### 3.1971 $$\int \frac{1}{(d+e x) (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx$$

Optimal. Leaf size=192 $\frac{128 c^2 d^2 e \left (a e^2+c d^2+2 c d e x\right )}{15 \left (c d^2-a e^2\right )^5 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{16 c d \left (a e^2+c d^2+2 c d e x\right )}{15 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{2}{5 (d+e x) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}$

[Out]

2/(5*(c*d^2 - a*e^2)*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (16*c*d*(c*d^2 + a*e^2 + 2*c*d
*e*x))/(15*(c*d^2 - a*e^2)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (128*c^2*d^2*e*(c*d^2 + a*e^2 +
2*c*d*e*x))/(15*(c*d^2 - a*e^2)^5*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])

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Rubi [A]  time = 0.0624783, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 37, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.081, Rules used = {658, 614, 613} $\frac{128 c^2 d^2 e \left (a e^2+c d^2+2 c d e x\right )}{15 \left (c d^2-a e^2\right )^5 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{16 c d \left (a e^2+c d^2+2 c d e x\right )}{15 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{2}{5 (d+e x) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

2/(5*(c*d^2 - a*e^2)*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (16*c*d*(c*d^2 + a*e^2 + 2*c*d
*e*x))/(15*(c*d^2 - a*e^2)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (128*c^2*d^2*e*(c*d^2 + a*e^2 +
2*c*d*e*x))/(15*(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 614

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^(p + 1))/((p +
1)*(b^2 - 4*a*c)), x] - Dist[(2*c*(2*p + 3))/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]

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) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx &=\frac{2}{5 \left (c d^2-a e^2\right ) (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{(8 c d) \int \frac{1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx}{5 \left (c d^2-a e^2\right )}\\ &=\frac{2}{5 \left (c d^2-a e^2\right ) (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{16 c d \left (c d^2+a e^2+2 c d e x\right )}{15 \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 (64 c^2 d^2 e\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}{15 \left (c d^2-a e^2\right )^3}\\ &=\frac{2}{5 \left (c d^2-a e^2\right ) (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{16 c d \left (c d^2+a e^2+2 c d e x\right )}{15 \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{128 c^2 d^2 e \left (c d^2+a e^2+2 c d e x\right )}{15 \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 = 0.0958727, size = 193, normalized size = 1.01 $\frac{2 \left (6 a^2 c^2 d^2 e^4 \left (15 d^2+20 d e x+8 e^2 x^2\right )-4 a^3 c d e^6 (5 d+2 e x)+3 a^4 e^8+12 a c^3 d^3 e^2 \left (30 d^2 e x+5 d^3+40 d e^2 x^2+16 e^3 x^3\right )+c^4 d^4 \left (240 d^2 e^2 x^2+40 d^3 e x-5 d^4+320 d e^3 x^3+128 e^4 x^4\right )\right )}{15 (d+e x) \left (c d^2-a e^2\right )^5 ((d+e x) (a e+c d x))^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*(3*a^4*e^8 - 4*a^3*c*d*e^6*(5*d + 2*e*x) + 6*a^2*c^2*d^2*e^4*(15*d^2 + 20*d*e*x + 8*e^2*x^2) + 12*a*c^3*d^3
*e^2*(5*d^3 + 30*d^2*e*x + 40*d*e^2*x^2 + 16*e^3*x^3) + c^4*d^4*(-5*d^4 + 40*d^3*e*x + 240*d^2*e^2*x^2 + 320*d
*e^3*x^3 + 128*e^4*x^4)))/(15*(c*d^2 - a*e^2)^5*(d + e*x)*((a*e + c*d*x)*(d + e*x))^(3/2))

