### 3.1973 $$\int \frac{1}{(d+e x)^3 (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx$$

Optimal. Leaf size=312 $\frac{1024 c^4 d^4 e \left (a e^2+c d^2+2 c d e x\right )}{63 \left (c d^2-a e^2\right )^7 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{128 c^3 d^3 \left (a e^2+c d^2+2 c d e x\right )}{63 \left (c d^2-a e^2\right )^5 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{16 c^2 d^2}{21 (d+e x) \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{8 c d}{21 (d+e x)^2 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{2}{9 (d+e x)^3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}$

[Out]

2/(9*(c*d^2 - a*e^2)*(d + e*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (8*c*d)/(21*(c*d^2 - a*e^2)^
2*(d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (16*c^2*d^2)/(21*(c*d^2 - a*e^2)^3*(d + e*x)*(a
*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (128*c^3*d^3*(c*d^2 + a*e^2 + 2*c*d*e*x))/(63*(c*d^2 - a*e^2)^5
*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (1024*c^4*d^4*e*(c*d^2 + a*e^2 + 2*c*d*e*x))/(63*(c*d^2 - a*
e^2)^7*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])

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Rubi [A]  time = 0.151707, antiderivative size = 312, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 37, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.081, Rules used = {658, 614, 613} $\frac{1024 c^4 d^4 e \left (a e^2+c d^2+2 c d e x\right )}{63 \left (c d^2-a e^2\right )^7 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{128 c^3 d^3 \left (a e^2+c d^2+2 c d e x\right )}{63 \left (c d^2-a e^2\right )^5 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{16 c^2 d^2}{21 (d+e x) \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{8 c d}{21 (d+e x)^2 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{2}{9 (d+e x)^3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

2/(9*(c*d^2 - a*e^2)*(d + e*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (8*c*d)/(21*(c*d^2 - a*e^2)^
2*(d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (16*c^2*d^2)/(21*(c*d^2 - a*e^2)^3*(d + e*x)*(a
*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (128*c^3*d^3*(c*d^2 + a*e^2 + 2*c*d*e*x))/(63*(c*d^2 - a*e^2)^5
*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (1024*c^4*d^4*e*(c*d^2 + a*e^2 + 2*c*d*e*x))/(63*(c*d^2 - a*
e^2)^7*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)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx &=\frac{2}{9 \left (c d^2-a e^2\right ) (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}}+\frac{(4 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 )^{5/2}} \, dx}{3 \left (c d^2-a e^2\right )}\\ &=\frac{2}{9 \left (c d^2-a e^2\right ) (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}}+\frac{8 c d}{21 \left (c d^2-a e^2\right )^2 (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{\left (40 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 )^{5/2}} \, dx}{21 \left (c d^2-a e^2\right )^2}\\ &=\frac{2}{9 \left (c d^2-a e^2\right ) (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}}+\frac{8 c d}{21 \left (c d^2-a e^2\right )^2 (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{16 c^2 d^2}{21 \left (c d^2-a e^2\right )^3 (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 (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 )^{5/2}} \, dx}{21 \left (c d^2-a e^2\right )^3}\\ &=\frac{2}{9 \left (c d^2-a e^2\right ) (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}}+\frac{8 c d}{21 \left (c d^2-a e^2\right )^2 (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{16 c^2 d^2}{21 \left (c d^2-a e^2\right )^3 (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{128 c^3 d^3 \left (c d^2+a e^2+2 c d e x\right )}{63 \left (c d^2-a e^2\right )^5 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac{\left (512 c^4 d^4 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}{63 \left (c d^2-a e^2\right )^5}\\ &=\frac{2}{9 \left (c d^2-a e^2\right ) (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}}+\frac{8 c d}{21 \left (c d^2-a e^2\right )^2 (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{16 c^2 d^2}{21 \left (c d^2-a e^2\right )^3 (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{128 c^3 d^3 \left (c d^2+a e^2+2 c d e x\right )}{63 \left (c d^2-a e^2\right )^5 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{1024 c^4 d^4 e \left (c d^2+a e^2+2 c d e x\right )}{63 \left (c d^2-a e^2\right )^7 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ \end{align*}

Mathematica [A]  time = 0.153942, size = 336, normalized size = 1.08 $\frac{2 \left (3 a^4 c^2 d^2 e^8 \left (63 d^2+36 d e x+8 e^2 x^2\right )-4 a^3 c^3 d^3 e^6 \left (126 d^2 e x+105 d^3+72 d e^2 x^2+16 e^3 x^3\right )+3 a^2 c^4 d^4 e^4 \left (1008 d^2 e^2 x^2+840 d^3 e x+315 d^4+576 d e^3 x^3+128 e^4 x^4\right )-6 a^5 c d e^{10} (9 d+2 e x)+7 a^6 e^{12}+6 a c^5 d^5 e^2 \left (1680 d^3 e^2 x^2+2016 d^2 e^3 x^3+630 d^4 e x+63 d^5+1152 d e^4 x^4+256 e^5 x^5\right )+c^6 d^6 \left (2520 d^4 e^2 x^2+6720 d^3 e^3 x^3+8064 d^2 e^4 x^4+252 d^5 e x-21 d^6+4608 d e^5 x^5+1024 e^6 x^6\right )\right )}{63 (d+e x)^3 \left (c d^2-a e^2\right )^7 ((d+e x) (a e+c d x))^{3/2}}$

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)^(5/2)),x]

[Out]

