3.21.70 \(\int \frac {1}{(d+e x)^{3/2} (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [2070]

Optimal. Leaf size=269 \[ \frac {1}{2 \left (c d^2-a e^2\right ) (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {5 c d}{4 \left (c d^2-a e^2\right )^2 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {15 c^2 d^2 \sqrt {d+e x}}{4 \left (c d^2-a e^2\right )^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {15 c^2 d^2 \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{4 \left (c d^2-a e^2\right )^{7/2}} \]

[Out]

-15/4*c^2*d^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))*e^(1/
2)/(-a*e^2+c*d^2)^(7/2)+1/2/(-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)^(1/2)+5/4*c*d/(-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)-15/4*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)^(1/2)

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Rubi [A]
time = 0.12, antiderivative size = 269, 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 = {686, 680, 674, 211} \begin {gather*} -\frac {15 c^2 d^2 \sqrt {e} \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 )^{7/2}}-\frac {15 c^2 d^2 \sqrt {d+e x}}{4 \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 c d}{4 \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 {1}{2 (d+e x)^{3/2} \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^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)^(3/2)),x]

[Out]

1/(2*(c*d^2 - a*e^2)*(d + e*x)^(3/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (5*c*d)/(4*(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]) - (15*c^2*d^2*Sqrt[d + e*x])/(4*(c*d^2 - a*e^2)^
3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - (15*c^2*d^2*Sqrt[e]*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)^(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 {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 )^{3/2}} \, dx &=\frac {1}{2 \left (c d^2-a e^2\right ) (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {(5 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 )^{3/2}} \, dx}{4 \left (c d^2-a e^2\right )}\\ &=\frac {1}{2 \left (c d^2-a e^2\right ) (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {5 c d}{4 \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 {\left (15 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 )^{3/2}} \, dx}{8 \left (c d^2-a e^2\right )^2}\\ &=\frac {1}{2 \left (c d^2-a e^2\right ) (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {5 c d}{4 \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 {15 c^2 d^2 \sqrt {d+e x}}{4 \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 (15 c^2 d^2 e\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 )^3}\\ &=\frac {1}{2 \left (c d^2-a e^2\right ) (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {5 c d}{4 \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 {15 c^2 d^2 \sqrt {d+e x}}{4 \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 (15 c^2 d^2 e^2\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 )^3}\\ &=\frac {1}{2 \left (c d^2-a e^2\right ) (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {5 c d}{4 \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 {15 c^2 d^2 \sqrt {d+e x}}{4 \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 {15 c^2 d^2 \sqrt {e} \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 )^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 0.45, size = 183, normalized size = 0.68 \begin {gather*} \frac {-\sqrt {c d^2-a e^2} \left (-2 a^2 e^4+a c d e^2 (9 d+5 e x)+c^2 d^2 \left (8 d^2+25 d e x+15 e^2 x^2\right )\right )-15 c^2 d^2 \sqrt {e} \sqrt {a e+c d x} (d+e x)^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2}}\right )}{4 \left (c d^2-a e^2\right )^{7/2} (d+e x)^{3/2} \sqrt {(a e+c d x) (d+e x)}} \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)^(3/2)),x]

[Out]

(-(Sqrt[c*d^2 - a*e^2]*(-2*a^2*e^4 + a*c*d*e^2*(9*d + 5*e*x) + c^2*d^2*(8*d^2 + 25*d*e*x + 15*e^2*x^2))) - 15*
c^2*d^2*Sqrt[e]*Sqrt[a*e + c*d*x]*(d + e*x)^2*ArcTan[(Sqrt[e]*Sqrt[a*e + c*d*x])/Sqrt[c*d^2 - a*e^2]])/(4*(c*d
^2 - a*e^2)^(7/2)*(d + e*x)^(3/2)*Sqrt[(a*e + c*d*x)*(d + e*x)])

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Maple [A]
time = 0.73, size = 374, normalized size = 1.39

method	result	size
default	\(-\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (15 \sqrt {c d x +a e}\, \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{2} d^{2} e^{3} x^{2}+30 \sqrt {c d x +a e}\, \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{2} d^{3} e^{2} x +15 \sqrt {c d x +a e}\, \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{2} d^{4} e -15 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{2} d^{2} e^{2} x^{2}-5 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a c d \,e^{3} x -25 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{2} d^{3} e x +2 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{2} e^{4}-9 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a c \,d^{2} e^{2}-8 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{2} d^{4}\right )}{4 \left (e x +d \right )^{\frac {5}{2}} \left (c d x +a e \right ) \left (e^{2} a -c \,d^{2}\right )^{3} \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\)	\(374\)