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Maple [A]  time = 0.057, size = 300, normalized size = 1.6 \begin{align*} -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 128\,{c}^{4}{d}^{4}{e}^{4}{x}^{4}+192\,a{c}^{3}{d}^{3}{e}^{5}{x}^{3}+320\,{c}^{4}{d}^{5}{e}^{3}{x}^{3}+48\,{a}^{2}{c}^{2}{d}^{2}{e}^{6}{x}^{2}+480\,a{c}^{3}{d}^{4}{e}^{4}{x}^{2}+240\,{c}^{4}{d}^{6}{e}^{2}{x}^{2}-8\,{a}^{3}cd{e}^{7}x+120\,{a}^{2}{c}^{2}{d}^{3}{e}^{5}x+360\,a{c}^{3}{d}^{5}{e}^{3}x+40\,{c}^{4}{d}^{7}ex+3\,{a}^{4}{e}^{8}-20\,{a}^{3}c{d}^{2}{e}^{6}+90\,{a}^{2}{c}^{2}{d}^{4}{e}^{4}+60\,a{c}^{3}{d}^{6}{e}^{2}-5\,{c}^{4}{d}^{8} \right ) }{15\,{a}^{5}{e}^{10}-75\,{a}^{4}c{d}^{2}{e}^{8}+150\,{a}^{3}{c}^{2}{d}^{4}{e}^{6}-150\,{a}^{2}{c}^{3}{d}^{6}{e}^{4}+75\,a{c}^{4}{d}^{8}{e}^{2}-15\,{c}^{5}{d}^{10}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{-{\frac{5}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15*(c*d*x+a*e)*(128*c^4*d^4*e^4*x^4+192*a*c^3*d^3*e^5*x^3+320*c^4*d^5*e^3*x^3+48*a^2*c^2*d^2*e^6*x^2+480*a*
c^3*d^4*e^4*x^2+240*c^4*d^6*e^2*x^2-8*a^3*c*d*e^7*x+120*a^2*c^2*d^3*e^5*x+360*a*c^3*d^5*e^3*x+40*c^4*d^7*e*x+3
*a^4*e^8-20*a^3*c*d^2*e^6+90*a^2*c^2*d^4*e^4+60*a*c^3*d^6*e^2-5*c^4*d^8)/(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)^(5/2)

