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*(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])

________________________________________________________________________________________

Rubi [A]  time = 0.111302, antiderivative size = 241, normalized size of antiderivative = 1., 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 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]

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]

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*}

Mathematica [A]  time = 0.0893301, size = 193, normalized size = 0.8 \[ -\frac{2 \left (-2 a^2 c^2 d^2 e^4 \left (35 d^2+28 d e x+8 e^2 x^2\right )+4 a^3 c d e^6 (7 d+2 e x)-5 a^4 e^8+4 a c^3 d^3 e^2 \left (70 d^2 e x+35 d^3+56 d e^2 x^2+16 e^3 x^3\right )+c^4 d^4 \left (560 d^2 e^2 x^2+280 d^3 e x+35 d^4+448 d e^3 x^3+128 e^4 x^4\right )\right )}{35 (d+e x)^3 \left (c d^2-a e^2\right )^5 \sqrt{(d+e x) (a e+c d x)}} \]

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)])

________________________________________________________________________________________

Maple [A]  time = 0.051, size = 307, normalized size = 1.3 \begin{align*} -{\frac{ \left ( 2\,cdx+2\,ae \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}cd{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}ex+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 ({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 ( ex+d \right ) ^{2}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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

________________________________________________________________________________________

Fricas [B]  time = 115.071, size = 1509, normalized size = 6.26 \begin{align*} -\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 )}} \end{align*}

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)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, 1\right ] \end{align*}

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]

[undef, undef, undef, 1]