3.21.78 \(\int \frac {1}{(d+e x)^{3/2} (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx\) [2078]

Optimal. Leaf size=395 \[ \frac {1}{3 \left (c d^2-a e^2\right ) (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {3 c d}{4 \left (c d^2-a e^2\right )^2 \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {7 c^2 d^2 \sqrt {d+e x}}{4 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {35 c^2 d^2 e}{8 \left (c d^2-a e^2\right )^4 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {105 c^3 d^3 e \sqrt {d+e x}}{8 \left (c d^2-a e^2\right )^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {105 c^3 d^3 e^{3/2} \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{8 \left (c d^2-a e^2\right )^{11/2}} \]

[Out]

1/3/(-a*e^2+c*d^2)/(e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+105/8*c^3*d^3*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)^(11/2)+3/4*c*d/(-a*
e^2+c*d^2)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(1/2)-7/4*c^2*d^2*(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)^(3/2)-35/8*c^2*d^2*e/(-a*e^2+c*d^2)^4/(e*x+d)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c
*d*e*x^2)^(1/2)+105/8*c^3*d^3*e*(e*x+d)^(1/2)/(-a*e^2+c*d^2)^5/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)

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Rubi [A]
time = 0.22, antiderivative size = 395, normalized size of antiderivative = 1.00, number of steps used = 7, 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 {105 c^3 d^3 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 )}{8 \left (c d^2-a e^2\right )^{11/2}}+\frac {105 c^3 d^3 e \sqrt {d+e x}}{8 \left (c d^2-a e^2\right )^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {35 c^2 d^2 e}{8 \sqrt {d+e x} \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 {7 c^2 d^2 \sqrt {d+e x}}{4 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac {3 c d}{4 \sqrt {d+e x} \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}{3 (d+e x)^{3/2} \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/((d + e*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]

[Out]

1/(3*(c*d^2 - a*e^2)*(d + e*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (3*c*d)/(4*(c*d^2 - a*e^
2)^2*Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (7*c^2*d^2*Sqrt[d + e*x])/(4*(c*d^2 - a*e^
2)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (35*c^2*d^2*e)/(8*(c*d^2 - a*e^2)^4*Sqrt[d + e*x]*Sqrt[a
*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (105*c^3*d^3*e*Sqrt[d + e*x])/(8*(c*d^2 - a*e^2)^5*Sqrt[a*d*e + (c*d^
2 + a*e^2)*x + c*d*e*x^2]) + (105*c^3*d^3*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])])/(8*(c*d^2 - a*e^2)^(11/2))

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}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx &=\frac {1}{3 \left (c d^2-a e^2\right ) (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {(3 c d) \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}{2 \left (c d^2-a e^2\right )}\\ &=\frac {1}{3 \left (c d^2-a e^2\right ) (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {3 c d}{4 \left (c d^2-a e^2\right )^2 \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 {\left (21 c^2 d^2\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 )^{5/2}} \, dx}{8 \left (c d^2-a e^2\right )^2}\\ &=\frac {1}{3 \left (c d^2-a e^2\right ) (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {3 c d}{4 \left (c d^2-a e^2\right )^2 \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^2 d^2 \sqrt {d+e x}}{4 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {\left (35 c^2 d^2 e\right ) \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}{8 \left (c d^2-a e^2\right )^3}\\ &=\frac {1}{3 \left (c d^2-a e^2\right ) (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {3 c d}{4 \left (c d^2-a e^2\right )^2 \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^2 d^2 \sqrt {d+e x}}{4 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {35 c^2 d^2 e}{8 \left (c d^2-a e^2\right )^4 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (105 c^3 d^3 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}{16 \left (c d^2-a e^2\right )^4}\\ &=\frac {1}{3 \left (c d^2-a e^2\right ) (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {3 c d}{4 \left (c d^2-a e^2\right )^2 \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^2 d^2 \sqrt {d+e x}}{4 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {35 c^2 d^2 e}{8 \left (c d^2-a e^2\right )^4 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {105 c^3 d^3 e \sqrt {d+e x}}{8 \left (c d^2-a e^2\right )^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (105 c^3 d^3 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}{16 \left (c d^2-a e^2\right )^5}\\ &=\frac {1}{3 \left (c d^2-a e^2\right ) (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {3 c d}{4 \left (c d^2-a e^2\right )^2 \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^2 d^2 \sqrt {d+e x}}{4 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {35 c^2 d^2 e}{8 \left (c d^2-a e^2\right )^4 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {105 c^3 d^3 e \sqrt {d+e x}}{8 \left (c d^2-a e^2\right )^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (105 c^3 d^3 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 )}{8 \left (c d^2-a e^2\right )^5}\\ &=\frac {1}{3 \left (c d^2-a e^2\right ) (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {3 c d}{4 \left (c d^2-a e^2\right )^2 \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^2 d^2 \sqrt {d+e x}}{4 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {35 c^2 d^2 e}{8 \left (c d^2-a e^2\right )^4 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {105 c^3 d^3 e \sqrt {d+e x}}{8 \left (c d^2-a e^2\right )^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {105 c^3 d^3 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 )}{8 \left (c d^2-a e^2\right )^{11/2}}\\ \end {align*}