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)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/4/(e*x+d)^(5/2)*((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+30*(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*(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^4*e-15*((a*e^2-c*d^2)*e)
^(1/2)*c^2*d^2*e^2*x^2-5*((a*e^2-c*d^2)*e)^(1/2)*a*c*d*e^3*x-25*((a*e^2-c*d^2)*e)^(1/2)*c^2*d^3*e*x+2*((a*e^2-
c*d^2)*e)^(1/2)*a^2*e^4-9*((a*e^2-c*d^2)*e)^(1/2)*a*c*d^2*e^2-8*((a*e^2-c*d^2)*e)^(1/2)*c^2*d^4)/(c*d*x+a*e)/(
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(1/(e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="maxima")

[Out]

integrate(1/((c*d*x^2*e + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*(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. 540 vs. \(2 (238) = 476\).
time = 2.64, size = 1118, normalized size = 4.16 \begin {gather*} \left [\frac {15 \, {\left (c^{3} d^{6} x + a c^{2} d^{2} x^{3} e^{4} + {\left (c^{3} d^{3} x^{4} + 3 \, a c^{2} d^{3} x^{2}\right )} e^{3} + 3 \, {\left (c^{3} d^{4} x^{3} + a c^{2} d^{4} x\right )} e^{2} + {\left (3 \, c^{3} d^{5} x^{2} + a c^{2} d^{5}\right )} e\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 (25 \, c^{2} d^{3} x e + 8 \, c^{2} d^{4} + 5 \, a c d x e^{3} - 2 \, a^{2} e^{4} + 3 \, {\left (5 \, c^{2} d^{2} x^{2} + 3 \, 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}}{8 \, {\left (3 \, c^{4} d^{8} x^{3} e^{2} + c^{4} d^{10} x - a^{4} x^{3} e^{10} - 3 \, a^{4} d^{2} x e^{8} - {\left (a^{3} c d x^{4} + 3 \, a^{4} d x^{2}\right )} e^{9} + {\left (3 \, a^{2} c^{2} d^{3} x^{4} + 6 \, a^{3} c d^{3} x^{2} - a^{4} d^{3}\right )} e^{7} + 2 \, {\left (3 \, a^{2} c^{2} d^{4} x^{3} + 4 \, a^{3} c d^{4} x\right )} e^{6} - 3 \, {\left (a c^{3} d^{5} x^{4} - a^{3} c d^{5}\right )} e^{5} - 2 \, {\left (4 \, a c^{3} d^{6} x^{3} + 3 \, a^{2} c^{2} d^{6} x\right )} e^{4} + {\left (c^{4} d^{7} x^{4} - 6 \, a c^{3} d^{7} x^{2} - 3 \, a^{2} c^{2} d^{7}\right )} e^{3} + {\left (3 \, c^{4} d^{9} x^{2} + a c^{3} d^{9}\right )} e\right )}}, -\frac {\frac {15 \, {\left (c^{3} d^{6} x + a c^{2} d^{2} x^{3} e^{4} + {\left (c^{3} d^{3} x^{4} + 3 \, a c^{2} d^{3} x^{2}\right )} e^{3} + 3 \, {\left (c^{3} d^{4} x^{3} + a c^{2} d^{4} x\right )} e^{2} + {\left (3 \, c^{3} d^{5} x^{2} + a c^{2} d^{5}\right )} e\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 (25 \, c^{2} d^{3} x e + 8 \, c^{2} d^{4} + 5 \, a c d x e^{3} - 2 \, a^{2} e^{4} + 3 \, {\left (5 \, c^{2} d^{2} x^{2} + 3 \, 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}}{4 \, {\left (3 \, c^{4} d^{8} x^{3} e^{2} + c^{4} d^{10} x - a^{4} x^{3} e^{10} - 3 \, a^{4} d^{2} x e^{8} - {\left (a^{3} c d x^{4} + 3 \, a^{4} d x^{2}\right )} e^{9} + {\left (3 \, a^{2} c^{2} d^{3} x^{4} + 6 \, a^{3} c d^{3} x^{2} - a^{4} d^{3}\right )} e^{7} + 2 \, {\left (3 \, a^{2} c^{2} d^{4} x^{3} + 4 \, a^{3} c d^{4} x\right )} e^{6} - 3 \, {\left (a c^{3} d^{5} x^{4} - a^{3} c d^{5}\right )} e^{5} - 2 \, {\left (4 \, a c^{3} d^{6} x^{3} + 3 \, a^{2} c^{2} d^{6} x\right )} e^{4} + {\left (c^{4} d^{7} x^{4} - 6 \, a c^{3} d^{7} x^{2} - 3 \, a^{2} c^{2} d^{7}\right )} e^{3} + {\left (3 \, c^{4} d^{9} x^{2} + a c^{3} d^{9}\right )} e\right )}}\right ] \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)^(3/2),x, algorithm="fricas")

