3}\\ &=\frac {35 e^2}{12 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2}}-\frac {1}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2 (d+e x)^{3/2}}+\frac {7 e}{4 \left (c d^2-a e^2\right )^2 (a e+c d x) (d+e x)^{3/2}}+\frac {35 c d e^2}{4 \left (c d^2-a e^2\right )^4 \sqrt {d+e x}}+\frac {\left (35 c^2 d^2 e^2\right ) \int \frac {1}{(a e+c d x) \sqrt {d+e x}} \, dx}{8 \left (c d^2-a e^2\right )^4}\\ &=\frac {35 e^2}{12 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2}}-\frac {1}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2 (d+e x)^{3/2}}+\frac {7 e}{4 \left (c d^2-a e^2\right )^2 (a e+c d x) (d+e x)^{3/2}}+\frac {35 c d e^2}{4 \left (c d^2-a e^2\right )^4 \sqrt {d+e x}}+\frac {\left (35 c^2 d^2 e\right ) \text {Subst}\left (\int \frac {1}{-\frac {c d^2}{e}+a e+\frac {c d x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{4 \left (c d^2-a e^2\right )^4}\\ &=\frac {35 e^2}{12 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2}}-\frac {1}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2 (d+e x)^{3/2}}+\frac {7 e}{4 \left (c d^2-a e^2\right )^2 (a e+c d x) (d+e x)^{3/2}}+\frac {35 c d e^2}{4 \left (c d^2-a e^2\right )^4 \sqrt {d+e x}}-\frac {35 c^{3/2} d^{3/2} e^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{4 \left (c d^2-a e^2\right )^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 0.61, size = 202, normalized size = 0.97 \begin {gather*} \frac {-8 a^3 e^6+8 a^2 c d e^4 (10 d+7 e x)+a c^2 d^2 e^2 \left (39 d^2+238 d e x+175 e^2 x^2\right )+c^3 d^3 \left (-6 d^3+21 d^2 e x+140 d e^2 x^2+105 e^3 x^3\right )}{12 \left (c d^2-a e^2\right )^4 (a e+c d x)^2 (d+e x)^{3/2}}+\frac {35 c^{3/2} d^{3/2} e^2 \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {-c d^2+a e^2}}\right )}{4 \left (-c d^2+a e^2\right )^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[d + e*x]/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]

[Out]

(-8*a^3*e^6 + 8*a^2*c*d*e^4*(10*d + 7*e*x) + a*c^2*d^2*e^2*(39*d^2 + 238*d*e*x + 175*e^2*x^2) + c^3*d^3*(-6*d^
3 + 21*d^2*e*x + 140*d*e^2*x^2 + 105*e^3*x^3))/(12*(c*d^2 - a*e^2)^4*(a*e + c*d*x)^2*(d + e*x)^(3/2)) + (35*c^
(3/2)*d^(3/2)*e^2*ArcTan[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[-(c*d^2) + a*e^2]])/(4*(-(c*d^2) + a*e^2)^(9/2))

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Maple [A]
time = 0.75, size = 180, normalized size = 0.87

method	result	size
derivativedivides	\(2 e^{2} \left (-\frac {1}{3 \left (e^{2} a -c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {3}{2}}}+\frac {3 c d}{\left (e^{2} a -c \,d^{2}\right )^{4} \sqrt {e x +d}}+\frac {c^{2} d^{2} \left (\frac {\frac {11 c d \left (e x +d \right )^{\frac {3}{2}}}{8}+\left (\frac {13 e^{2} a}{8}-\frac {13 c \,d^{2}}{8}\right ) \sqrt {e x +d}}{\left (c d \left (e x +d \right )+e^{2} a -c \,d^{2}\right )^{2}}+\frac {35 \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{8 \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{\left (e^{2} a -c \,d^{2}\right )^{4}}\right )\)	\(180\)
default	\(2 e^{2} \left (-\frac {1}{3 \left (e^{2} a -c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {3}{2}}}+\frac {3 c d}{\left (e^{2} a -c \,d^{2}\right )^{4} \sqrt {e x +d}}+\frac {c^{2} d^{2} \left (\frac {\frac {11 c d \left (e x +d \right )^{\frac {3}{2}}}{8}+\left (\frac {13 e^{2} a}{8}-\frac {13 c \,d^{2}}{8}\right ) \sqrt {e x +d}}{\left (c d \left (e x +d \right )+e^{2} a -c \,d^{2}\right )^{2}}+\frac {35 \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{8 \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{\left (e^{2} a -c \,d^{2}\right )^{4}}\right )\)	\(180\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x,method=_RETURNVERBOSE)

