3.1675 \(\int \frac{1}{(d+e x)^{7/2} (a^2+2 a b x+b^2 x^2)^3} \, dx\)

Optimal. Leaf size=295 \[ -\frac{9009 b^2 e^5}{128 \sqrt{d+e x} (b d-a e)^8}+\frac{9009 b^{5/2} e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 (b d-a e)^{17/2}}-\frac{3003 b e^5}{128 (d+e x)^{3/2} (b d-a e)^7}-\frac{9009 e^5}{640 (d+e x)^{5/2} (b d-a e)^6}-\frac{1287 e^4}{128 (a+b x) (d+e x)^{5/2} (b d-a e)^5}+\frac{143 e^3}{64 (a+b x)^2 (d+e x)^{5/2} (b d-a e)^4}-\frac{13 e^2}{16 (a+b x)^3 (d+e x)^{5/2} (b d-a e)^3}+\frac{3 e}{8 (a+b x)^4 (d+e x)^{5/2} (b d-a e)^2}-\frac{1}{5 (a+b x)^5 (d+e x)^{5/2} (b d-a e)} \]

[Out]

(-9009*e^5)/(640*(b*d - a*e)^6*(d + e*x)^(5/2)) - 1/(5*(b*d - a*e)*(a + b*x)^5*(d + e*x)^(5/2)) + (3*e)/(8*(b*
d - a*e)^2*(a + b*x)^4*(d + e*x)^(5/2)) - (13*e^2)/(16*(b*d - a*e)^3*(a + b*x)^3*(d + e*x)^(5/2)) + (143*e^3)/
(64*(b*d - a*e)^4*(a + b*x)^2*(d + e*x)^(5/2)) - (1287*e^4)/(128*(b*d - a*e)^5*(a + b*x)*(d + e*x)^(5/2)) - (3
003*b*e^5)/(128*(b*d - a*e)^7*(d + e*x)^(3/2)) - (9009*b^2*e^5)/(128*(b*d - a*e)^8*Sqrt[d + e*x]) + (9009*b^(5
/2)*e^5*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*(b*d - a*e)^(17/2))

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Rubi [A]  time = 0.270057, antiderivative size = 295, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {27, 51, 63, 208} \[ -\frac{9009 b^2 e^5}{128 \sqrt{d+e x} (b d-a e)^8}+\frac{9009 b^{5/2} e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 (b d-a e)^{17/2}}-\frac{3003 b e^5}{128 (d+e x)^{3/2} (b d-a e)^7}-\frac{9009 e^5}{640 (d+e x)^{5/2} (b d-a e)^6}-\frac{1287 e^4}{128 (a+b x) (d+e x)^{5/2} (b d-a e)^5}+\frac{143 e^3}{64 (a+b x)^2 (d+e x)^{5/2} (b d-a e)^4}-\frac{13 e^2}{16 (a+b x)^3 (d+e x)^{5/2} (b d-a e)^3}+\frac{3 e}{8 (a+b x)^4 (d+e x)^{5/2} (b d-a e)^2}-\frac{1}{5 (a+b x)^5 (d+e x)^{5/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]

[Out]

(-9009*e^5)/(640*(b*d - a*e)^6*(d + e*x)^(5/2)) - 1/(5*(b*d - a*e)*(a + b*x)^5*(d + e*x)^(5/2)) + (3*e)/(8*(b*
d - a*e)^2*(a + b*x)^4*(d + e*x)^(5/2)) - (13*e^2)/(16*(b*d - a*e)^3*(a + b*x)^3*(d + e*x)^(5/2)) + (143*e^3)/
(64*(b*d - a*e)^4*(a + b*x)^2*(d + e*x)^(5/2)) - (1287*e^4)/(128*(b*d - a*e)^5*(a + b*x)*(d + e*x)^(5/2)) - (3
003*b*e^5)/(128*(b*d - a*e)^7*(d + e*x)^(3/2)) - (9009*b^2*e^5)/(128*(b*d - a*e)^8*Sqrt[d + e*x]) + (9009*b^(5
/2)*e^5*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*(b*d - a*e)^(17/2))

