3.21.77 \(\int \frac {1}{\sqrt {d+e x} (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx\) [2077]

Optimal. Leaf size=329 \[ \frac {1}{2 \left (c d^2-a e^2\right ) \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {7 c d \sqrt {d+e x}}{6 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {35 c d e}{12 \left (c d^2-a e^2\right )^3 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 c^2 d^2 e \sqrt {d+e x}}{4 \left (c d^2-a e^2\right )^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 c^2 d^2 e^{3/2} \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{4 \left (c d^2-a e^2\right )^{9/2}} \]

[Out]

35/4*c^2*d^2*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)^(9/2)+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)/(e*x+d)^(1/2)-7/6*c*d*(e*x+d
)^(1/2)/(-a*e^2+c*d^2)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-35/12*c*d*e/(-a*e^2+c*d^2)^3/(e*x+d)^(1/2)/(a
*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+35/4*c^2*d^2*e*(e*x+d)^(1/2)/(-a*e^2+c*d^2)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d
*e*x^2)^(1/2)

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Rubi [A]
time = 0.18, antiderivative size = 329, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {686, 680, 674, 211} \begin {gather*} \frac {35 c^2 d^2 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 )}{4 \left (c d^2-a e^2\right )^{9/2}}+\frac {35 c^2 d^2 e \sqrt {d+e x}}{4 \left (c d^2-a e^2\right )^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {35 c d e}{12 \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 {7 c d \sqrt {d+e x}}{6 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac {1}{2 \sqrt {d+e x} \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[1/(Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]

[Out]

1/(2*(c*d^2 - a*e^2)*Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (7*c*d*Sqrt[d + e*x])/(6*(
c*d^2 - a*e^2)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (35*c*d*e)/(12*(c*d^2 - a*e^2)^3*Sqrt[d + e*
x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (35*c^2*d^2*e*Sqrt[d + e*x])/(4*(c*d^2 - a*e^2)^4*Sqrt[a*d*e
+ (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (35*c^2*d^2*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])])/(4*(c*d^2 - a*e^2)^(9/2))

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 {1}{\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 {1}{2 \left (c d^2-a e^2\right ) \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}}+\frac {(7 c d) \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}{4 \left (c d^2-a e^2\right )}\\ &=\frac {1}{2 \left (c d^2-a e^2\right ) \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}}-\frac {7 c d \sqrt {d+e x}}{6 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {(35 c d 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}{12 \left (c d^2-a e^2\right )^2}\\ &=\frac {1}{2 \left (c d^2-a e^2\right ) \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}}-\frac {7 c d \sqrt {d+e x}}{6 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {35 c d e}{12 \left (c d^2-a e^2\right )^3 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (35 c^2 d^2 e\right ) \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}{8 \left (c d^2-a e^2\right )^3}\\ &=\frac {1}{2 \left (c d^2-a e^2\right ) \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}}-\frac {7 c d \sqrt {d+e x}}{6 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {35 c d e}{12 \left (c d^2-a e^2\right )^3 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 c^2 d^2 e \sqrt {d+e x}}{4 \left (c d^2-a e^2\right )^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (35 c^2 d^2 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}{8 \left (c d^2-a e^2\right )^4}\\ &=\frac {1}{2 \left (c d^2-a e^2\right ) \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}}-\frac {7 c d \sqrt {d+e x}}{6 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {35 c d e}{12 \left (c d^2-a e^2\right )^3 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 c^2 d^2 e \sqrt {d+e x}}{4 \left (c d^2-a e^2\right )^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (35 c^2 d^2 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 )}{4 \left (c d^2-a e^2\right )^4}\\ &=\frac {1}{2 \left (c d^2-a e^2\right ) \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}}-\frac {7 c d \sqrt {d+e x}}{6 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {35 c d e}{12 \left (c d^2-a e^2\right )^3 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 c^2 d^2 e \sqrt {d+e x}}{4 \left (c d^2-a e^2\right )^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 c^2 d^2 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 )}{4 \left (c d^2-a e^2\right )^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 0.69, size = 240, normalized size = 0.73 \begin {gather*} \frac {c^2 d^2 (d+e x)^{5/2} \left (-\frac {(a e+c d x) \left (6 a^3 e^6-3 a^2 c d e^4 (13 d+7 e x)-2 a c^2 d^2 e^2 \left (40 d^2+119 d e x+70 e^2 x^2\right )+c^3 d^3 \left (8 d^3-56 d^2 e x-175 d e^2 x^2-105 e^3 x^3\right )\right )}{c^2 d^2 \left (c d^2-a e^2\right )^4 (d+e x)^2}+\frac {105 e^{3/2} (a e+c d x)^{5/2} \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{9/2}}\right )}{12 ((a e+c d x) (d+e x))^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]