[Out]

2*e^2*(-1/3/(a*e^2-c*d^2)^3/(e*x+d)^(3/2)+3/(a*e^2-c*d^2)^4*c*d/(e*x+d)^(1/2)+1/(a*e^2-c*d^2)^4*c^2*d^2*((11/8
*c*d*(e*x+d)^(3/2)+(13/8*e^2*a-13/8*c*d^2)*(e*x+d)^(1/2))/(c*d*(e*x+d)+e^2*a-c*d^2)^2+35/8/((a*e^2-c*d^2)*c*d)
^(1/2)*arctan(c*d*(e*x+d)^(1/2)/((a*e^2-c*d^2)*c*d)^(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)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,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. 663 vs. \(2 (175) = 350\).
time = 3.28, size = 1341, normalized size = 6.45 \begin {gather*} \left [\frac {105 \, {\left (c^{3} d^{5} x^{2} e^{2} + a^{2} c d x^{2} e^{6} + 2 \, {\left (a c^{2} d^{2} x^{3} + a^{2} c d^{2} x\right )} e^{5} + {\left (c^{3} d^{3} x^{4} + 4 \, a c^{2} d^{3} x^{2} + a^{2} c d^{3}\right )} e^{4} + 2 \, {\left (c^{3} d^{4} x^{3} + a c^{2} d^{4} x\right )} e^{3}\right )} \sqrt {\frac {c d}{c d^{2} - a e^{2}}} \log \left (\frac {c d x e + 2 \, c d^{2} - 2 \, {\left (c d^{2} - a e^{2}\right )} \sqrt {x e + d} \sqrt {\frac {c d}{c d^{2} - a e^{2}}} - a e^{2}}{c d x + a e}\right ) + 2 \, {\left (21 \, c^{3} d^{5} x e - 6 \, c^{3} d^{6} + 56 \, a^{2} c d x e^{5} - 8 \, a^{3} e^{6} + 5 \, {\left (35 \, a c^{2} d^{2} x^{2} + 16 \, a^{2} c d^{2}\right )} e^{4} + 7 \, {\left (15 \, c^{3} d^{3} x^{3} + 34 \, a c^{2} d^{3} x\right )} e^{3} + {\left (140 \, c^{3} d^{4} x^{2} + 39 \, a c^{2} d^{4}\right )} e^{2}\right )} \sqrt {x e + d}}{24 \, {\left (c^{6} d^{12} x^{2} + a^{6} x^{2} e^{12} + 2 \, {\left (a^{5} c d x^{3} + a^{6} d x\right )} e^{11} + {\left (a^{4} c^{2} d^{2} x^{4} + a^{6} d^{2}\right )} e^{10} - 6 \, {\left (a^{4} c^{2} d^{3} x^{3} + a^{5} c d^{3} x\right )} e^{9} - {\left (4 \, a^{3} c^{3} d^{4} x^{4} + 9 \, a^{4} c^{2} d^{4} x^{2} + 4 \, a^{5} c d^{4}\right )} e^{8} + 4 \, {\left (a^{3} c^{3} d^{5} x^{3} + a^{4} c^{2} d^{5} x\right )} e^{7} + 2 \, {\left (3 \, a^{2} c^{4} d^{6} x^{4} + 8 \, a^{3} c^{3} d^{6} x^{2} + 3 \, a^{4} c^{2} d^{6}\right )} e^{6} + 4 \, {\left (a^{2} c^{4} d^{7} x^{3} + a^{3} c^{3} d^{7} x\right )} e^{5} - {\left (4 \, a c^{5} d^{8} x^{4} + 9 \, a^{2} c^{4} d^{8} x^{2} + 