### 3.1972 $$\int \frac{1}{(d+e x)^2 (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx$$

Optimal. Leaf size=252 $\frac{256 c^3 d^3 e \left (a e^2+c d^2+2 c d e x\right )}{21 \left (c d^2-a e^2\right )^6 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{32 c^2 d^2 \left (a e^2+c d^2+2 c d e x\right )}{21 \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{4 c d}{7 (d+e x) \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{2}{7 (d+e x)^2 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}$

[Out]

2/(7*(c*d^2 - a*e^2)*(d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (4*c*d)/(7*(c*d^2 - a*e^2)^2
*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (32*c^2*d^2*(c*d^2 + a*e^2 + 2*c*d*e*x))/(21*(c*d^
2 - a*e^2)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (256*c^3*d^3*e*(c*d^2 + a*e^2 + 2*c*d*e*x))/(21*
(c*d^2 - a*e^2)^6*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])

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Rubi [A]  time = 0.101749, antiderivative size = 252, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 37, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.081, Rules used = {658, 614, 613} $\frac{256 c^3 d^3 e \left (a e^2+c d^2+2 c d e x\right )}{21 \left (c d^2-a e^2\right )^6 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{32 c^2 d^2 \left (a e^2+c d^2+2 c d e x\right )}{21 \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{4 c d}{7 (d+e x) \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{2}{7 (d+e x)^2 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

2/(7*(c*d^2 - a*e^2)*(d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (4*c*d)/(7*(c*d^2 - a*e^2)^2
*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (32*c^2*d^2*(c*d^2 + a*e^2 + 2*c*d*e*x))/(21*(c*d^
2 - a*e^2)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (256*c^3*d^3*e*(c*d^2 + a*e^2 + 2*c*d*e*x))/(21*
(c*d^2 - a*e^2)^6*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)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx &=\frac{2}{7 \left (c d^2-a e^2\right ) (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}}+\frac{(10 c d) \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}{7 \left (c d^2-a e^2\right )}\\ &=\frac{2}{7 \left (c d^2-a e^2\right ) (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}}+\frac{4 c d}{7 \left (c d^2-a e^2\right )^2 (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{\left (16 c^2 d^2\right ) \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}{7 \left (c d^2-a e^2\right )^2}\\ &=\frac{2}{7 \left (c d^2-a e^2\right ) (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}}+\frac{4 c d}{7 \left (c d^2-a e^2\right )^2 (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{32 c^2 d^2 \left (c d^2+a e^2+2 c d e x\right )}{21 \left (c d^2-a e^2\right )^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac{\left (128 c^3 d^3 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}{21 \left (c d^2-a e^2\right )^4}\\ &=\frac{2}{7 \left (c d^2-a e^2\right ) (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}}+\frac{4 c d}{7 \left (c d^2-a e^2\right )^2 (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{32 c^2 d^2 \left (c d^2+a e^2+2 c d e x\right )}{21 \left (c d^2-a e^2\right )^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{256 c^3 d^3 e \left (c d^2+a e^2+2 c d e x\right )}{21 \left (c d^2-a e^2\right )^6 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ \end{align*}

Mathematica [A]  time = 0.125281, size = 259, normalized size = 1.03 $\frac{2 \left (-2 a^3 c^2 d^2 e^6 \left (35 d^2+28 d e x+8 e^2 x^2\right )+6 a^2 c^3 d^3 e^4 \left (70 d^2 e x+35 d^3+56 d e^2 x^2+16 e^3 x^3\right )+3 a^4 c d e^8 (7 d+2 e x)-3 a^5 e^{10}+3 a c^4 d^4 e^2 \left (560 d^2 e^2 x^2+280 d^3 e x+35 d^4+448 d e^3 x^3+128 e^4 x^4\right )+c^5 d^5 \left (560 d^3 e^2 x^2+1120 d^2 e^3 x^3+70 d^4 e x-7 d^5+896 d e^4 x^4+256 e^5 x^5\right )\right )}{21 (d+e x)^2 \left (c d^2-a e^2\right )^6 ((d+e x) (a e+c d x))^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*(-3*a^5*e^10 + 3*a^4*c*d*e^8*(7*d + 2*e*x) - 2*a^3*c^2*d^2*e^6*(35*d^2 + 28*d*e*x + 8*e^2*x^2) + 6*a^2*c^3*
d^3*e^4*(35*d^3 + 70*d^2*e*x + 56*d*e^2*x^2 + 16*e^3*x^3) + 3*a*c^4*d^4*e^2*(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) + c^5*d^5*(-7*d^5 + 70*d^4*e*x + 560*d^3*e^2*x^2 + 1120*d^2*e^3*x^3 + 896
*d*e^4*x^4 + 256*e^5*x^5)))/(21*(c*d^2 - a*e^2)^6*(d + e*x)^2*((a*e + c*d*x)*(d + e*x))^(3/2))

