3.21.76 \(\int \frac {\sqrt {d+e x}}{(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx\) [2076]

Optimal. Leaf size=257 \[ -\frac {2 \sqrt {d+e x}}{3 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {5 e}{3 \left (c d^2-a e^2\right )^2 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {5 c d e \sqrt {d+e x}}{\left (c d^2-a e^2\right )^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {5 c d e^{3/2} \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{\left (c d^2-a e^2\right )^{7/2}} \]

[Out]

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

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Rubi [A]
time = 0.13, antiderivative size = 257, 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 = {680, 686, 674, 211} \begin {gather*} \frac {5 c d e^{3/2} \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 )}{\left (c d^2-a e^2\right )^{7/2}}+\frac {5 c d e \sqrt {d+e x}}{\left (c d^2-a e^2\right )^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {5 e}{3 \sqrt {d+e x} \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {2 \sqrt {d+e x}}{3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \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)^(5/2),x]

[Out]

(-2*Sqrt[d + e*x])/(3*(c*d^2 - a*e^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (5*e)/(3*(c*d^2 - a*e^2
)^2*Sqrt[d + e*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (5*c*d*e*Sqrt[d + e*x])/((c*d^2 - a*e^2)^3*Sq
rt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (5*c*d*e^(3/2)*ArcTan[(Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c
*d*e*x^2])/(Sqrt[c*d^2 - a*e^2]*Sqrt[d + e*x])])/(c*d^2 - a*e^2)^(7/2)

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 680

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(2*c*d - b*e)*(d + e
*x)^m*((a + b*x + c*x^2)^(p + 1)/(e*(p + 1)*(b^2 - 4*a*c))), x] - Dist[(2*c*d - b*e)*((m + 2*p + 2)/((p + 1)*(
b^2 - 4*a*c))), Int[(d + e*x)^(m - 1)*(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] && LtQ[p, -1] && LtQ[0, m, 1] && 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 {d+e x}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx &=-\frac {2 \sqrt {d+e x}}{3 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {(5 e) \int \frac {1}{\sqrt {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}{3 \left (c d^2-a e^2\right )}\\ &=-\frac {2 \sqrt {d+e x}}{3 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {5 e}{3 \left (c d^2-a e^2\right )^2 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {(5 c d e) \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 )^{3/2}} \, dx}{2 \left (c d^2-a e^2\right )^2}\\ &=-\frac {2 \sqrt {d+e x}}{3 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {5 e}{3 \left (c d^2-a e^2\right )^2 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {5 c d e \sqrt {d+e x}}{\left (c d^2-a e^2\right )^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (5 c d e^2\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}{2 \left (c d^2-a e^2\right )^3}\\ &=-\frac {2 \sqrt {d+e x}}{3 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {5 e}{3 \left (c d^2-a e^2\right )^2 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {5 c d e \sqrt {d+e x}}{\left (c d^2-a e^2\right )^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (5 c d e^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 )}{\left (c d^2-a e^2\right )^3}\\ &=-\frac {2 \sqrt {d+e x}}{3 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {5 e}{3 \left (c d^2-a e^2\right )^2 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {5 c d e \sqrt {d+e x}}{\left (c d^2-a e^2\right )^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {5 c d e^{3/2} \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 )}{\left (c d^2-a e^2\right )^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 0.35, size = 177, normalized size = 0.69 \begin {gather*} \frac {\sqrt {d+e x} \left (\sqrt {c d^2-a e^2} \left (3 a^2 e^4+2 a c d e^2 (7 d+10 e x)+c^2 d^2 \left (-2 d^2+10 d e x+15 e^2 x^2\right )\right )+15 c d e^{3/2} (a e+c d x)^{3/2} (d+e x) \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2}}\right )\right )}{3 \left (c d^2-a e^2\right )^{7/2} ((a e+c d x) (d+e x))^{3/2}} \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)^(5/2),x]

[Out]