4 \, a^{3} c^{3} d^{8}\right )} e^{4} - 6 \, {\left (a c^{5} d^{9} x^{3} + a^{2} c^{4} d^{9} x\right )} e^{3} + {\left (c^{6} d^{10} x^{4} + a^{2} c^{4} d^{10}\right )} e^{2} + 2 \, {\left (c^{6} d^{11} x^{3} + a c^{5} d^{11} x\right )} e\right )}}, -\frac {105 \, {\left (c^{3} d^{5} x^{2} e^{2} + a^{2} c d x^{2} e^{6} + 2 \, {\left (a c^{2} d^{2} x^{3} + a^{2} c d^{2} x\right )} e^{5} + {\left (c^{3} d^{3} x^{4} + 4 \, a c^{2} d^{3} x^{2} + a^{2} c d^{3}\right )} e^{4} + 2 \, {\left (c^{3} d^{4} x^{3} + a c^{2} d^{4} x\right )} e^{3}\right )} \sqrt {-\frac {c d}{c d^{2} - a e^{2}}} \arctan \left (-\frac {{\left (c d^{2} - a e^{2}\right )} \sqrt {x e + d} \sqrt {-\frac {c d}{c d^{2} - a e^{2}}}}{c d x e + c d^{2}}\right ) - {\left (21 \, c^{3} d^{5} x e - 6 \, c^{3} d^{6} + 56 \, a^{2} c d x e^{5} - 8 \, a^{3} e^{6} + 5 \, {\left (35 \, a c^{2} d^{2} x^{2} + 16 \, a^{2} c d^{2}\right )} e^{4} + 7 \, {\left (15 \, c^{3} d^{3} x^{3} + 34 \, a c^{2} d^{3} x\right )} e^{3} + {\left (140 \, c^{3} d^{4} x^{2} + 39 \, a c^{2} d^{4}\right )} e^{2}\right )} \sqrt {x e + d}}{12 \, {\left (c^{6} d^{12} x^{2} + a^{6} x^{2} e^{12} + 2 \, {\left (a^{5} c d x^{3} + a^{6} d x\right )} e^{11} + {\left (a^{4} c^{2} d^{2} x^{4} + a^{6} d^{2}\right )} e^{10} - 6 \, {\left (a^{4} c^{2} d^{3} x^{3} + a^{5} c d^{3} x\right )} e^{9} - {\left (4 \, a^{3} c^{3} d^{4} x^{4} + 9 \, a^{4} c^{2} d^{4} x^{2} + 4 \, a^{5} c d^{4}\right )} e^{8} + 4 \, {\left (a^{3} c^{3} d^{5} x^{3} + a^{4} c^{2} d^{5} x\right )} e^{7} + 2 \, {\left (3 \, a^{2} c^{4} d^{6} x^{4} + 8 \, a^{3} c^{3} d^{6} x^{2} + 3 \, a^{4} c^{2} d^{6}\right )} e^{6} + 4 \, {\left (a^{2} c^{4} d^{7} x^{3} + a^{3} c^{3} d^{7} x\right )} e^{5} - {\left (4 \, a c^{5} d^{8} x^{4} + 9 \, a^{2} c^{4} d^{8} x^{2} + 4 \, a^{3} c^{3} d^{8}\right )} e^{4} - 6 \, {\left (a c^{5} d^{9} x^{3} + a^{2} c^{4} d^{9} x\right )} e^{3} + {\left (c^{6} d^{10} x^{4} + a^{2} c^{4} d^{10}\right )} e^{2} + 2 \, {\left (c^{6} d^{11} x^{3} + a c^{5} d^{11} x\right )} e\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="fricas")