[Out]

(c^2*d^2*(d + e*x)^(5/2)*(-(((a*e + c*d*x)*(6*a^3*e^6 - 3*a^2*c*d*e^4*(13*d + 7*e*x) - 2*a*c^2*d^2*e^2*(40*d^2
+ 119*d*e*x + 70*e^2*x^2) + c^3*d^3*(8*d^3 - 56*d^2*e*x - 175*d*e^2*x^2 - 105*e^3*x^3)))/(c^2*d^2*(c*d^2 - a*
e^2)^4*(d + e*x)^2)) + (105*e^(3/2)*(a*e + c*d*x)^(5/2)*ArcTan[(Sqrt[e]*Sqrt[a*e + c*d*x])/Sqrt[c*d^2 - a*e^2]
])/(c*d^2 - a*e^2)^(9/2)))/(12*((a*e + c*d*x)*(d + e*x))^(5/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(657\) vs. \(2(291)=582\).
time = 0.72, size = 658, normalized size = 2.00

method	result	size
default	\(-\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (105 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) \sqrt {c d x +a e}\, c^{3} d^{3} e^{4} x^{3}+105 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) a \,c^{2} d^{2} e^{5} x^{2} \sqrt {c d x +a e}+210 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) \sqrt {c d x +a e}\, c^{3} d^{4} e^{3} x^{2}+210 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) a \,c^{2} d^{3} e^{4} x \sqrt {c d x +a e}+105 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) \sqrt {c d x +a e}\, c^{3} d^{5} e^{2} x -105 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{3} d^{3} e^{3} x^{3}+105 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) a \,c^{2} d^{4} e^{3} \sqrt {c d x +a e}-140 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a \,c^{2} d^{2} e^{4} x^{2}-175 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{3} d^{4} e^{2} x^{2}-21 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{2} c d \,e^{5} x -238 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a \,c^{2} d^{3} e^{3} x -56 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{3} d^{5} e x +6 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{3} e^{6}-39 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{2} c \,d^{2} e^{4}-80 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a \,c^{2} d^{4} e^{2}+8 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{3} d^{6}\right )}{12 \left (e x +d \right )^{\frac {5}{2}} \left (c d x +a e \right )^{2} \left (e^{2} a -c \,d^{2}\right )^{4} \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\)	\(658\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(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]