(Sqrt[d + e*x]*(Sqrt[c*d^2 - a*e^2]*(3*a^2*e^4 + 2*a*c*d*e^2*(7*d + 10*e*x) + c^2*d^2*(-2*d^2 + 10*d*e*x + 15*
e^2*x^2)) + 15*c*d*e^(3/2)*(a*e + c*d*x)^(3/2)*(d + e*x)*ArcTan[(Sqrt[e]*Sqrt[a*e + c*d*x])/Sqrt[c*d^2 - a*e^2
]]))/(3*(c*d^2 - a*e^2)^(7/2)*((a*e + c*d*x)*(d + e*x))^(3/2))

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Maple [A]
time = 0.74, size = 423, normalized size = 1.65

method	result	size
default	\(\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (15 \sqrt {c d x +a e}\, \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{2} d^{2} e^{3} x^{2}+15 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) a c d \,e^{4} x \sqrt {c d x +a e}+15 \sqrt {c d x +a e}\, \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{2} d^{3} e^{2} x +15 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) a c \,d^{2} e^{3} \sqrt {c d x +a e}-15 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{2} d^{2} e^{2} x^{2}-20 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a c d \,e^{3} x -10 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{2} d^{3} e x -3 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{2} e^{4}-14 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a c \,d^{2} e^{2}+2 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{2} d^{4}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} \left (c d x +a e \right )^{2} \left (e^{2} a -c \,d^{2}\right )^{3} \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\)	\(423\)