[Out]

[1/24*(105*(c^3*d^5*x^2*e^2 + a^2*c*d*x^2*e^6 + 2*(a*c^2*d^2*x^3 + a^2*c*d^2*x)*e^5 + (c^3*d^3*x^4 + 4*a*c^2*d
^3*x^2 + a^2*c*d^3)*e^4 + 2*(c^3*d^4*x^3 + a*c^2*d^4*x)*e^3)*sqrt(c*d/(c*d^2 - a*e^2))*log((c*d*x*e + 2*c*d^2
- 2*(c*d^2 - a*e^2)*sqrt(x*e + d)*sqrt(c*d/(c*d^2 - a*e^2)) - a*e^2)/(c*d*x + a*e)) + 2*(21*c^3*d^5*x*e - 6*c^
3*d^6 + 56*a^2*c*d*x*e^5 - 8*a^3*e^6 + 5*(35*a*c^2*d^2*x^2 + 16*a^2*c*d^2)*e^4 + 7*(15*c^3*d^3*x^3 + 34*a*c^2*
d^3*x)*e^3 + (140*c^3*d^4*x^2 + 39*a*c^2*d^4)*e^2)*sqrt(x*e + d))/(c^6*d^12*x^2 + a^6*x^2*e^12 + 2*(a^5*c*d*x^
3 + a^6*d*x)*e^11 + (a^4*c^2*d^2*x^4 + a^6*d^2)*e^10 - 6*(a^4*c^2*d^3*x^3 + a^5*c*d^3*x)*e^9 - (4*a^3*c^3*d^4*
x^4 + 9*a^4*c^2*d^4*x^2 + 4*a^5*c*d^4)*e^8 + 4*(a^3*c^3*d^5*x^3 + a^4*c^2*d^5*x)*e^7 + 2*(3*a^2*c^4*d^6*x^4 +
8*a^3*c^3*d^6*x^2 + 3*a^4*c^2*d^6)*e^6 + 4*(a^2*c^4*d^7*x^3 + a^3*c^3*d^7*x)*e^5 - (4*a*c^5*d^8*x^4 + 9*a^2*c^
4*d^8*x^2 + 4*a^3*c^3*d^8)*e^4 - 6*(a*c^5*d^9*x^3 + a^2*c^4*d^9*x)*e^3 + (c^6*d^10*x^4 + a^2*c^4*d^10)*e^2 + 2
*(c^6*d^11*x^3 + a*c^5*d^11*x)*e), -1/12*(105*(c^3*d^5*x^2*e^2 + a^2*c*d*x^2*e^6 + 2*(a*c^2*d^2*x^3 + a^2*c*d^
2*x)*e^5 + (c^3*d^3*x^4 + 4*a*c^2*d^3*x^2 + a^2*c*d^3)*e^4 + 2*(c^3*d^4*x^3 + a*c^2*d^4*x)*e^3)*sqrt(-c*d/(c*d
^2 - a*e^2))*arctan(-(c*d^2 - a*e^2)*sqrt(x*e + d)*sqrt(-c*d/(c*d^2 - a*e^2))/(c*d*x*e + c*d^2)) - (21*c^3*d^5
*x*e - 6*c^3*d^6 + 56*a^2*c*d*x*e^5 - 8*a^3*e^6 + 5*(35*a*c^2*d^2*x^2 + 16*a^2*c*d^2)*e^4 + 7*(15*c^3*d^3*x^3
+ 34*a*c^2*d^3*x)*e^3 + (140*c^3*d^4*x^2 + 39*a*c^2*d^4)*e^2)*sqrt(x*e + d))/(c^6*d^12*x^2 + a^6*x^2*e^12 + 2*
(a^5*c*d*x^3 + a^6*d*x)*e^11 + (a^4*c^2*d^2*x^4 + a^6*d^2)*e^10 - 6*(a^4*c^2*d^3*x^3 + a^5*c*d^3*x)*e^9 - (4*a
^3*c^3*d^4*x^4 + 9*a^4*c^2*d^4*x^2 + 4*a^5*c*d^4)*e^8 + 4*(a^3*c^3*d^5*x^3 + a^4*c^2*d^5*x)*e^7 + 2*(3*a^2*c^4
*d^6*x^4 + 8*a^3*c^3*d^6*x^2 + 3*a^4*c^2*d^6)*e^6 + 4*(a^2*c^4*d^7*x^3 + a^3*c^3*d^7*x)*e^5 - (4*a*c^5*d^8*x^4
+ 9*a^2*c^4*d^8*x^2 + 4*a^3*c^3*d^8)*e^4 - 6*(a*c^5*d^9*x^3 + a^2*c^4*d^9*x)*e^3 + (c^6*d^10*x^4 + a^2*c^4*d^
10)*e^2 + 2*(c^6*d^11*x^3 + a*c^5*d^11*x)*e)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1826 vs. \(2 (187) = 374\).