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Mathematica [A]
time = 1.05, size = 294, normalized size = 0.74 \begin {gather*} \frac {c^3 d^3 (d+e x)^{5/2} \left (\frac {(a e+c d x) \left (8 a^4 e^8-2 a^3 c d e^6 (25 d+9 e x)+3 a^2 c^2 d^2 e^4 \left (55 d^2+60 d e x+21 e^2 x^2\right )+2 a c^3 d^3 e^2 \left (104 d^3+477 d^2 e x+567 d e^2 x^2+210 e^3 x^3\right )+c^4 d^4 \left (-16 d^4+144 d^3 e x+693 d^2 e^2 x^2+840 d e^3 x^3+315 e^4 x^4\right )\right )}{c^3 d^3 \left (c d^2-a e^2\right )^5 (d+e x)^3}+\frac {315 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 )^{11/2}}\right )}{24 ((a e+c d x) (d+e x))^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]

[Out]

(c^3*d^3*(d + e*x)^(5/2)*(((a*e + c*d*x)*(8*a^4*e^8 - 2*a^3*c*d*e^6*(25*d + 9*e*x) + 3*a^2*c^2*d^2*e^4*(55*d^2
+ 60*d*e*x + 21*e^2*x^2) + 2*a*c^3*d^3*e^2*(104*d^3 + 477*d^2*e*x + 567*d*e^2*x^2 + 210*e^3*x^3) + c^4*d^4*(-
16*d^4 + 144*d^3*e*x + 693*d^2*e^2*x^2 + 840*d*e^3*x^3 + 315*e^4*x^4)))/(c^3*d^3*(c*d^2 - a*e^2)^5*(d + e*x)^3
) + (315*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)^
(11/2)))/(24*((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. \(919\) vs. \(2(351)=702\).
time = 0.75, size = 920, normalized size = 2.33