-1/12*((c*d*x+a*e)*(e*x+d))^(1/2)*(105*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*(c*d*x+a*e)^(1/2)*
c^3*d^3*e^4*x^3+105*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*a*c^2*d^2*e^5*x^2*(c*d*x+a*e)^(1/2)+2
10*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*(c*d*x+a*e)^(1/2)*c^3*d^4*e^3*x^2+210*arctanh(e*(c*d*x
+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*a*c^2*d^3*e^4*x*(c*d*x+a*e)^(1/2)+105*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2
-c*d^2)*e)^(1/2))*(c*d*x+a*e)^(1/2)*c^3*d^5*e^2*x-105*((a*e^2-c*d^2)*e)^(1/2)*c^3*d^3*e^3*x^3+105*arctanh(e*(c
*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*a*c^2*d^4*e^3*(c*d*x+a*e)^(1/2)-140*((a*e^2-c*d^2)*e)^(1/2)*a*c^2*d^2
*e^4*x^2-175*((a*e^2-c*d^2)*e)^(1/2)*c^3*d^4*e^2*x^2-21*((a*e^2-c*d^2)*e)^(1/2)*a^2*c*d*e^5*x-238*((a*e^2-c*d^
2)*e)^(1/2)*a*c^2*d^3*e^3*x-56*((a*e^2-c*d^2)*e)^(1/2)*c^3*d^5*e*x+6*((a*e^2-c*d^2)*e)^(1/2)*a^3*e^6-39*((a*e^
2-c*d^2)*e)^(1/2)*a^2*c*d^2*e^4-80*((a*e^2-c*d^2)*e)^(1/2)*a*c^2*d^4*e^2+8*((a*e^2-c*d^2)*e)^(1/2)*c^3*d^6)/(e
*x+d)^(5/2)/(c*d*x+a*e)^2/(a*e^2-c*d^2)^4/((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(1/(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(1/((c*d*x^2*e + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)*sqrt(x*e + d)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 864 vs. \(2 (295) = 590\).
time = 3.92, size = 1766, normalized size = 5.37 \begin {gather*} \left [\frac {105 \, {\left (c^{4} d^{7} x^{2} e + a^{2} c^{2} d^{2} x^{3} e^{6} + {\left (2 \, a c^{3} d^{3} x^{4} + 3 \, a^{2} c^{2} d^{3} x^{2}\right )} e^{5} + {\left (c^{4} d^{4} x^{5} + 6 \, a c^{3} d^{4} x^{3} + 3 \, a^{2} c^{2} d^{4} x\right )} e^{4} + {\left (3 \, c^{4} d^{5} x^{4} + 6 \, a c^{3} d^{5} x^{2} + a^{2} c^{2} d^{5}\right )} e^{3} + {\left (3 \, c^{4} d^{6} x^{3} + 2 \, a c^{3} d^{6} 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 (56 \, c^{3} d^{5} x e - 8 \, c^{3} d^{6} + 21 \, a^{2} c d x e^{5} - 6 \, a^{3} e^{6} + {\left (140 \, a c^{2} d^{2} x^{2} + 39 \, 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} + 5 \, {\left (35 \, c^{3} d^{4} x^{2} + 16 \, a c^{2} d^{4}\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}}{24 \, {\left (c^{6} d^{13} x^{2} + a^{6} x^{3} e^{13} + {\left (2 \, a^{5} c d x^{4} + 3 \, a^{6} d x^{2}\right )} e^{12} + {\left (a^{4} c^{2} d^{2} x^{5} + 2 \, a^{5} c d^{2} x^{3} + 3 \, a^{6} d^{2} x\right )} e^{11} - {\left (5 \, a^{4} c^{2} d^{3} x^{4} + 6 \, a^{5} c d^{3} x^{2} - a^{6} d^{3}\right )} e^{10} - {\left (4 \, a^{3} c^{3} d^{4} x^{5} + 15 \, a^{4} c^{2} d^{4} x^{3} + 10 \, a^{5} c d^{4} x\right )} e^{9} - {\left (5 \, a^{4} c^{2} d^{5} x^{2} + 4 \, a^{5} c d^{5}\right )} e^{8} + 2 \, {\left (3 \, a^{2} c^{4} d^{6} x^{5} + 10 \, a^{3} c^{3} d^{6} x^{3} + 5 \, a^{4} c^{2} d^{6} x\right )} e^{7} + 2 \, {\left (5 \, a^{2} c^{4} d^{7} x^{4} + 10 \, a^{3} c^{3} d^{7} x^{2} + 3 \, a^{4} c^{2} d^{7}\right )} e^{6} - {\left (4 \, a c^{5} d^{8} x^{5} + 5 \, a^{2} c^{4} d^{8} x^{3}\right )} e^{5} - {\left (10 \, a c^{5} d^{9} x^{4} + 15 \, a^{2} c^{4} d^{9} x^{2} + 4 \, a^{3} c^{3} d^{9}\right )} e^{4} + {\left (c^{6} d^{10} x^{5} - 6 \, a c^{5} d^{10} x^{3} - 5 \, a^{2} c^{4} d^{10} x\right )} e^{3} + {\left (3 \, c^{6} d^{11} x^{4} + 2 \, a c^{5} d^{11} x^{2} + a^{2} c^{4} d^{11}\right )} e^{2} + {\left (3 \, c^{6} d^{12} x^{3} + 2 \, a c^{5} d^{12} x\right )} e\right )}}, \frac {\frac {105 \, {\left (c^{4} d^{7} x^{2} e + a^{2} c^{2} d^{2} x^{3} e^{6} + {\left (2 \, a c^{3} d^{3} x^{4} + 3 \, a^{2} c^{2} d^{3} x^{2}\right )} e^{5} + {\left (c^{4} d^{4} x^{5} + 6 \, a c^{3} d^{4} x^{3} + 3 \, a^{2} c^{2} d^{4} x\right )} e^{4} + {\left (3 \, c^{4} d^{5} x^{4} + 6 \, a c^{3} d^{5} x^{2} + a^{2} c^{2} d^{5}\right )} e^{3} + {\left (3 \, c^{4} d^{6} x^{3} + 2 \, a c^{3} d^{6} 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 (56 \, c^{3} d^{5} x e - 8 \, c^{3} d^{6} + 21 \, a^{2} c d x e^{5} - 6 \, a^{3} e^{6} + {\left (140 \, a c^{2} d^{2} x^{2} + 39 \, 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} + 5 \, {\left (35 \, c^{3} d^{4} x^{2} + 16 \, a c^{2} d^{4}\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}}{12 \, {\left (c^{6} d^{13} x^{2} + a^{6} x^{3} e^{13} + {\left (2 \, a^{5} c d x^{4} + 3 \, a^{6} d x^{2}\right )} e^{12} + {\left (a^{4} c^{2} d^{2} x^{5} + 2 \, a^{5} c d^{2} x^{3} + 3 \, a^{6} d^{2} x\right )} e^{11} - {\left (5 \, a^{4} c^{2} d^{3} x^{4} + 6 \, a^{5} c d^{3} x^{2} - a^{6} d^{3}\right )} e^{10} - {\left (4 \, a^{3} c^{3} d^{4} x^{5} + 15 \, a^{4} c^{2} d^{4} x^{3} + 10 \, a^{5} c d^{4} x\right )} e^{9} - {\left (5 \, a^{4} c^{2} d^{5} x^{2} + 4 \, a^{5} c d^{5}\right )} e^{8} + 2 \, {\left (3 \, a^{2} c^{4} d^{6} x^{5} + 10 \, a^{3} c^{3} d^{6} x^{3} + 5 \, a^{4} c^{2} d^{6} x\right )} e^{7} + 2 \, {\left (5 \, a^{2} c^{4} d^{7} x^{4} + 10 \, a^{3} c^{3} d^{7} x^{2} + 3 \, a^{4} c^{2} d^{7}\right )} e^{6} - {\left (4 \, a c^{5} d^{8} x^{5} + 5 \, a^{2} c^{4} d^{8} x^{3}\right )} e^{5} - {\left (10 \, a c^{5} d^{9} x^{4} + 15 \, a^{2} c^{4} d^{9} x^{2} + 4 \, a^{3} c^{3} d^{9}\right )} e^{4} + {\left (c^{6} d^{10} x^{5} - 6 \, a c^{5} d^{10} x^{3} - 5 \, a^{2} c^{4} d^{10} x\right )} e^{3} + {\left (3 \, c^{6} d^{11} x^{4} + 2 \, a c^{5} d^{11} x^{2} + a^{2} c^{4} d^{11}\right )} e^{2} + {\left (3 \, c^{6} d^{12} x^{3} + 2 \, a c^{5} d^{12} x\right )} e\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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]