time = 41.25, size = 1826, normalized size = 8.78 \begin {gather*} \frac {10 a c^{2} d^{2} e^{4} \sqrt {d + e x}}{8 a^{4} e^{8} \left (a e^{2} - c d^{2}\right )^{2} - 16 a^{3} c d^{2} e^{6} \left (a e^{2} - c d^{2}\right )^{2} + 16 a^{3} c d e^{7} x \left (a e^{2} - c d^{2}\right )^{2} - 48 a^{2} c^{2} d^{3} e^{5} x \left (a e^{2} - c d^{2}\right )^{2} + 8 a^{2} c^{2} d^{2} e^{4} \left (d + e x\right )^{2} \left (a e^{2} - c d^{2}\right )^{2} + 16 a c^{3} d^{6} e^{2} \left (a e^{2} - c d^{2}\right )^{2} + 48 a c^{3} d^{5} e^{3} x \left (a e^{2} - c d^{2}\right )^{2} - 16 a c^{3} d^{4} e^{2} \left (d + e x\right )^{2} \left (a e^{2} - c d^{2}\right )^{2} - 8 c^{4} d^{8} \left (a e^{2} - c d^{2}\right )^{2} - 16 c^{4} d^{7} e x \left (a e^{2} - c d^{2}\right )^{2} + 8 c^{4} d^{6} \left (d + e x\right )^{2} \left (a e^{2} - c d^{2}\right )^{2}} - \frac {10 c^{3} d^{4} e^{2} \sqrt {d + e x}}{8 a^{4} e^{8} \left (a e^{2} - c d^{2}\right )^{2} - 16 a^{3} c d^{2} e^{6} \left (a e^{2} - c d^{2}\right )^{2} + 16 a^{3} c d e^{7} x \left (a e^{2} - c d^{2}\right )^{2} - 48 a^{2} c^{2} d^{3} e^{5} x \left (a e^{2} - c d^{2}\right )^{2} + 8 a^{2} c^{2} d^{2} e^{4} \left (d + e x\right )^{2} \left (a e^{2} - c d^{2}\right )^{2} + 16 a c^{3} d^{6} e^{2} \left (a e^{2} - c d^{2}\right )^{2} + 48 a c^{3} d^{5} e^{3} x \left (a e^{2} - c d^{2}\right )^{2} - 16 a c^{3} d^{4} e^{2} \left (d + e x\right )^{2} \left (a e^{2} - c d^{2}\right )^{2} - 8 c^{4} d^{8} \left (a e^{2} - c d^{2}\right )^{2} - 16 c^{4} d^{7} e x \left (a e^{2} - c d^{2}\right )^{2} + 8 c^{4} d^{6} \left (d + e x\right )^{2} \left (a e^{2} - c d^{2}\right )^{2}} + \frac {6 c^{3} d^{3} e^{2} \left (d + e x\right )^{\frac {3}{2}}}{8 a^{4} e^{8} \left (a e^{2} - c d^{2}\right )^{2} - 16 a^{3} c d^{2} e^{6} \left (a e^{2} - c d^{2}\right )^{2} + 16 a^{3} c d e^{7} x \left (a e^{2} - c d^{2}\right )^{2} - 48 a^{2} c^{2} d^{3} e^{5} x \left (a e^{2} - c d^{2}\right )^{2} + 8 a^{2} c^{2} d^{2} e^{4} \left (d + e x\right )^{2} \left (a e^{2} - c d^{2}\right )^{2} + 16 a c^{3} d^{6} e^{2} \left (a e^{2} - c d^{2}\right )^{2} + 48 a c^{3} d^{5} e^{3} x \left (a e^{2} - c d^{2}\right )^{2} - 16 a c^{3} d^{4} e^{2} \left (d + e x\right )^{2} \left (a e^{2} - c d^{2}\right )^{2} - 8 c^{4} d^{8} \left (a e^{2} - c d^{2}\right )^{2} - 16 c^{4} d^{7} e x \left (a e^{2} - c d^{2}\right )^{2} + 8 c^{4} d^{6} \left (d + e x\right )^{2} \left (a e^{2} - c d^{2}\right )^{2}} - \frac {3 c^{2} d^{2} e^{2} \sqrt {- \frac {1}{c d \left (a e^{2} - c d^{2}\right )^{5}}} \log {\left (- a^{3} e^{6} \sqrt {- \frac {1}{c d \left (a