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)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/3*((c*d*x+a*e)*(e*x+d))^(1/2)*(15*(c*d*x+a*e)^(1/2)*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*c^2
*d^2*e^3*x^2+15*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*a*c*d*e^4*x*(c*d*x+a*e)^(1/2)+15*(c*d*x+a
*e)^(1/2)*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*c^2*d^3*e^2*x+15*arctanh(e*(c*d*x+a*e)^(1/2)/((
a*e^2-c*d^2)*e)^(1/2))*a*c*d^2*e^3*(c*d*x+a*e)^(1/2)-15*((a*e^2-c*d^2)*e)^(1/2)*c^2*d^2*e^2*x^2-20*((a*e^2-c*d
^2)*e)^(1/2)*a*c*d*e^3*x-10*((a*e^2-c*d^2)*e)^(1/2)*c^2*d^3*e*x-3*((a*e^2-c*d^2)*e)^(1/2)*a^2*e^4-14*((a*e^2-c
*d^2)*e)^(1/2)*a*c*d^2*e^2+2*((a*e^2-c*d^2)*e)^(1/2)*c^2*d^4)/(e*x+d)^(3/2)/(c*d*x+a*e)^2/(a*e^2-c*d^2)^3/((a*
e^2-c*d^2)*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((e*x+d)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 577 vs. \(2 (232) = 464\).
time = 3.68, size = 1192, normalized size = 4.64 \begin {gather*} \left [-\frac {15 \, {\left (c^{3} d^{5} x^{2} e + a^{2} c d x^{2} e^{5} + 2 \, {\left (a c^{2} d^{2} x^{3} + a^{2} c d^{2} x\right )} e^{4} + {\left (c^{3} d^{3} x^{4} + 4 \, a c^{2} d^{3} x^{2} + a^{2} c d^{3}\right )} e^{3} + 2 \, {\left (c^{3} d^{4} x^{3} + a c^{2} d^{4} x\right )} e^{2}\right )} \sqrt {-\frac {e}{c d^{2} - a e^{2}}} \log \left (\frac {c d^{3} - 2 \, a x e^{3} + 2 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (c d^{2} - a e^{2}\right )} \sqrt {x e + d} \sqrt {-\frac {e}{c d^{2} - a e^{2}}} - {\left (c d x^{2} + 2 \, a d\right )} e^{2}}{x^{2} e^{2} + 2 \, d x e + d^{2}}\right ) - 2 \, {\left (10 \, c^{2} d^{3} x e - 2 \, c^{2} d^{4} + 20 \, a c d x e^{3} + 3 \, a^{2} e^{4} + {\left (15 \, c^{2} d^{2} x^{2} + 14 \, a c d^{2}\right )} e^{2}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{6 \, {\left (c^{5} d^{10} x^{2} - a^{5} x^{2} e^{10} - 2 \, {\left (a^{4} c d x^{3} + a^{5} d x\right )} e^{9} - {\left (a^{3} c^{2} d^{2} x^{4} + a^{4} c d^{2} x^{2} + a^{5} d^{2}\right )} e^{8} + 4 \, {\left (a^{3} c^{2} d^{3} x^{3} + a^{4} c d^{3} x\right )} e^{7} + {\left (3 \, a^{2} c^{3} d^{4} x^{4} + 8 \, a^{3} c^{2} d^{4} x^{2} + 3 \, a^{4} c d^{4}\right )} e^{6} - {\left (3 \, a c^{4} d^{6} x^{4} + 8 \, a^{2} c^{3} d^{6} x^{2} + 3 \, a^{3} c^{2} d^{6}\right )} e^{4} - 4 \, {\left (a c^{4} d^{7} x^{3} + a^{2} c^{3} d^{7} x\right )} e^{3} + {\left (c^{5} d^{8} x^{4} + a c^{4} d^{8} x^{2} + a^{2} c^{3} d^{8}\right )} e^{2} + 2 \, {\left (c^{5} d^{9} x^{3} + a c^{4} d^{9} x\right )} e\right )}}, \frac {\frac {15 \, {\left (c^{3} d^{5} x^{2} e + a^{2} c d x^{2} e^{5} + 2 \, {\left (a c^{2} d^{2} x^{3} + a^{2} c d^{2} x\right )} e^{4} + {\left (c^{3} d^{3} x^{4} + 4 \, a c^{2} d^{3} x^{2} + a^{2} c d^{3}\right )} e^{3} + 2 \, {\left (c^{3} d^{4} x^{3} + a c^{2} d^{4} x\right )} e^{2}\right )} \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} - a e^{2}} \sqrt {x e + d} e^{\frac {1}{2}}}{c d^{2} x e + a x e^{3} + {\left (c d x^{2} + a d\right )} e^{2}}\right ) e^{\frac {1}{2}}}{\sqrt {c d^{2} - a e^{2}}} + {\left (10 \, c^{2} d^{3} x e - 2 \, c^{2} d^{4} + 20 \, a c d x e^{3} + 3 \, a^{2} e^{4} + {\left (15 \, c^{2} d^{2} x^{2} + 14 \, a c d^{2}\right )} e^{2}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{3 \, {\left (c^{5} d^{10} x^{2} - a^{5} x^{2} e^{10} - 2 \, {\left (a^{4} c d x^{3} + a^{5} d x\right )} e^{9} - {\left (a^{3} c^{2} d^{2} x^{4} + a^{4} c d^{2} x^{2} + a^{5} d^{2}\right )} e^{8} + 4 \, {\left (a^{3} c^{2} d^{3} x^{3} + a^{4} c d^{3} x\right )} e^{7} + {\left (3 \, a^{2} c^{3} d^{4} x^{4} + 8 \, a^{3} c^{2} d^{4} x^{2} + 3 \, a^{4} c d^{4}\right )} e^{6} - {\left (3 \, a c^{4} d^{6} x^{4} + 8 \, a^{2} c^{3} d^{6} x^{2} + 3 \, a^{3} c^{2} d^{6}\right )} e^{4} - 4 \, {\left (a c^{4} d^{7} x^{3} + a^{2} c^{3} d^{7} x\right )} e^{3} + {\left (c^{5} d^{8} x^{4} + a c^{4} d^{8} x^{2} + a^{2} c^{3} d^{8}\right )} e^{2} + 2 \, {\left (c^{5} d^{9} x^{3} + a c^{4} d^{9} 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)^(5/2),x, algorithm="fricas")

[Out]

