3.1962 \(\int \frac {1}{(d+e x)^3 (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\)

Optimal. Leaf size=241 \[ -\frac {128 c^3 d^3 \left (a e^2+c d^2+2 c d e x\right )}{35 \left (c d^2-a e^2\right )^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {32 c^2 d^2}{35 (d+e x) \left (c d^2-a e^2\right )^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {16 c d}{35 (d+e x)^2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {2}{7 (d+e x)^3 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

[Out]

2/7/(-a*e^2+c*d^2)/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+16/35*c*d/(-a*e^2+c*d^2)^2/(e*x+d)^2/(a*d
*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+32/35*c^2*d^2/(-a*e^2+c*d^2)^3/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(
1/2)-128/35*c^3*d^3*(2*c*d*e*x+a*e^2+c*d^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 = 0.11, antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {658, 613} \[ -\frac {128 c^3 d^3 \left (a e^2+c d^2+2 c d e x\right )}{35 \left (c d^2-a e^2\right )^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {32 c^2 d^2}{35 (d+e x) \left (c d^2-a e^2\right )^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {16 c d}{35 (d+e x)^2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {2}{7 (d+e x)^3 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

2/(7*(c*d^2 - a*e^2)*(d + e*x)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (16*c*d)/(35*(c*d^2 - a*e^2)^2
*(d + e*x)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (32*c^2*d^2)/(35*(c*d^2 - a*e^2)^3*(d + e*x)*Sqrt[
a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - (128*c^3*d^3*(c*d^2 + a*e^2 + 2*c*d*e*x))/(35*(c*d^2 - a*e^2)^5*Sqrt
[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])

Rule 613

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[(-2*(b + 2*c*x))/((b^2 - 4*a*c)*Sqrt[a + b*x
+ c*x^2]), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 658

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*Simplify[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, m, p}, x] && NeQ[b^2 - 4*a*c
, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[Simplify[m + 2*p + 2], 0]

Rubi steps

\begin {align*} \int \frac {1}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx &=\frac {2}{7 \left (c d^2-a e^2\right ) (d+e x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {(8 c d) \int \frac {1}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{7 \left (c d^2-a e^2\right )}\\ &=\frac {2}{7 \left (c d^2-a e^2\right ) (d+e x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {16 c d}{35 \left (c d^2-a e^2\right )^2 (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (48 c^2 d^2\right ) \int \frac {1}{(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}{35 \left (c d^2-a e^2\right )^2}\\ &=\frac {2}{7 \left (c d^2-a e^2\right ) (d+e x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {16 c d}{35 \left (c d^2-a e^2\right )^2 (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {32 c^2 d^2}{35 \left (c d^2-a e^2\right )^3 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (64 c^3 d^3\right ) \int \frac {1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{35 \left (c d^2-a e^2\right )^3}\\ &=\frac {2}{7 \left (c d^2-a e^2\right ) (d+e x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {16 c d}{35 \left (c d^2-a e^2\right )^2 (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {32 c^2 d^2}{35 \left (c d^2-a e^2\right )^3 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {128 c^3 d^3 \left (c d^2+a e^2+2 c d e x\right )}{35 \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}}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 193, normalized size = 0.80 \[ -\frac {2 \left (-5 a^4 e^8+4 a^3 c d e^6 (7 d+2 e x)-2 a^2 c^2 d^2 e^4 \left (35 d^2+28 d e x+8 e^2 x^2\right )+4 a c^3 d^3 e^2 \left (35 d^3+70 d^2 e x+56 d e^2 x^2+16 e^3 x^3\right )+c^4 d^4 \left (35 d^4+280 d^3 e x+560 d^2 e^2 x^2+448 d e^3 x^3+128 e^4 x^4\right )\right )}{35 (d+e x)^3 \left (c d^2-a e^2\right )^5 \sqrt {(d+e x) (a e+c d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(-5*a^4*e^8 + 4*a^3*c*d*e^6*(7*d + 2*e*x) - 2*a^2*c^2*d^2*e^4*(35*d^2 + 28*d*e*x + 8*e^2*x^2) + 4*a*c^3*d^
3*e^2*(35*d^3 + 70*d^2*e*x + 56*d*e^2*x^2 + 16*e^3*x^3) + c^4*d^4*(35*d^4 + 280*d^3*e*x + 560*d^2*e^2*x^2 + 44
8*d*e^3*x^3 + 128*e^4*x^4)))/(35*(c*d^2 - a*e^2)^5*(d + e*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)])