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{1}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac{1}{(a+b x)^6 (d+e x)^{7/2}} \, dx\\ &=-\frac{1}{5 (b d-a e) (a+b x)^5 (d+e x)^{5/2}}-\frac{(3 e) \int \frac{1}{(a+b x)^5 (d+e x)^{7/2}} \, dx}{2 (b d-a e)}\\ &=-\frac{1}{5 (b d-a e) (a+b x)^5 (d+e x)^{5/2}}+\frac{3 e}{8 (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}+\frac{\left (39 e^2\right ) \int \frac{1}{(a+b x)^4 (d+e x)^{7/2}} \, dx}{16 (b d-a e)^2}\\ &=-\frac{1}{5 (b d-a e) (a+b x)^5 (d+e x)^{5/2}}+\frac{3 e}{8 (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}-\frac{13 e^2}{16 (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}-\frac{\left (143 e^3\right ) \int \frac{1}{(a+b x)^3 (d+e x)^{7/2}} \, dx}{32 (b d-a e)^3}\\ &=-\frac{1}{5 (b d-a e) (a+b x)^5 (d+e x)^{5/2}}+\frac{3 e}{8 (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}-\frac{13 e^2}{16 (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}+\frac{143 e^3}{64 (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}+\frac{\left (1287 e^4\right ) \int \frac{1}{(a+b x)^2 (d+e x)^{7/2}} \, dx}{128 (b d-a e)^4}\\ &=-\frac{1}{5 (b d-a e) (a+b x)^5 (d+e x)^{5/2}}+\frac{3 e}{8 (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}-\frac{13 e^2}{16 (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}+\frac{143 e^3}{64 (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}-\frac{1287 e^4}{128 (b d-a e)^5 (a+b x) (d+e x)^{5/2}}-\frac{\left (9009 e^5\right ) \int \frac{1}{(a+b x) (d+e x)^{7/2}} \, dx}{256 (b d-a e)^5}\\ &=-\frac{9009 e^5}{640 (b d-a e)^6 (d+e x)^{5/2}}-\frac{1}{5 (b d-a e) (a+b x)^5 (d+e x)^{5/2}}+\frac{3 e}{8 (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}-\frac{13 e^2}{16 (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}+\frac{143 e^3}{64 (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}-\frac{1287 e^4}{128 (b d-a e)^5 (a+b x) (d+e x)^{5/2}}-\frac{\left (9009 b e^5\right ) \int \frac{1}{(a+b x) (d+e x)^{5/2}} \, dx}{256 (b d-a e)^6}\\ &=-\frac{9009 e^5}{640 (b d-a e)^6 (d+e x)^{5/2}}-\frac{1}{5 (b d-a e) (a+b x)^5 (d+e x)^{5/2}}+\frac{3 e}{8 (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}-\frac{13 e^2}{16 (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}+\frac{143 e^3}{64 (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}-\frac{1287 e^4}{128 (b d-a e)^5 (a+b x) (d+e x)^{5/2}}-\frac{3003 b e^5}{128 (b d-a e)^7 (d+e x)^{3/2}}-\frac{\left (9009 b^2 e^5\right ) \int \frac{1}{(a+b x) (d+e x)^{3/2}} \, dx}{256 (b d-a e)^7}\\ &=-\frac{9009 e^5}{640 (b d-a e)^6 (d+e x)^{5/2}}-\frac{1}{5 (b d-a e) (a+b x)^5 (d+e x)^{5/2}}+\frac{3 e}{8 (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}-\frac{13 e^2}{16 (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}+\frac{143 e^3}{64 (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}-\frac{1287 e^4}{128 (b d-a e)^5 (a+b x) (d+e x)^{5/2}}-\frac{3003 b e^5}{128 (b d-a e)^7 (d+e x)^{3/2}}-\frac{9009 b^2 e^5}{128 (b d-a e)^8 \sqrt{d+e x}}-\frac{\left (9009 b^3 e^5\right ) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{256 (b d-a e)^8}\\ &=-\frac{9009 e^5}{640 (b d-a e)^6 (d+e x)^{5/2}}-\frac{1}{5 (b d-a e) (a+b x)^5 (d+e x)^{5/2}}+\frac{3 e}{8 (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}-\frac{13 e^2}{16 (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}+\frac{143 e^3}{64 (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}-\frac{1287 e^4}{128 (b d-a e)^5 (a+b x) (d+e x)^{5/2}}-\frac{3003 b e^5}{128 (b d-a e)^7 (d+e x)^{3/2}}-\frac{9009 b^2 e^5}{128 (b d-a e)^8 \sqrt{d+e x}}-\frac{\left (9009 b^3 e^4\right ) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b d}{e}+\frac{b x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{128 (b d-a e)^8}\\ &=-\frac{9009 e^5}{640 (b d-a e)^6 (d+e x)^{5/2}}-\frac{1}{5 (b d-a e) (a+b x)^5 (d+e x)^{5/2}}+\frac{3 e}{8 (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}-\frac{13 e^2}{16 (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}+\frac{143 e^3}{64 (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}-\frac{1287 e^4}{128 (b d-a e)^5 (a+b x) (d+e x)^{5/2}}-\frac{3003 b e^5}{128 (b d-a e)^7 (d+e x)^{3/2}}-\frac{9009 b^2 e^5}{128 (b d-a e)^8 \sqrt{d+e x}}+\frac{9009 b^{5/2} e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 (b d-a e)^{17/2}}\\ \end{align*}

Mathematica [C]  time = 0.0263282, size = 52, normalized size = 0.18 \[ -\frac{2 e^5 \, _2F_1\left (-\frac{5}{2},6;-\frac{3}{2};-\frac{b (d+e x)}{a e-b d}\right )}{5 (d+e x)^{5/2} (a e-b d)^6} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]

