3.21.71 \(\int \frac {1}{(d+e x)^{5/2} (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [2071]

Optimal. Leaf size=331 \[ \frac {1}{3 \left (c d^2-a e^2\right ) (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {7 c d}{12 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 c^2 d^2}{24 \left (c d^2-a e^2\right )^3 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {35 c^3 d^3 \sqrt {d+e x}}{8 \left (c d^2-a e^2\right )^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {35 c^3 d^3 \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{8 \left (c d^2-a e^2\right )^{9/2}} \]

[Out]

-35/8*c^3*d^3*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)^(9/2)+1/3/(-a*e^2+c*d^2)/(e*x+d)^(5/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+7/12*c*d/(-a*
e^2+c*d^2)^2/(e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+35/24*c^2*d^2/(-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)-35/8*c^3*d^3*(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 = 0.17, antiderivative size = 331, 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^3 d^3 \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 )}{8 \left (c d^2-a e^2\right )^{9/2}}-\frac {35 c^3 d^3 \sqrt {d+e x}}{8 \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^2 d^2}{24 \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}{12 (d+e x)^{3/2} \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}{3 (d+e x)^{5/2} \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)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]

[Out]

1/(3*(c*d^2 - a*e^2)*(d + e*x)^(5/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (7*c*d)/(12*(c*d^2 - a*e^2
)^2*(d + e*x)^(3/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (35*c^2*d^2)/(24*(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^3*d^3*Sqrt[d + e*x])/(8*(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^3*d^3*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])])/(8*(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}{(d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx &=\frac {1}{3 \left (c d^2-a e^2\right ) (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {(7 c d) \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}{6 \left (c d^2-a e^2\right )}\\ &=\frac {1}{3 \left (c d^2-a e^2\right ) (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {7 c d}{12 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2} \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\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}{24 \left (c d^2-a e^2\right )^2}\\ &=\frac {1}{3 \left (c d^2-a e^2\right ) (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {7 c d}{12 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 c^2 d^2}{24 \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^3 d^3\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 )^3}\\ &=\frac {1}{3 \left (c d^2-a e^2\right ) (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {7 c d}{12 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 c^2 d^2}{24 \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^3 d^3 \sqrt {d+e x}}{8 \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^3 d^3 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}{16 \left (c d^2-a e^2\right )^4}\\ &=\frac {1}{3 \left (c d^2-a e^2\right ) (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {7 c d}{12 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 c^2 d^2}{24 \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^3 d^3 \sqrt {d+e x}}{8 \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^3 d^3 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 )}{8 \left (c d^2-a e^2\right )^4}\\ &=\frac {1}{3 \left (c d^2-a e^2\right ) (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {7 c d}{12 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 c^2 d^2}{24 \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^3 d^3 \sqrt {d+e x}}{8 \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^3 d^3 \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 )}{8 \left (c d^2-a e^2\right )^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 0.79, size = 227, normalized size = 0.69 \begin {gather*} \frac {-\sqrt {c d^2-a e^2} \left (8 a^3 e^6-2 a^2 c d e^4 (19 d+7 e x)+a c^2 d^2 e^2 \left (87 d^2+98 d e x+35 e^2 x^2\right )+c^3 d^3 \left (48 d^3+231 d^2 e x+280 d e^2 x^2+105 e^3 x^3\right )\right )-105 c^3 d^3 \sqrt {e} \sqrt {a e+c d x} (d+e x)^3 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2}}\right )}{24 \left (c d^2-a e^2\right )^{9/2} (d+e x)^{5/2} \sqrt {(a e+c d x) (d+e x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^(5/2)*(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]*(8*a^3*e^6 - 2*a^2*c*d*e^4*(19*d + 7*e*x) + a*c^2*d^2*e^2*(87*d^2 + 98*d*e*x + 35*e^2*x
^2) + c^3*d^3*(48*d^3 + 231*d^2*e*x + 280*d*e^2*x^2 + 105*e^3*x^3))) - 105*c^3*d^3*Sqrt[e]*Sqrt[a*e + c*d*x]*(
d + e*x)^3*ArcTan[(Sqrt[e]*Sqrt[a*e + c*d*x])/Sqrt[c*d^2 - a*e^2]])/(24*(c*d^2 - a*e^2)^(9/2)*(d + e*x)^(5/2)*
Sqrt[(a*e + c*d*x)*(d + e*x)])