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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)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 172.752, size = 1566, normalized size = 8.16 \begin{align*} \frac{2 \,{\left (128 \, c^{4} d^{4} e^{4} x^{4} - 5 \, c^{4} d^{8} + 60 \, a c^{3} d^{6} e^{2} + 90 \, a^{2} c^{2} d^{4} e^{4} - 20 \, a^{3} c d^{2} e^{6} + 3 \, a^{4} e^{8} + 64 \,{\left (5 \, c^{4} d^{5} e^{3} + 3 \, a c^{3} d^{3} e^{5}\right )} x^{3} + 48 \,{\left (5 \, c^{4} d^{6} e^{2} + 10 \, a c^{3} d^{4} e^{4} + a^{2} c^{2} d^{2} e^{6}\right )} x^{2} + 8 \,{\left (5 \, c^{4} d^{7} e + 45 \, a c^{3} d^{5} e^{3} + 15 \, 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}}{15 \,{\left (a^{2} c^{5} d^{13} e^{2} - 5 \, a^{3} c^{4} d^{11} e^{4} + 10 \, a^{4} c^{3} d^{9} e^{6} - 10 \, a^{5} c^{2} d^{7} e^{8} + 5 \, a^{6} c d^{5} e^{10} - a^{7} d^{3} e^{12} +{\left (c^{7} d^{12} e^{3} - 5 \, a c^{6} d^{10} e^{5} + 10 \, a^{2} c^{5} d^{8} e^{7} - 10 \, a^{3} c^{4} d^{6} e^{9} + 5 \, a^{4} c^{3} d^{4} e^{11} - a^{5} c^{2} d^{2} e^{13}\right )} x^{5} +{\left (3 \, c^{7} d^{13} e^{2} - 13 \, a c^{6} d^{11} e^{4} + 20 \, a^{2} c^{5} d^{9} e^{6} - 10 \, a^{3} c^{4} d^{7} e^{8} - 5 \, a^{4} c^{3} d^{5} e^{10} + 7 \, a^{5} c^{2} d^{3} e^{12} - 2 \, a^{6} c d e^{14}\right )} x^{4} +{\left (3 \, c^{7} d^{14} e - 9 \, a c^{6} d^{12} e^{3} + a^{2} c^{5} d^{10} e^{5} + 25 \, a^{3} c^{4} d^{8} e^{7} - 35 \, a^{4} c^{3} d^{6} e^{9} + 17 \, a^{5} c^{2} d^{4} e^{11} - a^{6} c d^{2} e^{13} - a^{7} e^{15}\right )} x^{3} +{\left (c^{7} d^{15} + a c^{6} d^{13} e^{2} - 17 \, a^{2} c^{5} d^{11} e^{4} + 35 \, a^{3} c^{4} d^{9} e^{6} - 25 \, a^{4} c^{3} d^{7} e^{8} - a^{5} c^{2} d^{5} e^{10} + 9 \, a^{6} c d^{3} e^{12} - 3 \, a^{7} d e^{14}\right )} x^{2} +{\left (2 \, a c^{6} d^{14} e - 7 \, a^{2} c^{5} d^{12} e^{3} + 5 \, a^{3} c^{4} d^{10} e^{5} + 10 \, a^{4} c^{3} d^{8} e^{7} - 20 \, a^{5} c^{2} d^{6} e^{9} + 13 \, a^{6} c d^{4} e^{11} - 3 \, a^{7} d^{2} e^{13}\right )} x\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(128*c^4*d^4*e^4*x^4 - 5*c^4*d^8 + 60*a*c^3*d^6*e^2 + 90*a^2*c^2*d^4*e^4 - 20*a^3*c*d^2*e^6 + 3*a^4*e^8 +
64*(5*c^4*d^5*e^3 + 3*a*c^3*d^3*e^5)*x^3 + 48*(5*c^4*d^6*e^2 + 10*a*c^3*d^4*e^4 + a^2*c^2*d^2*e^6)*x^2 + 8*(5
*c^4*d^7*e + 45*a*c^3*d^5*e^3 + 15*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^2*c^5*d^13*e^2 - 5*a^3*c^4*d^11*e^4 + 10*a^4*c^3*d^9*e^6 - 10*a^5*c^2*d^7*e^8 + 5*a^6*c*d^5*e^10 - a^7*d
^3*e^12 + (c^7*d^12*e^3 - 5*a*c^6*d^10*e^5 + 10*a^2*c^5*d^8*e^7 - 10*a^3*c^4*d^6*e^9 + 5*a^4*c^3*d^4*e^11 - a^
5*c^2*d^2*e^13)*x^5 + (3*c^7*d^13*e^2 - 13*a*c^6*d^11*e^4 + 20*a^2*c^5*d^9*e^6 - 10*a^3*c^4*d^7*e^8 - 5*a^4*c^
3*d^5*e^10 + 7*a^5*c^2*d^3*e^12 - 2*a^6*c*d*e^14)*x^4 + (3*c^7*d^14*e - 9*a*c^6*d^12*e^3 + a^2*c^5*d^10*e^5 +
25*a^3*c^4*d^8*e^7 - 35*a^4*c^3*d^6*e^9 + 17*a^5*c^2*d^4*e^11 - a^6*c*d^2*e^13 - a^7*e^15)*x^3 + (c^7*d^15 + a
*c^6*d^13*e^2 - 17*a^2*c^5*d^11*e^4 + 35*a^3*c^4*d^9*e^6 - 25*a^4*c^3*d^7*e^8 - a^5*c^2*d^5*e^10 + 9*a^6*c*d^3
*e^12 - 3*a^7*d*e^14)*x^2 + (2*a*c^6*d^14*e - 7*a^2*c^5*d^12*e^3 + 5*a^3*c^4*d^10*e^5 + 10*a^4*c^3*d^8*e^7 - 2
0*a^5*c^2*d^6*e^9 + 13*a^6*c*d^4*e^11 - 3*a^7*d^2*e^13)*x)

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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)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \left [\mathit{undef}, \mathit{undef}, \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)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="giac")

[Out]

[undef, undef, undef, undef, undef, 1]