method	result	size
default	\(\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (315 \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^{4} d^{4} e^{5} x^{4}+315 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) a \,c^{3} d^{3} e^{6} x^{3} \sqrt {c d x +a e}+945 \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^{4} d^{5} e^{4} x^{3}+945 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) a \,c^{3} d^{4} e^{5} x^{2} \sqrt {c d x +a e}+945 \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^{4} d^{6} e^{3} x^{2}-315 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{4} d^{4} e^{4} x^{4}+945 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) a \,c^{3} d^{5} e^{4} x \sqrt {c d x +a e}+315 \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^{4} d^{7} e^{2} x -420 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a \,c^{3} d^{3} e^{5} x^{3}-840 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{4} d^{5} e^{3} x^{3}+315 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) a \,c^{3} d^{6} e^{3} \sqrt {c d x +a e}-63 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{2} c^{2} d^{2} e^{6} x^{2}-1134 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a \,c^{3} d^{4} e^{4} x^{2}-693 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{4} d^{6} e^{2} x^{2}+18 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{3} c d \,e^{7} x -180 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{2} c^{2} d^{3} e^{5} x -954 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a \,c^{3} d^{5} e^{3} x -144 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{4} d^{7} e x -8 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{4} e^{8}+50 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{3} c \,d^{2} e^{6}-165 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{2} c^{2} d^{4} e^{4}-208 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a \,c^{3} d^{6} e^{2}+16 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{4} d^{8}\right )}{24 \left (e x +d \right )^{\frac {7}{2}} \left (c d x +a e \right )^{2} \left (e^{2} a -c \,d^{2}\right )^{5} \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\)	\(920\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/24*((c*d*x+a*e)*(e*x+d))^(1/2)*(315*(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
^4*d^4*e^5*x^4+315*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*a*c^3*d^3*e^6*x^3*(c*d*x+a*e)^(1/2)+94
5*(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^4*d^5*e^4*x^3+945*arctanh(e*(c*d*x+
a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*a*c^3*d^4*e^5*x^2*(c*d*x+a*e)^(1/2)+945*(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^4*d^6*e^3*x^2-315*((a*e^2-c*d^2)*e)^(1/2)*c^4*d^4*e^4*x^4+945*arctanh(e
*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*a*c^3*d^5*e^4*x*(c*d*x+a*e)^(1/2)+315*(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^4*d^7*e^2*x-420*((a*e^2-c*d^2)*e)^(1/2)*a*c^3*d^3*e^5*x^3-840*((a
*e^2-c*d^2)*e)^(1/2)*c^4*d^5*e^3*x^3+315*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*a*c^3*d^6*e^3*(c
*d*x+a*e)^(1/2)-63*((a*e^2-c*d^2)*e)^(1/2)*a^2*c^2*d^2*e^6*x^2-1134*((a*e^2-c*d^2)*e)^(1/2)*a*c^3*d^4*e^4*x^2-
693*((a*e^2-c*d^2)*e)^(1/2)*c^4*d^6*e^2*x^2+18*((a*e^2-c*d^2)*e)^(1/2)*a^3*c*d*e^7*x-180*((a*e^2-c*d^2)*e)^(1/
2)*a^2*c^2*d^3*e^5*x-954*((a*e^2-c*d^2)*e)^(1/2)*a*c^3*d^5*e^3*x-144*((a*e^2-c*d^2)*e)^(1/2)*c^4*d^7*e*x-8*((a
*e^2-c*d^2)*e)^(1/2)*a^4*e^8+50*((a*e^2-c*d^2)*e)^(1/2)*a^3*c*d^2*e^6-165*((a*e^2-c*d^2)*e)^(1/2)*a^2*c^2*d^4*
e^4-208*((a*e^2-c*d^2)*e)^(1/2)*a*c^3*d^6*e^2+16*((a*e^2-c*d^2)*e)^(1/2)*c^4*d^8)/(e*x+d)^(7/2)/(c*d*x+a*e)^2/
(a*e^2-c*d^2)^5/((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)^(3/2)/(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)*(x*e + d)^(3/2)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1170 vs. \(2 (356) = 712\).
time = 4.46, size = 2378, normalized size = 6.02 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="fricas")

[Out]