[1/24*(105*(c^4*d^7*x^2*e + a^2*c^2*d^2*x^3*e^6 + (2*a*c^3*d^3*x^4 + 3*a^2*c^2*d^3*x^2)*e^5 + (c^4*d^4*x^5 + 6
*a*c^3*d^4*x^3 + 3*a^2*c^2*d^4*x)*e^4 + (3*c^4*d^5*x^4 + 6*a*c^3*d^5*x^2 + a^2*c^2*d^5)*e^3 + (3*c^4*d^6*x^3 +
2*a*c^3*d^6*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*(56*c^3*d^5*x*e - 8*c^3*d^6 + 21*a^2*c*d*x*e^5 - 6*a^3*e^6 + (140*a*c^2*d^2*x^2 + 39*a^2*c*d^2)*e^4 +
7*(15*c^3*d^3*x^3 + 34*a*c^2*d^3*x)*e^3 + 5*(35*c^3*d^4*x^2 + 16*a*c^2*d^4)*e^2)*sqrt(c*d^2*x + a*x*e^2 + (c*d
*x^2 + a*d)*e)*sqrt(x*e + d))/(c^6*d^13*x^2 + a^6*x^3*e^13 + (2*a^5*c*d*x^4 + 3*a^6*d*x^2)*e^12 + (a^4*c^2*d^2
*x^5 + 2*a^5*c*d^2*x^3 + 3*a^6*d^2*x)*e^11 - (5*a^4*c^2*d^3*x^4 + 6*a^5*c*d^3*x^2 - a^6*d^3)*e^10 - (4*a^3*c^3
*d^4*x^5 + 15*a^4*c^2*d^4*x^3 + 10*a^5*c*d^4*x)*e^9 - (5*a^4*c^2*d^5*x^2 + 4*a^5*c*d^5)*e^8 + 2*(3*a^2*c^4*d^6
*x^5 + 10*a^3*c^3*d^6*x^3 + 5*a^4*c^2*d^6*x)*e^7 + 2*(5*a^2*c^4*d^7*x^4 + 10*a^3*c^3*d^7*x^2 + 3*a^4*c^2*d^7)*
e^6 - (4*a*c^5*d^8*x^5 + 5*a^2*c^4*d^8*x^3)*e^5 - (10*a*c^5*d^9*x^4 + 15*a^2*c^4*d^9*x^2 + 4*a^3*c^3*d^9)*e^4
+ (c^6*d^10*x^5 - 6*a*c^5*d^10*x^3 - 5*a^2*c^4*d^10*x)*e^3 + (3*c^6*d^11*x^4 + 2*a*c^5*d^11*x^2 + a^2*c^4*d^11
)*e^2 + (3*c^6*d^12*x^3 + 2*a*c^5*d^12*x)*e), 1/12*(105*(c^4*d^7*x^2*e + a^2*c^2*d^2*x^3*e^6 + (2*a*c^3*d^3*x^
4 + 3*a^2*c^2*d^3*x^2)*e^5 + (c^4*d^4*x^5 + 6*a*c^3*d^4*x^3 + 3*a^2*c^2*d^4*x)*e^4 + (3*c^4*d^5*x^4 + 6*a*c^3*
d^5*x^2 + a^2*c^2*d^5)*e^3 + (3*c^4*d^6*x^3 + 2*a*c^3*d^6*x)*e^2)*arctan(-sqrt(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) + (56*c^3*d^5*x*e - 8*c^3*d^6 + 21*a^2*c*d*x*e^5 - 6*a^3*e^6 + (140*a*c^2*d^2*x^2 + 39*a^2*c*d^2)
*e^4 + 7*(15*c^3*d^3*x^3 + 34*a*c^2*d^3*x)*e^3 + 5*(35*c^3*d^4*x^2 + 16*a*c^2*d^4)*e^2)*sqrt(c*d^2*x + a*x*e^2
+ (c*d*x^2 + a*d)*e)*sqrt(x*e + d))/(c^6*d^13*x^2 + a^6*x^3*e^13 + (2*a^5*c*d*x^4 + 3*a^6*d*x^2)*e^12 + (a^4*
c^2*d^2*x^5 + 2*a^5*c*d^2*x^3 + 3*a^6*d^2*x)*e^11 - (5*a^4*c^2*d^3*x^4 + 6*a^5*c*d^3*x^2 - a^6*d^3)*e^10 - (4*
a^3*c^3*d^4*x^5 + 15*a^4*c^2*d^4*x^3 + 10*a^5*c*d^4*x)*e^9 - (5*a^4*c^2*d^5*x^2 + 4*a^5*c*d^5)*e^8 + 2*(3*a^2*
c^4*d^6*x^5 + 10*a^3*c^3*d^6*x^3 + 5*a^4*c^2*d^6*x)*e^7 + 2*(5*a^2*c^4*d^7*x^4 + 10*a^3*c^3*d^7*x^2 + 3*a^4*c^
2*d^7)*e^6 - (4*a*c^5*d^8*x^5 + 5*a^2*c^4*d^8*x^3)*e^5 - (10*a*c^5*d^9*x^4 + 15*a^2*c^4*d^9*x^2 + 4*a^3*c^3*d^
9)*e^4 + (c^6*d^10*x^5 - 6*a*c^5*d^10*x^3 - 5*a^2*c^4*d^10*x)*e^3 + (3*c^6*d^11*x^4 + 2*a*c^5*d^11*x^2 + a^2*c
^4*d^11)*e^2 + (3*c^6*d^12*x^3 + 2*a*c^5*d^12*x)*e)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {5}{2}} \sqrt {d + e x}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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(1/(((d + e*x)*(a*e + c*d*x))**(5/2)*sqrt(d + e*x)), x)

