3.1963 \(\int \frac{1}{(d+e x)^4 (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\)

Optimal. Leaf size=301 \[ -\frac{256 c^4 d^4 \left (a e^2+c d^2+2 c d e x\right )}{63 \left (c d^2-a e^2\right )^6 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{64 c^3 d^3}{63 (d+e x) \left (c d^2-a e^2\right )^4 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{32 c^2 d^2}{63 (d+e x)^2 \left (c d^2-a e^2\right )^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{20 c d}{63 (d+e x)^3 \left (c d^2-a e^2\right )^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{2}{9 (d+e x)^4 \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

[Out]

2/(9*(c*d^2 - a*e^2)*(d + e*x)^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (20*c*d)/(63*(c*d^2 - a*e^2)^2
*(d + e*x)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (32*c^2*d^2)/(63*(c*d^2 - a*e^2)^3*(d + e*x)^2*Sqr
t[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (64*c^3*d^3)/(63*(c*d^2 - a*e^2)^4*(d + e*x)*Sqrt[a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2]) - (256*c^4*d^4*(c*d^2 + a*e^2 + 2*c*d*e*x))/(63*(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 = 0.158645, antiderivative size = 301, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.054, Rules used = {658, 613} \[ -\frac{256 c^4 d^4 \left (a e^2+c d^2+2 c d e x\right )}{63 \left (c d^2-a e^2\right )^6 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{64 c^3 d^3}{63 (d+e x) \left (c d^2-a e^2\right )^4 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{32 c^2 d^2}{63 (d+e x)^2 \left (c d^2-a e^2\right )^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{20 c d}{63 (d+e x)^3 \left (c d^2-a e^2\right )^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{2}{9 (d+e x)^4 \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]

[Out]

2/(9*(c*d^2 - a*e^2)*(d + e*x)^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (20*c*d)/(63*(c*d^2 - a*e^2)^2
*(d + e*x)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (32*c^2*d^2)/(63*(c*d^2 - a*e^2)^3*(d + e*x)^2*Sqr
t[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (64*c^3*d^3)/(63*(c*d^2 - a*e^2)^4*(d + e*x)*Sqrt[a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2]) - (256*c^4*d^4*(c*d^2 + a*e^2 + 2*c*d*e*x))/(63*(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 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)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx &=\frac{2}{9 \left (c d^2-a e^2\right ) (d+e x)^4 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{(10 c d) \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}{9 \left (c d^2-a e^2\right )}\\ &=\frac{2}{9 \left (c d^2-a e^2\right ) (d+e x)^4 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{20 c d}{63 \left (c d^2-a e^2\right )^2 (d+e x)^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{\left (80 c^2 d^2\right ) \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}{63 \left (c d^2-a e^2\right )^2}\\ &=\frac{2}{9 \left (c d^2-a e^2\right ) (d+e x)^4 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{20 c d}{63 \left (c d^2-a e^2\right )^2 (d+e x)^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{32 c^2 d^2}{63 \left (c d^2-a e^2\right )^3 (d+e x)^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{\left (32 c^3 d^3\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}{21 \left (c d^2-a e^2\right )^3}\\ &=\frac{2}{9 \left (c d^2-a e^2\right ) (d+e x)^4 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{20 c d}{63 \left (c d^2-a e^2\right )^2 (d+e x)^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{32 c^2 d^2}{63 \left (c d^2-a e^2\right )^3 (d+e x)^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{64 c^3 d^3}{63 \left (c d^2-a e^2\right )^4 (d+e x) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{\left (128 c^4 d^4\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 )^4}\\ &=\frac{2}{9 \left (c d^2-a e^2\right ) (d+e x)^4 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{20 c d}{63 \left (c d^2-a e^2\right )^2 (d+e x)^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{32 c^2 d^2}{63 \left (c d^2-a e^2\right )^3 (d+e x)^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{64 c^3 d^3}{63 \left (c d^2-a e^2\right )^4 (d+e x) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac{256 c^4 d^4 \left (c d^2+a e^2+2 c d e x\right )}{63 \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 = 0.121169, size = 258, normalized size = 0.86 \[ -\frac{2 \left (2 a^3 c^2 d^2 e^6 \left (63 d^2+36 d e x+8 e^2 x^2\right )-2 a^2 c^3 d^3 e^4 \left (126 d^2 e x+105 d^3+72 d e^2 x^2+16 e^3 x^3\right )-5 a^4 c d e^8 (9 d+2 e x)+7 a^5 e^{10}+a c^4 d^4 e^2 \left (1008 d^2 e^2 x^2+840 d^3 e x+315 d^4+576 d e^3 x^3+128 e^4 x^4\right )+c^5 d^5 \left (1680 d^3 e^2 x^2+2016 d^2 e^3 x^3+630 d^4 e x+63 d^5+1152 d e^4 x^4+256 e^5 x^5\right )\right )}{63 (d+e x)^4 \left (c d^2-a e^2\right )^6 \sqrt{(d+e x) (a e+c d x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]