[-1/6*(15*(c^3*d^5*x^2*e + a^2*c*d*x^2*e^5 + 2*(a*c^2*d^2*x^3 + a^2*c*d^2*x)*e^4 + (c^3*d^3*x^4 + 4*a*c^2*d^3*
x^2 + a^2*c*d^3)*e^3 + 2*(c^3*d^4*x^3 + a*c^2*d^4*x)*e^2)*sqrt(-e/(c*d^2 - a*e^2))*log((c*d^3 - 2*a*x*e^3 + 2*
sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*(c*d^2 - a*e^2)*sqrt(x*e + d)*sqrt(-e/(c*d^2 - a*e^2)) - (c*d*x^2
+ 2*a*d)*e^2)/(x^2*e^2 + 2*d*x*e + d^2)) - 2*(10*c^2*d^3*x*e - 2*c^2*d^4 + 20*a*c*d*x*e^3 + 3*a^2*e^4 + (15*c^
2*d^2*x^2 + 14*a*c*d^2)*e^2)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(x*e + d))/(c^5*d^10*x^2 - a^5*x^
2*e^10 - 2*(a^4*c*d*x^3 + a^5*d*x)*e^9 - (a^3*c^2*d^2*x^4 + a^4*c*d^2*x^2 + a^5*d^2)*e^8 + 4*(a^3*c^2*d^3*x^3
+ a^4*c*d^3*x)*e^7 + (3*a^2*c^3*d^4*x^4 + 8*a^3*c^2*d^4*x^2 + 3*a^4*c*d^4)*e^6 - (3*a*c^4*d^6*x^4 + 8*a^2*c^3*
d^6*x^2 + 3*a^3*c^2*d^6)*e^4 - 4*(a*c^4*d^7*x^3 + a^2*c^3*d^7*x)*e^3 + (c^5*d^8*x^4 + a*c^4*d^8*x^2 + a^2*c^3*
d^8)*e^2 + 2*(c^5*d^9*x^3 + a*c^4*d^9*x)*e), 1/3*(15*(c^3*d^5*x^2*e + a^2*c*d*x^2*e^5 + 2*(a*c^2*d^2*x^3 + a^2
*c*d^2*x)*e^4 + (c^3*d^3*x^4 + 4*a*c^2*d^3*x^2 + a^2*c*d^3)*e^3 + 2*(c^3*d^4*x^3 + a*c^2*d^4*x)*e^2)*arctan(-s
qrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(c*d^2 - a*e^2)*sqrt(x*e + d)*e^(1/2)/(c*d^2*x*e + a*x*e^3 + (c
*d*x^2 + a*d)*e^2))*e^(1/2)/sqrt(c*d^2 - a*e^2) + (10*c^2*d^3*x*e - 2*c^2*d^4 + 20*a*c*d*x*e^3 + 3*a^2*e^4 + (
15*c^2*d^2*x^2 + 14*a*c*d^2)*e^2)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(x*e + d))/(c^5*d^10*x^2 - a
^5*x^2*e^10 - 2*(a^4*c*d*x^3 + a^5*d*x)*e^9 - (a^3*c^2*d^2*x^4 + a^4*c*d^2*x^2 + a^5*d^2)*e^8 + 4*(a^3*c^2*d^3
*x^3 + a^4*c*d^3*x)*e^7 + (3*a^2*c^3*d^4*x^4 + 8*a^3*c^2*d^4*x^2 + 3*a^4*c*d^4)*e^6 - (3*a*c^4*d^6*x^4 + 8*a^2
*c^3*d^6*x^2 + 3*a^3*c^2*d^6)*e^4 - 4*(a*c^4*d^7*x^3 + a^2*c^3*d^7*x)*e^3 + (c^5*d^8*x^4 + a*c^4*d^8*x^2 + a^2
*c^3*d^8)*e^2 + 2*(c^5*d^9*x^3 + a*c^4*d^9*x)*e)]

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

[Out]

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

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Giac [A]
time = 1.36, size = 297, normalized size = 1.16 \begin {gather*} \frac {1}{3} \, {\left (\frac {15 \, c d \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^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \sqrt {c d^{2} e - a e^{3}}} - \frac {2 \, {\left (c^{2} d^{3} e^{2} - a c d e^{4} - 6 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )} c d e\right )}}{{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}}} + \frac {3 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}}}{{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} {\left (x e + d\right )}}\right )} e \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)^(5/2),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {d+e\,x}}{{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}} \,d x \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)^(5/2),x)

[Out]

int((d + e*x)^(1/2)/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2), x)

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