e^{2} - c d^{2}\right )^{5}}} + 3 a^{2} c d^{2} e^{4} \sqrt {- \frac {1}{c d \left (a e^{2} - c d^{2}\right )^{5}}} - 3 a c^{2} d^{4} e^{2} \sqrt {- \frac {1}{c d \left (a e^{2} - c d^{2}\right )^{5}}} + c^{3} d^{6} \sqrt {- \frac {1}{c d \left (a e^{2} - c d^{2}\right )^{5}}} + \sqrt {d + e x} \right )}}{8 \left (a e^{2} - c d^{2}\right )^{2}} + \frac {3 c^{2} d^{2} e^{2} \sqrt {- \frac {1}{c d \left (a e^{2} - c d^{2}\right )^{5}}} \log {\left (a^{3} e^{6} \sqrt {- \frac {1}{c d \left (a e^{2} - c d^{2}\right )^{5}}} - 3 a^{2} c d^{2} e^{4} \sqrt {- \frac {1}{c d \left (a e^{2} - c d^{2}\right )^{5}}} + 3 a c^{2} d^{4} e^{2} \sqrt {- \frac {1}{c d \left (a e^{2} - c d^{2}\right )^{5}}} - c^{3} d^{6} \sqrt {- \frac {1}{c d \left (a e^{2} - c d^{2}\right )^{5}}} + \sqrt {d + e x} \right )}}{8 \left (a e^{2} - c d^{2}\right )^{2}} - \frac {c^{2} d^{2} e^{2} \sqrt {- \frac {1}{c d \left (a e^{2} - c d^{2}\right )^{3}}} \log {\left (- a^{2} e^{4} \sqrt {- \frac {1}{c d \left (a e^{2} - c d^{2}\right )^{3}}} + 2 a c d^{2} e^{2} \sqrt {- \frac {1}{c d \left (a e^{2} - c d^{2}\right )^{3}}} - c^{2} d^{4} \sqrt {- \frac {1}{c d \left (a e^{2} - c d^{2}\right )^{3}}} + \sqrt {d + e x} \right )}}{\left (a e^{2} - c d^{2}\right )^{3}} + \frac {c^{2} d^{2} e^{2} \sqrt {- \frac {1}{c d \left (a e^{2} - c d^{2}\right )^{3}}} \log {\left (a^{2} e^{4} \sqrt {- \frac {1}{c d \left (a e^{2} - c d^{2}\right )^{3}}} - 2 a c d^{2} e^{2} \sqrt {- \frac {1}{c d \left (a e^{2} - c d^{2}\right )^{3}}} + c^{2} d^{4} \sqrt {- \frac {1}{c d \left (a e^{2} - c d^{2}\right )^{3}}} + \sqrt {d + e x} \right )}}{\left (a e^{2} - c d^{2}\right )^{3}} + \frac {4 c^{2} d^{2} e^{2} \sqrt {d + e x}}{2 a^{2} e^{4} \left (a e^{2} - c d^{2}\right )^{3} - 2 a c d^{2} e^{2} \left (a e^{2} - c d^{2}\right )^{3} + 2 a c d e^{3} x \left (a e^{2} - c d^{2}\right )^{3} - 2 c^{2} d^{3} e x \left (a e^{2} - c d^{2}\right )^{3}} + \frac {6 c d e^{2} \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {a e^{2}}{c d} - d}} \right )}}{\left (a e^{2} - c d^{2}\right )^{4} \sqrt {\frac {a e^{2}}{c d} - d}} + \frac {6 c d e^{2}}{\sqrt {d + e x} \left (a e^{2} - c d^{2}\right )^{4}} - \frac {2 e^{2}}{3 \left (d + e x\right )^{\frac {3}{2}} \left (a e^{2} - c d^{2}\right )^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(1/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)