[1/48*(315*(c^5*d^9*x^2*e + a^2*c^3*d^3*x^4*e^7 + 2*(a*c^4*d^4*x^5 + 2*a^2*c^3*d^4*x^3)*e^6 + (c^5*d^5*x^6 + 8
*a*c^4*d^5*x^4 + 6*a^2*c^3*d^5*x^2)*e^5 + 4*(c^5*d^6*x^5 + 3*a*c^4*d^6*x^3 + a^2*c^3*d^6*x)*e^4 + (6*c^5*d^7*x
^4 + 8*a*c^4*d^7*x^2 + a^2*c^3*d^7)*e^3 + 2*(2*c^5*d^8*x^3 + a*c^4*d^8*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*(144*c^4*d^7*x*e - 16*c^4*d^8 - 18*a^3*c*d*x
*e^7 + 8*a^4*e^8 + (63*a^2*c^2*d^2*x^2 - 50*a^3*c*d^2)*e^6 + 60*(7*a*c^3*d^3*x^3 + 3*a^2*c^2*d^3*x)*e^5 + 3*(1
05*c^4*d^4*x^4 + 378*a*c^3*d^4*x^2 + 55*a^2*c^2*d^4)*e^4 + 6*(140*c^4*d^5*x^3 + 159*a*c^3*d^5*x)*e^3 + (693*c^
4*d^6*x^2 + 208*a*c^3*d^6)*e^2)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(x*e + d))/(c^7*d^16*x^2 - a^7
*x^4*e^16 - 2*(a^6*c*d*x^5 + 2*a^7*d*x^3)*e^15 - (a^5*c^2*d^2*x^6 + 3*a^6*c*d^2*x^4 + 6*a^7*d^2*x^2)*e^14 + 2*
(3*a^5*c^2*d^3*x^5 + 4*a^6*c*d^3*x^3 - 2*a^7*d^3*x)*e^13 + (5*a^4*c^3*d^4*x^6 + 24*a^5*c^2*d^4*x^4 + 22*a^6*c*
d^4*x^2 - a^7*d^4)*e^12 + 2*(8*a^5*c^2*d^5*x^3 + 9*a^6*c*d^5*x)*e^11 - (10*a^3*c^4*d^6*x^6 + 40*a^4*c^3*d^6*x^
4 + 21*a^5*c^2*d^6*x^2 - 5*a^6*c*d^6)*e^10 - 10*(2*a^3*c^4*d^7*x^5 + 6*a^4*c^3*d^7*x^3 + 3*a^5*c^2*d^7*x)*e^9
+ 5*(2*a^2*c^5*d^8*x^6 + 3*a^3*c^4*d^8*x^4 - 3*a^4*c^3*d^8*x^2 - 2*a^5*c^2*d^8)*e^8 + 10*(3*a^2*c^5*d^9*x^5 +
6*a^3*c^4*d^9*x^3 + 2*a^4*c^3*d^9*x)*e^7 - (5*a*c^6*d^10*x^6 - 21*a^2*c^5*d^10*x^4 - 40*a^3*c^4*d^10*x^2 - 10*
a^4*c^3*d^10)*e^6 - 2*(9*a*c^6*d^11*x^5 + 8*a^2*c^5*d^11*x^3)*e^5 + (c^7*d^12*x^6 - 22*a*c^6*d^12*x^4 - 24*a^2
*c^5*d^12*x^2 - 5*a^3*c^4*d^12)*e^4 + 2*(2*c^7*d^13*x^5 - 4*a*c^6*d^13*x^3 - 3*a^2*c^5*d^13*x)*e^3 + (6*c^7*d^
14*x^4 + 3*a*c^6*d^14*x^2 + a^2*c^5*d^14)*e^2 + 2*(2*c^7*d^15*x^3 + a*c^6*d^15*x)*e), 1/24*(315*(c^5*d^9*x^2*e
+ a^2*c^3*d^3*x^4*e^7 + 2*(a*c^4*d^4*x^5 + 2*a^2*c^3*d^4*x^3)*e^6 + (c^5*d^5*x^6 + 8*a*c^4*d^5*x^4 + 6*a^2*c^
3*d^5*x^2)*e^5 + 4*(c^5*d^6*x^5 + 3*a*c^4*d^6*x^3 + a^2*c^3*d^6*x)*e^4 + (6*c^5*d^7*x^4 + 8*a*c^4*d^7*x^2 + a^
2*c^3*d^7)*e^3 + 2*(2*c^5*d^8*x^3 + a*c^4*d^8*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)
+ (144*c^4*d^7*x*e - 16*c^4*d^8 - 18*a^3*c*d*x*e^7 + 8*a^4*e^8 + (63*a^2*c^2*d^2*x^2 - 50*a^3*c*d^2)*e^6 + 60
*(7*a*c^3*d^3*x^3 + 3*a^2*c^2*d^3*x)*e^5 + 3*(105*c^4*d^4*x^4 + 378*a*c^3*d^4*x^2 + 55*a^2*c^2*d^4)*e^4 + 6*(1
40*c^4*d^5*x^3 + 159*a*c^3*d^5*x)*e^3 + (693*c^4*d^6*x^2 + 208*a*c^3*d^6)*e^2)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x
^2 + a*d)*e)*sqrt(x*e + d))/(c^7*d^16*x^2 - a^7*x^4*e^16 - 2*(a^6*c*d*x^5 + 2*a^7*d*x^3)*e^15 - (a^5*c^2*d^2*x
^6 + 3*a^6*c*d^2*x^4 + 6*a^7*d^2*x^2)*e^14 + 2*(3*a^5*c^2*d^3*x^5 + 4*a^6*c*d^3*x^3 - 2*a^7*d^3*x)*e^13 + (5*a
^4*c^3*d^4*x^6 + 24*a^5*c^2*d^4*x^4 + 22*a^6*c*d^4*x^2 - a^7*d^4)*e^12 + 2*(8*a^5*c^2*d^5*x^3 + 9*a^6*c*d^5*x)
*e^11 - (10*a^3*c^4*d^6*x^6 + 40*a^4*c^3*d^6*x^4 + 21*a^5*c^2*d^6*x^2 - 5*a^6*c*d^6)*e^10 - 10*(2*a^3*c^4*d^7*
x^5 + 6*a^4*c^3*d^7*x^3 + 3*a^5*c^2*d^7*x)*e^9 + 5*(2*a^2*c^5*d^8*x^6 + 3*a^3*c^4*d^8*x^4 - 3*a^4*c^3*d^8*x^2
- 2*a^5*c^2*d^8)*e^8 + 10*(3*a^2*c^5*d^9*x^5 + 6*a^3*c^4*d^9*x^3 + 2*a^4*c^3*d^9*x)*e^7 - (5*a*c^6*d^10*x^6 -
21*a^2*c^5*d^10*x^4 - 40*a^3*c^4*d^10*x^2 - 10*a^4*c^3*d^10)*e^6 - 2*(9*a*c^6*d^11*x^5 + 8*a^2*c^5*d^11*x^3)*e
^5 + (c^7*d^12*x^6 - 22*a*c^6*d^12*x^4 - 24*a^2*c^5*d^12*x^2 - 5*a^3*c^4*d^12)*e^4 + 2*(2*c^7*d^13*x^5 - 4*a*c
^6*d^13*x^3 - 3*a^2*c^5*d^13*x)*e^3 + (6*c^7*d^14*x^4 + 3*a*c^6*d^14*x^2 + a^2*c^5*d^14)*e^2 + 2*(2*c^7*d^15*x
^3 + a*c^6*d^15*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}} \left (d + e x\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 723 vs. \(2 (356) = 712\).
time = 2.26, size = 723, normalized size = 1.83 \begin {gather*} \frac {1}{24} \, {\left (\frac {315 \, c^{3} d^{3} \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^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}\right )} \sqrt {c d^{2} e - a e^{3}}} - \frac {16 \, c^{7} d^{11} e^{5} - 64 \, a c^{6} d^{9} e^{7} - 144 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )} c^{6} d^{9} e^{4} + 96 \, a^{2} c^{5} d^{7} e^{9} + 432 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )} a c^{5} d^{7} e^{6} - 693 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{2} c^{5} d^{7} e^{3} - 64 \, a^{3} c^{4} d^{5} e^{11} - 432 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )} a^{2} c^{4} d^{5} e^{8} + 1386 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{2} a c^{4} d^{5} e^{5} - 840 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{3} c^{4} d^{5} e^{2} + 16 \, a^{4} c^{3} d^{3} e^{13} + 144 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )} a^{3} c^{3} d^{3} e^{10} - 693 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{2} a^{2} c^{3} d^{3} e^{7} + 840 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{3} a c^{3} d^{3} e^{4} - 315 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{4} c^{3} d^{3} e}{{\left (c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}\right )} {\left (\sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} c d^{2} e - \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a e^{3} + {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}}\right )}^{3}}\right )} e \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="giac")