[Out]

(-2*(7*a^5*e^10 - 5*a^4*c*d*e^8*(9*d + 2*e*x) + 2*a^3*c^2*d^2*e^6*(63*d^2 + 36*d*e*x + 8*e^2*x^2) - 2*a^2*c^3*
d^3*e^4*(105*d^3 + 126*d^2*e*x + 72*d*e^2*x^2 + 16*e^3*x^3) + a*c^4*d^4*e^2*(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) + c^5*d^5*(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)))/(63*(c*d^2 - a*e^2)^6*(d + e*x)^4*Sqrt[(a*e + c*d*x)*(d + e*x)])

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Maple [A]  time = 0.051, size = 412, normalized size = 1.4 \begin{align*} -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 256\,{c}^{5}{d}^{5}{e}^{5}{x}^{5}+128\,a{c}^{4}{d}^{4}{e}^{6}{x}^{4}+1152\,{c}^{5}{d}^{6}{e}^{4}{x}^{4}-32\,{a}^{2}{c}^{3}{d}^{3}{e}^{7}{x}^{3}+576\,a{c}^{4}{d}^{5}{e}^{5}{x}^{3}+2016\,{c}^{5}{d}^{7}{e}^{3}{x}^{3}+16\,{a}^{3}{c}^{2}{d}^{2}{e}^{8}{x}^{2}-144\,{a}^{2}{c}^{3}{d}^{4}{e}^{6}{x}^{2}+1008\,a{c}^{4}{d}^{6}{e}^{4}{x}^{2}+1680\,{c}^{5}{d}^{8}{e}^{2}{x}^{2}-10\,{a}^{4}cd{e}^{9}x+72\,{a}^{3}{c}^{2}{d}^{3}{e}^{7}x-252\,{a}^{2}{c}^{3}{d}^{5}{e}^{5}x+840\,a{c}^{4}{d}^{7}{e}^{3}x+630\,{c}^{5}{d}^{9}ex+7\,{a}^{5}{e}^{10}-45\,{a}^{4}c{d}^{2}{e}^{8}+126\,{a}^{3}{c}^{2}{d}^{4}{e}^{6}-210\,{a}^{2}{c}^{3}{d}^{6}{e}^{4}+315\,a{c}^{4}{d}^{8}{e}^{2}+63\,{c}^{5}{d}^{10} \right ) }{63\, \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 ( ex+d \right ) ^{3}} \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)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x)

[Out]

-2/63*(c*d*x+a*e)*(256*c^5*d^5*e^5*x^5+128*a*c^4*d^4*e^6*x^4+1152*c^5*d^6*e^4*x^4-32*a^2*c^3*d^3*e^7*x^3+576*a
*c^4*d^5*e^5*x^3+2016*c^5*d^7*e^3*x^3+16*a^3*c^2*d^2*e^8*x^2-144*a^2*c^3*d^4*e^6*x^2+1008*a*c^4*d^6*e^4*x^2+16
80*c^5*d^8*e^2*x^2-10*a^4*c*d*e^9*x+72*a^3*c^2*d^3*e^7*x-252*a^2*c^3*d^5*e^5*x+840*a*c^4*d^7*e^3*x+630*c^5*d^9
*e*x+7*a^5*e^10-45*a^4*c*d^2*e^8+126*a^3*c^2*d^4*e^6-210*a^2*c^3*d^6*e^4+315*a*c^4*d^8*e^2+63*c^5*d^10)/(e*x+d
)^3/(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)^(3/2)

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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)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),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)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),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)**4/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),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}, 1\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^4/(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]