[Out]

[1/8*(15*(c^3*d^6*x + a*c^2*d^2*x^3*e^4 + (c^3*d^3*x^4 + 3*a*c^2*d^3*x^2)*e^3 + 3*(c^3*d^4*x^3 + a*c^2*d^4*x)*
e^2 + (3*c^3*d^5*x^2 + a*c^2*d^5)*e)*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*(25*c^2*d^3*x*e + 8*c^2*d^4 + 5*a*c*d*x*e^3 - 2*a^2*e^4 + 3*(5*c^2*d^2*x^2 + 3*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))/(3*c^4*d^8*x^3*e^2 + c^4*d^10*x - a^4*x^3*e^
10 - 3*a^4*d^2*x*e^8 - (a^3*c*d*x^4 + 3*a^4*d*x^2)*e^9 + (3*a^2*c^2*d^3*x^4 + 6*a^3*c*d^3*x^2 - a^4*d^3)*e^7 +
2*(3*a^2*c^2*d^4*x^3 + 4*a^3*c*d^4*x)*e^6 - 3*(a*c^3*d^5*x^4 - a^3*c*d^5)*e^5 - 2*(4*a*c^3*d^6*x^3 + 3*a^2*c^
2*d^6*x)*e^4 + (c^4*d^7*x^4 - 6*a*c^3*d^7*x^2 - 3*a^2*c^2*d^7)*e^3 + (3*c^4*d^9*x^2 + a*c^3*d^9)*e), -1/4*(15*
(c^3*d^6*x + a*c^2*d^2*x^3*e^4 + (c^3*d^3*x^4 + 3*a*c^2*d^3*x^2)*e^3 + 3*(c^3*d^4*x^3 + a*c^2*d^4*x)*e^2 + (3*
c^3*d^5*x^2 + a*c^2*d^5)*e)*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) + (25*c^2*d^3*x*e + 8*c^2
*d^4 + 5*a*c*d*x*e^3 - 2*a^2*e^4 + 3*(5*c^2*d^2*x^2 + 3*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))/(3*c^4*d^8*x^3*e^2 + c^4*d^10*x - a^4*x^3*e^10 - 3*a^4*d^2*x*e^8 - (a^3*c*d*x^4 + 3*a^4*d*x
^2)*e^9 + (3*a^2*c^2*d^3*x^4 + 6*a^3*c*d^3*x^2 - a^4*d^3)*e^7 + 2*(3*a^2*c^2*d^4*x^3 + 4*a^3*c*d^4*x)*e^6 - 3*
(a*c^3*d^5*x^4 - a^3*c*d^5)*e^5 - 2*(4*a*c^3*d^6*x^3 + 3*a^2*c^2*d^6*x)*e^4 + (c^4*d^7*x^4 - 6*a*c^3*d^7*x^2 -
3*a^2*c^2*d^7)*e^3 + (3*c^4*d^9*x^2 + a*c^3*d^9)*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 {3}{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)**(3/2),x)

[Out]

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

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Giac [A]
time = 1.19, size = 350, normalized size = 1.30 \begin {gather*} -\frac {15 \, 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}{4 \, {\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 \, c^{2} d^{2} 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 {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}}} - \frac {{\left (9 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} c^{3} d^{4} e^{2} - 9 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a c^{2} d^{2} e^{4} + 7 \, {\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 )}}{4 \, {\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 )}^{2} c^{2} d^{2}} \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)^(3/2),x, algorithm="giac")

[Out]

-15/4*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^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^2*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((x*e + d)*c*d*e - c*d^2*e + a*e^3)) - 1/4*(9*sqrt((x*e + d)*c*d*e - c*d^2*e + a*e^3)*c^3
*d^4*e^2 - 9*sqrt((x*e + d)*c*d*e - c*d^2*e + a*e^3)*a*c^2*d^2*e^4 + 7*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/
2)*c^2*d^2*e)*e^(-2)/((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)^2*c^2*d^2)

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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 )}^{3/2}} \,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)^(3/2)),x)

[Out]

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

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