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fricas [B]  time = 34.78, size = 733, normalized size = 3.04 \[ -\frac {2 \, {\left (128 \, c^{4} d^{4} e^{4} x^{4} + 35 \, c^{4} d^{8} + 140 \, a c^{3} d^{6} e^{2} - 70 \, a^{2} c^{2} d^{4} e^{4} + 28 \, a^{3} c d^{2} e^{6} - 5 \, a^{4} e^{8} + 64 \, {\left (7 \, c^{4} d^{5} e^{3} + a c^{3} d^{3} e^{5}\right )} x^{3} + 16 \, {\left (35 \, c^{4} d^{6} e^{2} + 14 \, a c^{3} d^{4} e^{4} - a^{2} c^{2} d^{2} e^{6}\right )} x^{2} + 8 \, {\left (35 \, c^{4} d^{7} e + 35 \, a c^{3} d^{5} e^{3} - 7 \, a^{2} c^{2} d^{3} e^{5} + a^{3} c d e^{7}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{35 \, {\left (a c^{5} d^{14} e - 5 \, a^{2} c^{4} d^{12} e^{3} + 10 \, a^{3} c^{3} d^{10} e^{5} - 10 \, a^{4} c^{2} d^{8} e^{7} + 5 \, a^{5} c d^{6} e^{9} - a^{6} d^{4} e^{11} + {\left (c^{6} d^{11} e^{4} - 5 \, a c^{5} d^{9} e^{6} + 10 \, a^{2} c^{4} d^{7} e^{8} - 10 \, a^{3} c^{3} d^{5} e^{10} + 5 \, a^{4} c^{2} d^{3} e^{12} - a^{5} c d e^{14}\right )} x^{5} + {\left (4 \, c^{6} d^{12} e^{3} - 19 \, a c^{5} d^{10} e^{5} + 35 \, a^{2} c^{4} d^{8} e^{7} - 30 \, a^{3} c^{3} d^{6} e^{9} + 10 \, a^{4} c^{2} d^{4} e^{11} + a^{5} c d^{2} e^{13} - a^{6} e^{15}\right )} x^{4} + 2 \, {\left (3 \, c^{6} d^{13} e^{2} - 13 \, a c^{5} d^{11} e^{4} + 20 \, a^{2} c^{4} d^{9} e^{6} - 10 \, a^{3} c^{3} d^{7} e^{8} - 5 \, a^{4} c^{2} d^{5} e^{10} + 7 \, a^{5} c d^{3} e^{12} - 2 \, a^{6} d e^{14}\right )} x^{3} + 2 \, {\left (2 \, c^{6} d^{14} e - 7 \, a c^{5} d^{12} e^{3} + 5 \, a^{2} c^{4} d^{10} e^{5} + 10 \, a^{3} c^{3} d^{8} e^{7} - 20 \, a^{4} c^{2} d^{6} e^{9} + 13 \, a^{5} c d^{4} e^{11} - 3 \, a^{6} d^{2} e^{13}\right )} x^{2} + {\left (c^{6} d^{15} - a c^{5} d^{13} e^{2} - 10 \, a^{2} c^{4} d^{11} e^{4} + 30 \, a^{3} c^{3} d^{9} e^{6} - 35 \, a^{4} c^{2} d^{7} e^{8} + 19 \, a^{5} c d^{5} e^{10} - 4 \, a^{6} d^{3} e^{12}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/35*(128*c^4*d^4*e^4*x^4 + 35*c^4*d^8 + 140*a*c^3*d^6*e^2 - 70*a^2*c^2*d^4*e^4 + 28*a^3*c*d^2*e^6 - 5*a^4*e^
8 + 64*(7*c^4*d^5*e^3 + a*c^3*d^3*e^5)*x^3 + 16*(35*c^4*d^6*e^2 + 14*a*c^3*d^4*e^4 - a^2*c^2*d^2*e^6)*x^2 + 8*
(35*c^4*d^7*e + 35*a*c^3*d^5*e^3 - 7*a^2*c^2*d^3*e^5 + a^3*c*d*e^7)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2
)*x)/(a*c^5*d^14*e - 5*a^2*c^4*d^12*e^3 + 10*a^3*c^3*d^10*e^5 - 10*a^4*c^2*d^8*e^7 + 5*a^5*c*d^6*e^9 - a^6*d^4
*e^11 + (c^6*d^11*e^4 - 5*a*c^5*d^9*e^6 + 10*a^2*c^4*d^7*e^8 - 10*a^3*c^3*d^5*e^10 + 5*a^4*c^2*d^3*e^12 - a^5*
c*d*e^14)*x^5 + (4*c^6*d^12*e^3 - 19*a*c^5*d^10*e^5 + 35*a^2*c^4*d^8*e^7 - 30*a^3*c^3*d^6*e^9 + 10*a^4*c^2*d^4
*e^11 + a^5*c*d^2*e^13 - a^6*e^15)*x^4 + 2*(3*c^6*d^13*e^2 - 13*a*c^5*d^11*e^4 + 20*a^2*c^4*d^9*e^6 - 10*a^3*c
^3*d^7*e^8 - 5*a^4*c^2*d^5*e^10 + 7*a^5*c*d^3*e^12 - 2*a^6*d*e^14)*x^3 + 2*(2*c^6*d^14*e - 7*a*c^5*d^12*e^3 +
5*a^2*c^4*d^10*e^5 + 10*a^3*c^3*d^8*e^7 - 20*a^4*c^2*d^6*e^9 + 13*a^5*c*d^4*e^11 - 3*a^6*d^2*e^13)*x^2 + (c^6*
d^15 - a*c^5*d^13*e^2 - 10*a^2*c^4*d^11*e^4 + 30*a^3*c^3*d^9*e^6 - 35*a^4*c^2*d^7*e^8 + 19*a^5*c*d^5*e^10 - 4*
a^6*d^3*e^12)*x)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Eval
uation time: 0.43Unable to transpose Error: Bad Argument Value