[Out]

10*a*c**2*d**2*e**4*sqrt(d + e*x)/(8*a**4*e**8*(a*e**2 - c*d**2)**2 - 16*a**3*c*d**2*e**6*(a*e**2 - c*d**2)**2
+ 16*a**3*c*d*e**7*x*(a*e**2 - c*d**2)**2 - 48*a**2*c**2*d**3*e**5*x*(a*e**2 - c*d**2)**2 + 8*a**2*c**2*d**2*
e**4*(d + e*x)**2*(a*e**2 - c*d**2)**2 + 16*a*c**3*d**6*e**2*(a*e**2 - c*d**2)**2 + 48*a*c**3*d**5*e**3*x*(a*e
**2 - c*d**2)**2 - 16*a*c**3*d**4*e**2*(d + e*x)**2*(a*e**2 - c*d**2)**2 - 8*c**4*d**8*(a*e**2 - c*d**2)**2 -
16*c**4*d**7*e*x*(a*e**2 - c*d**2)**2 + 8*c**4*d**6*(d + e*x)**2*(a*e**2 - c*d**2)**2) - 10*c**3*d**4*e**2*sqr
t(d + e*x)/(8*a**4*e**8*(a*e**2 - c*d**2)**2 - 16*a**3*c*d**2*e**6*(a*e**2 - c*d**2)**2 + 16*a**3*c*d*e**7*x*(
a*e**2 - c*d**2)**2 - 48*a**2*c**2*d**3*e**5*x*(a*e**2 - c*d**2)**2 + 8*a**2*c**2*d**2*e**4*(d + e*x)**2*(a*e*
*2 - c*d**2)**2 + 16*a*c**3*d**6*e**2*(a*e**2 - c*d**2)**2 + 48*a*c**3*d**5*e**3*x*(a*e**2 - c*d**2)**2 - 16*a
*c**3*d**4*e**2*(d + e*x)**2*(a*e**2 - c*d**2)**2 - 8*c**4*d**8*(a*e**2 - c*d**2)**2 - 16*c**4*d**7*e*x*(a*e**
2 - c*d**2)**2 + 8*c**4*d**6*(d + e*x)**2*(a*e**2 - c*d**2)**2) + 6*c**3*d**3*e**2*(d + e*x)**(3/2)/(8*a**4*e*
*8*(a*e**2 - c*d**2)**2 - 16*a**3*c*d**2*e**6*(a*e**2 - c*d**2)**2 + 16*a**3*c*d*e**7*x*(a*e**2 - c*d**2)**2 -
48*a**2*c**2*d**3*e**5*x*(a*e**2 - c*d**2)**2 + 8*a**2*c**2*d**2*e**4*(d + e*x)**2*(a*e**2 - c*d**2)**2 + 16*
a*c**3*d**6*e**2*(a*e**2 - c*d**2)**2 + 48*a*c**3*d**5*e**3*x*(a*e**2 - c*d**2)**2 - 16*a*c**3*d**4*e**2*(d +
e*x)**2*(a*e**2 - c*d**2)**2 - 8*c**4*d**8*(a*e**2 - c*d**2)**2 - 16*c**4*d**7*e*x*(a*e**2 - c*d**2)**2 + 8*c*
*4*d**6*(d + e*x)**2*(a*e**2 - c*d**2)**2) - 3*c**2*d**2*e**2*sqrt(-1/(c*d*(a*e**2 - c*d**2)**5))*log(-a**3*e*
*6*sqrt(-1/(c*d*(a*e**2 - c*d**2)**5)) + 3*a**2*c*d**2*e**4*sqrt(-1/(c*d*(a*e**2 - c*d**2)**5)) - 3*a*c**2*d**
4*e**2*sqrt(-1/(c*d*(a*e**2 - c*d**2)**5)) + c**3*d**6*sqrt(-1/(c*d*(a*e**2 - c*d**2)**5)) + sqrt(d + e*x))/(8
*(a*e**2 - c*d**2)**2) + 3*c**2*d**2*e**2*sqrt(-1/(c*d*(a*e**2 - c*d**2)**5))*log(a**3*e**6*sqrt(-1/(c*d*(a*e*
*2 - c*d**2)**5)) - 3*a**2*c*d**2*e**4*sqrt(-1/(c*d*(a*e**2 - c*d**2)**5)) + 3*a*c**2*d**4*e**2*sqrt(-1/(c*d*(
a*e**2 - c*d**2)**5)) - c**3*d**6*sqrt(-1/(c*d*(a*e**2 - c*d**2)**5)) + sqrt(d + e*x))/(8*(a*e**2 - c*d**2)**2
) - c**2*d**2*e**2*sqrt(-1/(c*d*(a*e**2 - c*d**2)**3))*log(-a**2*e**4*sqrt(-1/(c*d*(a*e**2 - c*d**2)**3)) + 2*
a*c*d**2*e**2*sqrt(-1/(c*d*(a*e**2 - c*d**2)**3)) - c**2*d**4*sqrt(-1/(c*d*(a*e**2 - c*d**2)**3)) + sqrt(d + e
*x))/(a*e**2 - c*d**2)**3 + c**2*d**2*e**2*sqrt(-1/(c*d*(a*e**2 - c*d**2)**3))*log(a**2*e**4*sqrt(-1/(c*d*(a*e
**2 - c*d**2)**3)) - 2*a*c*d**2*e**2*sqrt(-1/(c*d*(a*e**2 - c*d**2)**3)) + c**2*d**4*sqrt(-1/(c*d*(a*e**2 - c*
d**2)**3)) + sqrt(d + e*x))/(a*e**2 - c*d**2)**3 + 4*c**2*d**2*e**2*sqrt(d + e*x)/(2*a**2*e**4*(a*e**2 - c*d**
2)**3 - 2*a*c*d**2*e**2*(a*e**2 - c*d**2)**3 + 2*a*c*d*e**3*x*(a*e**2 - c*d**2)**3 - 2*c**2*d**3*e*x*(a*e**2 -
c*d**2)**3) + 6*c*d*e**2*atan(sqrt(d + e*x)/sqrt(a*e**2/(c*d) - d))/((a*e**2 - c*d**2)**4*sqrt(a*e**2/(c*d) -
d)) + 6*c*d*e**2/(sqrt(d + e*x)*(a*e**2 - c*d**2)**4) - 2*e**2/(3*(d + e*x)**(3/2)*(a*e**2 - c*d**2)**3)