(2*(7*a^6*e^12 - 6*a^5*c*d*e^10*(9*d + 2*e*x) + 3*a^4*c^2*d^2*e^8*(63*d^2 + 36*d*e*x + 8*e^2*x^2) - 4*a^3*c^3*
d^3*e^6*(105*d^3 + 126*d^2*e*x + 72*d*e^2*x^2 + 16*e^3*x^3) + 3*a^2*c^4*d^4*e^4*(315*d^4 + 840*d^3*e*x + 1008*
d^2*e^2*x^2 + 576*d*e^3*x^3 + 128*e^4*x^4) + 6*a*c^5*d^5*e^2*(63*d^5 + 630*d^4*e*x + 1680*d^3*e^2*x^2 + 2016*d
^2*e^3*x^3 + 1152*d*e^4*x^4 + 256*e^5*x^5) + c^6*d^6*(-21*d^6 + 252*d^5*e*x + 2520*d^4*e^2*x^2 + 6720*d^3*e^3*
x^3 + 8064*d^2*e^4*x^4 + 4608*d*e^5*x^5 + 1024*e^6*x^6)))/(63*(c*d^2 - a*e^2)^7*(d + e*x)^3*((a*e + c*d*x)*(d
+ e*x))^(3/2))

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Maple [A]  time = 0.055, size = 536, normalized size = 1.7 \begin{align*} -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 1024\,{c}^{6}{d}^{6}{e}^{6}{x}^{6}+1536\,a{c}^{5}{d}^{5}{e}^{7}{x}^{5}+4608\,{c}^{6}{d}^{7}{e}^{5}{x}^{5}+384\,{a}^{2}{c}^{4}{d}^{4}{e}^{8}{x}^{4}+6912\,a{c}^{5}{d}^{6}{e}^{6}{x}^{4}+8064\,{c}^{6}{d}^{8}{e}^{4}{x}^{4}-64\,{a}^{3}{c}^{3}{d}^{3}{e}^{9}{x}^{3}+1728\,{a}^{2}{c}^{4}{d}^{5}{e}^{7}{x}^{3}+12096\,a{c}^{5}{d}^{7}{e}^{5}{x}^{3}+6720\,{c}^{6}{d}^{9}{e}^{3}{x}^{3}+24\,{a}^{4}{c}^{2}{d}^{2}{e}^{10}{x}^{2}-288\,{a}^{3}{c}^{3}{d}^{4}{e}^{8}{x}^{2}+3024\,{a}^{2}{c}^{4}{d}^{6}{e}^{6}{x}^{2}+10080\,a{c}^{5}{d}^{8}{e}^{4}{x}^{2}+2520\,{c}^{6}{d}^{10}{e}^{2}{x}^{2}-12\,{a}^{5}cd{e}^{11}x+108\,{a}^{4}{c}^{2}{d}^{3}{e}^{9}x-504\,{a}^{3}{c}^{3}{d}^{5}{e}^{7}x+2520\,{a}^{2}{c}^{4}{d}^{7}{e}^{5}x+3780\,a{c}^{5}{d}^{9}{e}^{3}x+252\,{c}^{6}{d}^{11}ex+7\,{a}^{6}{e}^{12}-54\,{a}^{5}c{d}^{2}{e}^{10}+189\,{a}^{4}{c}^{2}{d}^{4}{e}^{8}-420\,{a}^{3}{c}^{3}{d}^{6}{e}^{6}+945\,{a}^{2}{c}^{4}{d}^{8}{e}^{4}+378\,a{c}^{5}{d}^{10}{e}^{2}-21\,{c}^{6}{d}^{12} \right ) }{63\, \left ({a}^{7}{e}^{14}-7\,{a}^{6}c{d}^{2}{e}^{12}+21\,{a}^{5}{c}^{2}{d}^{4}{e}^{10}-35\,{a}^{4}{c}^{3}{d}^{6}{e}^{8}+35\,{a}^{3}{c}^{4}{d}^{8}{e}^{6}-21\,{a}^{2}{c}^{5}{d}^{10}{e}^{4}+7\,a{c}^{6}{d}^{12}{e}^{2}-{c}^{7}{d}^{14} \right ) \left ( ex+d \right ) ^{2}} \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)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x)

[Out]

-2/63*(c*d*x+a*e)*(1024*c^6*d^6*e^6*x^6+1536*a*c^5*d^5*e^7*x^5+4608*c^6*d^7*e^5*x^5+384*a^2*c^4*d^4*e^8*x^4+69
12*a*c^5*d^6*e^6*x^4+8064*c^6*d^8*e^4*x^4-64*a^3*c^3*d^3*e^9*x^3+1728*a^2*c^4*d^5*e^7*x^3+12096*a*c^5*d^7*e^5*
x^3+6720*c^6*d^9*e^3*x^3+24*a^4*c^2*d^2*e^10*x^2-288*a^3*c^3*d^4*e^8*x^2+3024*a^2*c^4*d^6*e^6*x^2+10080*a*c^5*
d^8*e^4*x^2+2520*c^6*d^10*e^2*x^2-12*a^5*c*d*e^11*x+108*a^4*c^2*d^3*e^9*x-504*a^3*c^3*d^5*e^7*x+2520*a^2*c^4*d
^7*e^5*x+3780*a*c^5*d^9*e^3*x+252*c^6*d^11*e*x+7*a^6*e^12-54*a^5*c*d^2*e^10+189*a^4*c^2*d^4*e^8-420*a^3*c^3*d^
6*e^6+945*a^2*c^4*d^8*e^4+378*a*c^5*d^10*e^2-21*c^6*d^12)/(e*x+d)^2/(a^7*e^14-7*a^6*c*d^2*e^12+21*a^5*c^2*d^4*
e^10-35*a^4*c^3*d^6*e^8+35*a^3*c^4*d^8*e^6-21*a^2*c^5*d^10*e^4+7*a*c^6*d^12*e^2-c^7*d^14)/(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)^3/(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)^3/(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)**3/(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)^3/(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]