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Maple [A]
time = 0.75, size = 549, normalized size = 1.66

method	result	size
default	\(\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (105 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) \sqrt {c d x +a e}\, c^{3} d^{3} e^{4} x^{3}+315 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) \sqrt {c d x +a e}\, c^{3} d^{4} e^{3} x^{2}+315 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) \sqrt {c d x +a e}\, c^{3} d^{5} e^{2} x -105 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{3} d^{3} e^{3} x^{3}+105 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) \sqrt {c d x +a e}\, c^{3} d^{6} e -35 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a \,c^{2} d^{2} e^{4} x^{2}-280 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{3} d^{4} e^{2} x^{2}+14 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{2} c d \,e^{5} x -98 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a \,c^{2} d^{3} e^{3} x -231 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{3} d^{5} e x -8 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{3} e^{6}+38 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{2} c \,d^{2} e^{4}-87 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a \,c^{2} d^{4} e^{2}-48 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{3} d^{6}\right )}{24 \left (e x +d \right )^{\frac {7}{2}} \left (c d x +a e \right ) \left (e^{2} a -c \,d^{2}\right )^{4} \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\)	\(549\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*((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+315*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+315*
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))*(c*d*x+a*e)^(1/2)*c^3*d^6*e-35*((a
*e^2-c*d^2)*e)^(1/2)*a*c^2*d^2*e^4*x^2-280*((a*e^2-c*d^2)*e)^(1/2)*c^3*d^4*e^2*x^2+14*((a*e^2-c*d^2)*e)^(1/2)*
a^2*c*d*e^5*x-98*((a*e^2-c*d^2)*e)^(1/2)*a*c^2*d^3*e^3*x-231*((a*e^2-c*d^2)*e)^(1/2)*c^3*d^5*e*x-8*((a*e^2-c*d
^2)*e)^(1/2)*a^3*e^6+38*((a*e^2-c*d^2)*e)^(1/2)*a^2*c*d^2*e^4-87*((a*e^2-c*d^2)*e)^(1/2)*a*c^2*d^4*e^2-48*((a*
e^2-c*d^2)*e)^(1/2)*c^3*d^6)/(e*x+d)^(7/2)/(c*d*x+a*e)/(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)^(5/2)/(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)^(5/2)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 763 vs. \(2 (295) = 590\).
time = 3.57, size = 1564, normalized size = 4.73 \begin {gather*} \left [\frac {105 \, {\left (c^{4} d^{8} x + a c^{3} d^{3} x^{4} e^{5} + {\left (c^{4} d^{4} x^{5} + 4 \, a c^{3} d^{4} x^{3}\right )} e^{4} + 2 \, {\left (2 \, c^{4} d^{5} x^{4} + 3 \, a c^{3} d^{5} x^{2}\right )} e^{3} + 2 \, {\left (3 \, c^{4} d^{6} x^{3} + 2 \, a c^{3} d^{6} x\right )} e^{2} + {\left (4 \, c^{4} d^{7} x^{2} + a c^{3} d^{7}\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 (231 \, c^{3} d^{5} x e + 48 \, c^{3} d^{6} - 14 \, a^{2} c d x e^{5} + 8 \, a^{3} e^{6} + {\left (35 \, a c^{2} d^{2} x^{2} - 38 \, a^{2} c d^{2}\right )} e^{4} + 7 \, {\left (15 \, c^{3} d^{3} x^{3} + 14 \, a c^{2} d^{3} x\right )} e^{3} + {\left (280 \, c^{3} d^{4} x^{2} + 87 \, 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}}{48 \, {\left (6 \, c^{5} d^{11} x^{3} e^{2} + c^{5} d^{13} x + a^{5} x^{4} e^{13} + 6 \, a^{5} d^{2} x^{2} e^{11} + {\left (a^{4} c d x^{5} + 4 \, a^{5} d x^{3}\right )} e^{12} - 2 \, {\left (2 \, a^{3} c^{2} d^{3} x^{5} + 5 \, a^{4} c d^{3} x^{3} - 2 \, a^{5} d^{3} x\right )} e^{10} - {\left (10 \, a^{3} c^{2} d^{4} x^{4} + 20 \, a^{4} c d^{4} x^{2} - a^{5} d^{4}\right )} e^{9} + 3 \, {\left (2 \, a^{2} c^{3} d^{5} x^{5} - 5 \, a^{4} c d^{5} x\right )} e^{8} + 4 \, {\left (5 \, a^{2} c^{3} d^{6} x^{4} + 5 \, a^{3} c^{2} d^{6} x^{2} - a^{4} c d^{6}\right )} e^{7} - 4 \, {\left (a c^{4} d^{7} x^{5} - 5 \, a^{2} c^{3} d^{7} x^{3} - 5 \, a^{3} c^{2} d^{7} x\right )} e^{6} - 3 \, {\left (5 \, a c^{4} d^{8} x^{4} - 2 \, a^{3} c^{2} d^{8}\right )} e^{5} + {\left (c^{5} d^{9} x^{5} - 20 \, a c^{4} d^{9} x^{3} - 10 \, a^{2} c^{3} d^{9} x\right )} e^{4} + 2 \, {\left (2 \, c^{5} d^{10} x^{4} - 5 \, a c^{4} d^{10} x^{2} - 2 \, a^{2} c^{3} d^{10}\right )} e^{3} + {\left (4 \, c^{5} d^{12} x^{2} + a c^{4} d^{12}\right )} e\right )}}, -\frac {\frac {105 \, {\left (c^{4} d^{8} x + a c^{3} d^{3} x^{4} e^{5} + {\left (c^{4} d^{4} x^{5} + 4 \, a c^{3} d^{4} x^{3}\right )} e^{4} + 2 \, {\left (2 \, c^{4} d^{5} x^{4} + 3 \, a c^{3} d^{5} x^{2}\right )} e^{3} + 2 \, {\left (3 \, c^{4} d^{6} x^{3} + 2 \, a c^{3} d^{6} x\right )} e^{2} + {\left (4 \, c^{4} d^{7} x^{2} + a c^{3} d^{7}\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 (231 \, c^{3} d^{5} x e + 48 \, c^{3} d^{6} - 14 \, a^{2} c d x e^{5} + 8 \, a^{3} e^{6} + {\left (35 \, a c^{2} d^{2} x^{2} - 38 \, a^{2} c d^{2}\right )} e^{4} + 7 \, {\left (15 \, c^{3} d^{3} x^{3} + 14 \, a c^{2} d^{3} x\right )} e^{3} + {\left (280 \, c^{3} d^{4} x^{2} + 87 \, 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 (6 \, c^{5} d^{11} x^{3} e^{2} + c^{5} d^{13} x + a^{5} x^{4} e^{13} + 6 \, a^{5} d^{2} x^{2} e^{11} + {\left (a^{4} c d x^{5} + 4 \, a^{5} d x^{3}\right )} e^{12} - 2 \, {\left (2 \, a^{3} c^{2} d^{3} x^{5} + 5 \, a^{4} c d^{3} x^{3} - 2 \, a^{5} d^{3} x\right )} e^{10} - {\left (10 \, a^{3} c^{2} d^{4} x^{4} + 20 \, a^{4} c d^{4} x^{2} - a^{5} d^{4}\right )} e^{9} + 3 \, {\left (2 \, a^{2} c^{3} d^{5} x^{5} - 5 \, a^{4} c d^{5} x\right )} e^{8} + 4 \, {\left (5 \, a^{2} c^{3} d^{6} x^{4} + 5 \, a^{3} c^{2} d^{6} x^{2} - a^{4} c d^{6}\right )} e^{7} - 4 \, {\left (a c^{4} d^{7} x^{5} - 5 \, a^{2} c^{3} d^{7} x^{3} - 5 \, a^{3} c^{2} d^{7} x\right )} e^{6} - 3 \, {\left (5 \, a c^{4} d^{8} x^{4} - 2 \, a^{3} c^{2} d^{8}\right )} e^{5} + {\left (c^{5} d^{9} x^{5} - 20 \, a c^{4} d^{9} x^{3} - 10 \, a^{2} c^{3} d^{9} x\right )} e^{4} + 2 \, {\left (2 \, c^{5} d^{10} x^{4} - 5 \, a c^{4} d^{10} x^{2} - 2 \, a^{2} c^{3} d^{10}\right )} e^{3} + {\left (4 \, c^{5} d^{12} x^{2} + a c^{4} d^{12}\right )} e\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/48*(105*(c^4*d^8*x + a*c^3*d^3*x^4*e^5 + (c^4*d^4*x^5 + 4*a*c^3*d^4*x^3)*e^4 + 2*(2*c^4*d^5*x^4 + 3*a*c^3*d
^5*x^2)*e^3 + 2*(3*c^4*d^6*x^3 + 2*a*c^3*d^6*x)*e^2 + (4*c^4*d^7*x^2 + a*c^3*d^7)*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*(231*c^3*d^5*x*e + 48*c^3*d^6 - 14*a^2
*c*d*x*e^5 + 8*a^3*e^6 + (35*a*c^2*d^2*x^2 - 38*a^2*c*d^2)*e^4 + 7*(15*c^3*d^3*x^3 + 14*a*c^2*d^3*x)*e^3 + (28
0*c^3*d^4*x^2 + 87*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))/(6*c^5*d^11*x^3*
e^2 + c^5*d^13*x + a^5*x^4*e^13 + 6*a^5*d^2*x^2*e^11 + (a^4*c*d*x^5 + 4*a^5*d*x^3)*e^12 - 2*(2*a^3*c^2*d^3*x^5
+ 5*a^4*c*d^3*x^3 - 2*a^5*d^3*x)*e^10 - (10*a^3*c^2*d^4*x^4 + 20*a^4*c*d^4*x^2 - a^5*d^4)*e^9 + 3*(2*a^2*c^3*
d^5*x^5 - 5*a^4*c*d^5*x)*e^8 + 4*(5*a^2*c^3*d^6*x^4 + 5*a^3*c^2*d^6*x^2 - a^4*c*d^6)*e^7 - 4*(a*c^4*d^7*x^5 -