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Giac [A]
time = 1.41, size = 437, normalized size = 1.33 \begin {gather*} \frac {1}{12} \, {\left (\frac {105 \, c^{2} d^{2} \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^{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 d^{2} e - a e^{3}}} - \frac {8 \, {\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4} - 9 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )} c^{2} d^{2} e\right )}}{{\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 e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}}} + \frac {3 \, {\left (13 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} c^{3} d^{4} e^{2} - 13 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a c^{2} d^{2} e^{4} + 11 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{2} d^{2} e\right )} e^{\left (-2\right )}}{{\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 )}^{2} c^{2} d^{2}}\right )} e \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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]

1/12*(105*c^2*d^2*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^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*d^2*e - a*e^3)) - 8*(c^3*d^4*e^2 - a*c^2*d^2*e
^4 - 9*((x*e + d)*c*d*e - c*d^2*e + a*e^3)*c^2*d^2*e)/((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*e - c*d^2*e + a*e^3)^(3/2)) + 3*(13*sqrt((x*e + d)*c*d*e - c*d^2*e + a*e^3
)*c^3*d^4*e^2 - 13*sqrt((x*e + d)*c*d*e - c*d^2*e + a*e^3)*a*c^2*d^2*e^4 + 11*((x*e + d)*c*d*e - c*d^2*e + a*e
^3)^(3/2)*c^2*d^2*e)*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)*(x*e
+ d)^2*c^2*d^2))*e

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{\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(1/((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(1/((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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