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Giac [A]
time = 1.52, size = 325, normalized size = 1.56 \begin {gather*} \frac {35 \, c^{2} d^{2} \arctan \left (\frac {\sqrt {x e + d} c d}{\sqrt {-c^{2} d^{3} + a c d e^{2}}}\right ) e^{2}}{4 \, {\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} \sqrt {-c^{2} d^{3} + a c d e^{2}}} + \frac {2 \, {\left (9 \, {\left (x e + d\right )} c d e^{2} + c d^{2} e^{2} - a e^{4}\right )}}{3 \, {\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} {\left (x e + d\right )}^{\frac {3}{2}}} + \frac {11 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{3} d^{3} e^{2} - 13 \, \sqrt {x e + d} c^{3} d^{4} e^{2} + 13 \, \sqrt {x e + d} a c^{2} d^{2} e^{4}}{4 \, {\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} {\left ({\left (x e + d\right )} c d - c d^{2} + a e^{2}\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="giac")

[Out]

35/4*c^2*d^2*arctan(sqrt(x*e + d)*c*d/sqrt(-c^2*d^3 + a*c*d*e^2))*e^2/((c^4*d^8 - 4*a*c^3*d^6*e^2 + 6*a^2*c^2*
d^4*e^4 - 4*a^3*c*d^2*e^6 + a^4*e^8)*sqrt(-c^2*d^3 + a*c*d*e^2)) + 2/3*(9*(x*e + d)*c*d*e^2 + c*d^2*e^2 - a*e^
4)/((c^4*d^8 - 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 - 4*a^3*c*d^2*e^6 + a^4*e^8)*(x*e + d)^(3/2)) + 1/4*(11*(x*
e + d)^(3/2)*c^3*d^3*e^2 - 13*sqrt(x*e + d)*c^3*d^4*e^2 + 13*sqrt(x*e + d)*a*c^2*d^2*e^4)/((c^4*d^8 - 4*a*c^3*
d^6*e^2 + 6*a^2*c^2*d^4*e^4 - 4*a^3*c*d^2*e^6 + a^4*e^8)*((x*e + d)*c*d - c*d^2 + a*e^2)^2)

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Mupad [B]
time = 0.86, size = 296, normalized size = 1.42 \begin {gather*} \frac {\frac {14\,c\,d\,e^2\,\left (d+e\,x\right )}{3\,{\left (a\,e^2-c\,d^2\right )}^2}-\frac {2\,e^2}{3\,\left (a\,e^2-c\,d^2\right )}+\frac {175\,c^2\,d^2\,e^2\,{\left (d+e\,x\right )}^2}{12\,{\left (a\,e^2-c\,d^2\right )}^3}+\frac {35\,c^3\,d^3\,e^2\,{\left (d+e\,x\right )}^3}{4\,{\left (a\,e^2-c\,d^2\right )}^4}}{{\left (d+e\,x\right )}^{3/2}\,\left (a^2\,e^4-2\,a\,c\,d^2\,e^2+c^2\,d^4\right )-\left (2\,c^2\,d^3-2\,a\,c\,d\,e^2\right )\,{\left (d+e\,x\right )}^{5/2}+c^2\,d^2\,{\left (d+e\,x\right )}^{7/2}}+\frac {35\,c^{3/2}\,d^{3/2}\,e^2\,\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {d}\,\sqrt {d+e\,x}\,\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 (a\,e^2-c\,d^2\right )}^{9/2}}\right )}{4\,{\left (a\,e^2-c\,d^2\right )}^{9/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(1/2)/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^3,x)

[Out]

((14*c*d*e^2*(d + e*x))/(3*(a*e^2 - c*d^2)^2) - (2*e^2)/(3*(a*e^2 - c*d^2)) + (175*c^2*d^2*e^2*(d + e*x)^2)/(1
2*(a*e^2 - c*d^2)^3) + (35*c^3*d^3*e^2*(d + e*x)^3)/(4*(a*e^2 - c*d^2)^4))/((d + e*x)^(3/2)*(a^2*e^4 + c^2*d^4
- 2*a*c*d^2*e^2) - (2*c^2*d^3 - 2*a*c*d*e^2)*(d + e*x)^(5/2) + c^2*d^2*(d + e*x)^(7/2)) + (35*c^(3/2)*d^(3/2)
*e^2*atan((c^(1/2)*d^(1/2)*(d + e*x)^(1/2)*(a^4*e^8 + c^4*d^8 - 4*a*c^3*d^6*e^2 - 4*a^3*c*d^2*e^6 + 6*a^2*c^2*
d^4*e^4))/(a*e^2 - c*d^2)^(9/2)))/(4*(a*e^2 - c*d^2)^(9/2))

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