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maple [A]  time = 0.06, size = 307, normalized size = 1.27 \[ -\frac {2 \left (c d x +a e \right ) \left (-128 c^{4} d^{4} e^{4} x^{4}-64 a \,c^{3} d^{3} e^{5} x^{3}-448 c^{4} d^{5} e^{3} x^{3}+16 a^{2} c^{2} d^{2} e^{6} x^{2}-224 a \,c^{3} d^{4} e^{4} x^{2}-560 c^{4} d^{6} e^{2} x^{2}-8 a^{3} c d \,e^{7} x +56 a^{2} c^{2} d^{3} e^{5} x -280 a \,c^{3} d^{5} e^{3} x -280 c^{4} d^{7} e x +5 a^{4} e^{8}-28 a^{3} c \,d^{2} e^{6}+70 a^{2} c^{2} d^{4} e^{4}-140 a \,c^{3} d^{6} e^{2}-35 c^{4} d^{8}\right )}{35 \left (e x +d \right )^{2} \left (a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/35*(c*d*x+a*e)*(-128*c^4*d^4*e^4*x^4-64*a*c^3*d^3*e^5*x^3-448*c^4*d^5*e^3*x^3+16*a^2*c^2*d^2*e^6*x^2-224*a*
c^3*d^4*e^4*x^2-560*c^4*d^6*e^2*x^2-8*a^3*c*d*e^7*x+56*a^2*c^2*d^3*e^5*x-280*a*c^3*d^5*e^3*x-280*c^4*d^7*e*x+5
*a^4*e^8-28*a^3*c*d^2*e^6+70*a^2*c^2*d^4*e^4-140*a*c^3*d^6*e^2-35*c^4*d^8)/(e*x+d)^2/(a^5*e^10-5*a^4*c*d^2*e^8
+10*a^3*c^2*d^4*e^6-10*a^2*c^3*d^6*e^4+5*a*c^4*d^8*e^2-c^5*d^10)/(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(3/2)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*e^2-c*d^2>0)', see `assume?`
for more details)Is a*e^2-c*d^2 zero or nonzero?