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Maple [A]  time = 0.051, size = 412, normalized size = 1.6 \begin{align*} -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -256\,{c}^{5}{d}^{5}{e}^{5}{x}^{5}-384\,a{c}^{4}{d}^{4}{e}^{6}{x}^{4}-896\,{c}^{5}{d}^{6}{e}^{4}{x}^{4}-96\,{a}^{2}{c}^{3}{d}^{3}{e}^{7}{x}^{3}-1344\,a{c}^{4}{d}^{5}{e}^{5}{x}^{3}-1120\,{c}^{5}{d}^{7}{e}^{3}{x}^{3}+16\,{a}^{3}{c}^{2}{d}^{2}{e}^{8}{x}^{2}-336\,{a}^{2}{c}^{3}{d}^{4}{e}^{6}{x}^{2}-1680\,a{c}^{4}{d}^{6}{e}^{4}{x}^{2}-560\,{c}^{5}{d}^{8}{e}^{2}{x}^{2}-6\,{a}^{4}cd{e}^{9}x+56\,{a}^{3}{c}^{2}{d}^{3}{e}^{7}x-420\,{a}^{2}{c}^{3}{d}^{5}{e}^{5}x-840\,a{c}^{4}{d}^{7}{e}^{3}x-70\,{c}^{5}{d}^{9}ex+3\,{a}^{5}{e}^{10}-21\,{a}^{4}c{d}^{2}{e}^{8}+70\,{a}^{3}{c}^{2}{d}^{4}{e}^{6}-210\,{a}^{2}{c}^{3}{d}^{6}{e}^{4}-105\,a{c}^{4}{d}^{8}{e}^{2}+7\,{c}^{5}{d}^{10} \right ) }{ \left ( 21\,ex+21\,d \right ) \left ({a}^{6}{e}^{12}-6\,{a}^{5}c{d}^{2}{e}^{10}+15\,{a}^{4}{c}^{2}{d}^{4}{e}^{8}-20\,{a}^{3}{c}^{3}{d}^{6}{e}^{6}+15\,{a}^{2}{c}^{4}{d}^{8}{e}^{4}-6\,a{c}^{5}{d}^{10}{e}^{2}+{c}^{6}{d}^{12} \right ) } \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)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x)

[Out]

-2/21*(c*d*x+a*e)*(-256*c^5*d^5*e^5*x^5-384*a*c^4*d^4*e^6*x^4-896*c^5*d^6*e^4*x^4-96*a^2*c^3*d^3*e^7*x^3-1344*
a*c^4*d^5*e^5*x^3-1120*c^5*d^7*e^3*x^3+16*a^3*c^2*d^2*e^8*x^2-336*a^2*c^3*d^4*e^6*x^2-1680*a*c^4*d^6*e^4*x^2-5
60*c^5*d^8*e^2*x^2-6*a^4*c*d*e^9*x+56*a^3*c^2*d^3*e^7*x-420*a^2*c^3*d^5*e^5*x-840*a*c^4*d^7*e^3*x-70*c^5*d^9*e
*x+3*a^5*e^10-21*a^4*c*d^2*e^8+70*a^3*c^2*d^4*e^6-210*a^2*c^3*d^6*e^4-105*a*c^4*d^8*e^2+7*c^5*d^10)/(e*x+d)/(a
^6*e^12-6*a^5*c*d^2*e^10+15*a^4*c^2*d^4*e^8-20*a^3*c^3*d^6*e^6+15*a^2*c^4*d^8*e^4-6*a*c^5*d^10*e^2+c^6*d^12)/(
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)^2/(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 [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)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="fricas")

[Out]

Timed out

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

[Out]

Timed out