5*a^2*c^3*d^7*x^3 - 5*a^3*c^2*d^7*x)*e^6 - 3*(5*a*c^4*d^8*x^4 - 2*a^3*c^2*d^8)*e^5 + (c^5*d^9*x^5 - 20*a*c^4*d
^9*x^3 - 10*a^2*c^3*d^9*x)*e^4 + 2*(2*c^5*d^10*x^4 - 5*a*c^4*d^10*x^2 - 2*a^2*c^3*d^10)*e^3 + (4*c^5*d^12*x^2
+ a*c^4*d^12)*e), -1/24*(105*(c^4*d^8*x + a*c^3*d^3*x^4*e^5 + (c^4*d^4*x^5 + 4*a*c^3*d^4*x^3)*e^4 + 2*(2*c^4*d
^5*x^4 + 3*a*c^3*d^5*x^2)*e^3 + 2*(3*c^4*d^6*x^3 + 2*a*c^3*d^6*x)*e^2 + (4*c^4*d^7*x^2 + a*c^3*d^7)*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) + (231*c^3*d^5*x*e + 48*c^3*d^6 - 14*a^2*c*d*x*e^5 + 8*a^3*e
^6 + (35*a*c^2*d^2*x^2 - 38*a^2*c*d^2)*e^4 + 7*(15*c^3*d^3*x^3 + 14*a*c^2*d^3*x)*e^3 + (280*c^3*d^4*x^2 + 87*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))/(6*c^5*d^11*x^3*e^2 + c^5*d^13*x + a
^5*x^4*e^13 + 6*a^5*d^2*x^2*e^11 + (a^4*c*d*x^5 + 4*a^5*d*x^3)*e^12 - 2*(2*a^3*c^2*d^3*x^5 + 5*a^4*c*d^3*x^3 -
2*a^5*d^3*x)*e^10 - (10*a^3*c^2*d^4*x^4 + 20*a^4*c*d^4*x^2 - a^5*d^4)*e^9 + 3*(2*a^2*c^3*d^5*x^5 - 5*a^4*c*d^
5*x)*e^8 + 4*(5*a^2*c^3*d^6*x^4 + 5*a^3*c^2*d^6*x^2 - a^4*c*d^6)*e^7 - 4*(a*c^4*d^7*x^5 - 5*a^2*c^3*d^7*x^3 -
5*a^3*c^2*d^7*x)*e^6 - 3*(5*a*c^4*d^8*x^4 - 2*a^3*c^2*d^8)*e^5 + (c^5*d^9*x^5 - 20*a*c^4*d^9*x^3 - 10*a^2*c^3*
d^9*x)*e^4 + 2*(2*c^5*d^10*x^4 - 5*a*c^4*d^10*x^2 - 2*a^2*c^3*d^10)*e^3 + (4*c^5*d^12*x^2 + a*c^4*d^12)*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 {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Giac [A]
time = 1.79, size = 498, normalized size = 1.50 \begin {gather*} -\frac {35 \, 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}{8 \, {\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 {2 \, c^{3} d^{3} 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 {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}}} - \frac {{\left (87 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} c^{5} d^{7} e^{3} - 174 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a c^{4} d^{5} e^{5} + 136 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{4} d^{5} e^{2} + 87 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a^{2} c^{3} d^{3} e^{7} - 136 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a c^{3} d^{3} e^{4} + 57 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} c^{3} d^{3} e\right )} e^{\left (-3\right )}}{24 \, {\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 )}^{3} c^{3} d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-35/8*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^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)) - 2*c^3*d^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((x*e + d)*c*d*e - c*d^2*e + a*e^3)) - 1/24*(87*s
qrt((x*e + d)*c*d*e - c*d^2*e + a*e^3)*c^5*d^7*e^3 - 174*sqrt((x*e + d)*c*d*e - c*d^2*e + a*e^3)*a*c^4*d^5*e^5
+ 136*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*c^4*d^5*e^2 + 87*sqrt((x*e + d)*c*d*e - c*d^2*e + a*e^3)*a^2*
c^3*d^3*e^7 - 136*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a*c^3*d^3*e^4 + 57*((x*e + d)*c*d*e - c*d^2*e + a*
e^3)^(5/2)*c^3*d^3*e)*e^(-3)/((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*c^3*d^3)

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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 )}^{5/2}\,{\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)^(5/2)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)),x)

[Out]

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

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