[Out]

1/24*(315*c^3*d^3*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^5*d^10 - 5*a*c^4
*d^8*e^2 + 10*a^2*c^3*d^6*e^4 - 10*a^3*c^2*d^4*e^6 + 5*a^4*c*d^2*e^8 - a^5*e^10)*sqrt(c*d^2*e - a*e^3)) - (16*
c^7*d^11*e^5 - 64*a*c^6*d^9*e^7 - 144*((x*e + d)*c*d*e - c*d^2*e + a*e^3)*c^6*d^9*e^4 + 96*a^2*c^5*d^7*e^9 + 4
32*((x*e + d)*c*d*e - c*d^2*e + a*e^3)*a*c^5*d^7*e^6 - 693*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^2*c^5*d^7*e^3 -
64*a^3*c^4*d^5*e^11 - 432*((x*e + d)*c*d*e - c*d^2*e + a*e^3)*a^2*c^4*d^5*e^8 + 1386*((x*e + d)*c*d*e - c*d^2
*e + a*e^3)^2*a*c^4*d^5*e^5 - 840*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^3*c^4*d^5*e^2 + 16*a^4*c^3*d^3*e^13 + 14
4*((x*e + d)*c*d*e - c*d^2*e + a*e^3)*a^3*c^3*d^3*e^10 - 693*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^2*a^2*c^3*d^3
*e^7 + 840*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^3*a*c^3*d^3*e^4 - 315*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^4*c^3
*d^3*e)/((c^5*d^10 - 5*a*c^4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 - 10*a^3*c^2*d^4*e^6 + 5*a^4*c*d^2*e^8 - a^5*e^10)*(
sqrt((x*e + d)*c*d*e - c*d^2*e + a*e^3)*c*d^2*e - sqrt((x*e + d)*c*d*e - c*d^2*e + a*e^3)*a*e^3 + ((x*e + d)*c
*d*e - c*d^2*e + a*e^3)^(3/2))^3))*e

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (d+e\,x\right )}^{3/2}\,{\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)^(3/2)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)),x)

[Out]

int(1/((d + e*x)^(3/2)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)), x)

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