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mupad [B]  time = 2.59, size = 2121, normalized size = 8.80 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(((d*((24*c^4*d^5*e^3)/(35*(a*e^2 - c*d^2)^4*(3*a^2*e^5 + 3*c^2*d^4*e - 6*a*c*d^2*e^3)) + (4*c^3*d^3*e^3*(17*a
*e^2 - 29*c*d^2))/(35*(a*e^2 - c*d^2)^4*(3*a^2*e^5 + 3*c^2*d^4*e - 6*a*c*d^2*e^3))))/e - (e^2*(70*c^4*d^6 - 25
6*a*c^3*d^4*e^2 + 162*a^2*c^2*d^2*e^4))/(35*(a*e^2 - c*d^2)^4*(3*a^2*e^5 + 3*c^2*d^4*e - 6*a*c*d^2*e^3)))*(x*(
a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^2 - (((e^2*(14*c^2*d^3 - 26*a*c*d*e^2))/(7*(a*e^2 - c*d^2
)^2*(5*a^2*e^5 + 5*c^2*d^4*e - 10*a*c*d^2*e^3)) + (12*c^2*d^3*e^2)/(7*(a*e^2 - c*d^2)^2*(5*a^2*e^5 + 5*c^2*d^4
*e - 10*a*c*d^2*e^3)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^3 - (((d*((16*c^5*d^6*e^2)/(35
*(a*e^2 - c*d^2)^7) + (16*c^4*d^4*e^2*(7*a*e^2 - 13*c*d^2))/(105*(a*e^2 - c*d^2)^7)))/e + (4*c^3*d^3*e*(23*c^2
*d^4 - 17*a^2*e^4 + 6*a*c*d^2*e^2))/(105*(a*e^2 - c*d^2)^7))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d
+ e*x) + (((24*c^3*d^4*e^2)/(35*(a*e^2 - c*d^2)^4*(3*a*e^3 - 3*c*d^2*e)) - (12*c^2*d^2*e^2*(a*e^2 + c*d^2))/(
35*(a*e^2 - c*d^2)^4*(3*a*e^3 - 3*c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^2 - (2*e
^2*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/((d + e*x)^4*(7*a^2*e^5 + 7*c^2*d^4*e - 14*a*c*d^2*e^3)) - (
((a*(((a*e^2 + c*d^2)*((16*c^7*d^7*e^4*(a*e^2 + c*d^2))/(35*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a
^2*c*d*e^5)) + (32*c^7*d^7*e^4*(4*a*e^2 - 13*c*d^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2
*c*d*e^5))))/(c*d*e) + (16*c^6*d^6*e^3*(27*a^2*e^4 + 61*c^2*d^4 - 70*a*c*d^2*e^2))/(105*(a*e^2 - c*d^2)^6*(c^3
*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (32*a*c^7*d^8*e^5)/(35*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e
^3 + a^2*c*d*e^5)) - (16*c^6*d^6*e^3*(a*e^2 + c*d^2)*(4*a*e^2 - 13*c*d^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e -
2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/c - x*((a*((16*c^7*d^7*e^4*(a*e^2 + c*d^2))/(35*(a*e^2 - c*d^2)^6*(c^3*d^5*
e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (32*c^7*d^7*e^4*(4*a*e^2 - 13*c*d^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e
- 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/c - ((a*e^2 + c*d^2)*(((a*e^2 + c*d^2)*((16*c^7*d^7*e^4*(a*e^2 + c*d^2))/(
35*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (32*c^7*d^7*e^4*(4*a*e^2 - 13*c*d^2))/(105
*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) + (16*c^6*d^6*e^3*(27*a^2*e^4 + 61*c
^2*d^4 - 70*a*c*d^2*e^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (32*a*c^7*d^8*
e^5)/(35*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (16*c^6*d^6*e^3*(a*e^2 + c*d^2)*(4*a
*e^2 - 13*c*d^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) - (2*c^2*d^2*e
^2*(210*c^6*d^9 - 1118*a*c^5*d^7*e^2 + 1398*a^2*c^4*d^5*e^4 - 538*a^3*c^3*d^3*e^6))/(105*(a*e^2 - c*d^2)^6*(c^
3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (8*c^5*d^5*e^2*(a*e^2 + c*d^2)*(27*a^2*e^4 + 61*c^2*d^4 - 70*a*c*d
^2*e^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))) + (c*d*e*(a*e^2 + c*d^2)*(210*c^
6*d^9 - 1118*a*c^5*d^7*e^2 + 1398*a^2*c^4*d^5*e^4 - 538*a^3*c^3*d^3*e^6))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e -
2*a*c^2*d^3*e^3 + a^2*c*d*e^5)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/((a*e + c*d*x)*(d + e*x)) - (1
6*c^3*d^3*e*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(35*(a*e^2 - c*d^2)^5*(d + e*x))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**3/(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), x)

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