[Out]

(-2*e^5*Hypergeometric2F1[-5/2, 6, -3/2, -((b*(d + e*x))/(-(b*d) + a*e))])/(5*(-(b*d) + a*e)^6*(d + e*x)^(5/2)
)

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Maple [B]  time = 0.218, size = 693, normalized size = 2.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

-2/5*e^5/(a*e-b*d)^6/(e*x+d)^(5/2)-42*e^5/(a*e-b*d)^8*b^2/(e*x+d)^(1/2)+4*e^5/(a*e-b*d)^7*b/(e*x+d)^(3/2)-3633
/128*e^5/(a*e-b*d)^8*b^7/(b*e*x+a*e)^5*(e*x+d)^(9/2)-7837/64*e^6/(a*e-b*d)^8*b^6/(b*e*x+a*e)^5*(e*x+d)^(7/2)*a
+7837/64*e^5/(a*e-b*d)^8*b^7/(b*e*x+a*e)^5*(e*x+d)^(7/2)*d-1001/5*e^7/(a*e-b*d)^8*b^5/(b*e*x+a*e)^5*(e*x+d)^(5
/2)*a^2+2002/5*e^6/(a*e-b*d)^8*b^6/(b*e*x+a*e)^5*(e*x+d)^(5/2)*a*d-1001/5*e^5/(a*e-b*d)^8*b^7/(b*e*x+a*e)^5*(e
*x+d)^(5/2)*d^2-9443/64*e^8/(a*e-b*d)^8*b^4/(b*e*x+a*e)^5*(e*x+d)^(3/2)*a^3+28329/64*e^7/(a*e-b*d)^8*b^5/(b*e*
x+a*e)^5*(e*x+d)^(3/2)*a^2*d-28329/64*e^6/(a*e-b*d)^8*b^6/(b*e*x+a*e)^5*(e*x+d)^(3/2)*a*d^2+9443/64*e^5/(a*e-b
*d)^8*b^7/(b*e*x+a*e)^5*(e*x+d)^(3/2)*d^3-5327/128*e^9/(a*e-b*d)^8*b^3/(b*e*x+a*e)^5*(e*x+d)^(1/2)*a^4+5327/32
*e^8/(a*e-b*d)^8*b^4/(b*e*x+a*e)^5*(e*x+d)^(1/2)*a^3*d-15981/64*e^7/(a*e-b*d)^8*b^5/(b*e*x+a*e)^5*(e*x+d)^(1/2
)*d^2*a^2+5327/32*e^6/(a*e-b*d)^8*b^6/(b*e*x+a*e)^5*(e*x+d)^(1/2)*a*d^3-5327/128*e^5/(a*e-b*d)^8*b^7/(b*e*x+a*
e)^5*(e*x+d)^(1/2)*d^4-9009/128*e^5/(a*e-b*d)^8*b^3/((a*e-b*d)*b)^(1/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(
1/2))

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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)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.74879, size = 9030, normalized size = 30.61 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

[1/1280*(45045*(b^7*e^8*x^8 + a^5*b^2*d^3*e^5 + (3*b^7*d*e^7 + 5*a*b^6*e^8)*x^7 + (3*b^7*d^2*e^6 + 15*a*b^6*d*
e^7 + 10*a^2*b^5*e^8)*x^6 + (b^7*d^3*e^5 + 15*a*b^6*d^2*e^6 + 30*a^2*b^5*d*e^7 + 10*a^3*b^4*e^8)*x^5 + 5*(a*b^
6*d^3*e^5 + 6*a^2*b^5*d^2*e^6 + 6*a^3*b^4*d*e^7 + a^4*b^3*e^8)*x^4 + (10*a^2*b^5*d^3*e^5 + 30*a^3*b^4*d^2*e^6
+ 15*a^4*b^3*d*e^7 + a^5*b^2*e^8)*x^3 + (10*a^3*b^4*d^3*e^5 + 15*a^4*b^3*d^2*e^6 + 3*a^5*b^2*d*e^7)*x^2 + (5*a
^4*b^3*d^3*e^5 + 3*a^5*b^2*d^2*e^6)*x)*sqrt(b/(b*d - a*e))*log((b*e*x + 2*b*d - a*e + 2*(b*d - a*e)*sqrt(e*x +
d)*sqrt(b/(b*d - a*e)))/(b*x + a)) - 2*(45045*b^7*e^7*x^7 + 128*b^7*d^7 - 1136*a*b^6*d^6*e + 4648*a^2*b^5*d^5
*e^2 - 12110*a^3*b^4*d^4*e^3 + 26635*a^4*b^3*d^3*e^4 + 29696*a^5*b^2*d^2*e^5 - 3072*a^6*b*d*e^6 + 256*a^7*e^7
+ 105105*(b^7*d*e^6 + 2*a*b^6*e^7)*x^6 + 3003*(23*b^7*d^2*e^5 + 164*a*b^6*d*e^6 + 128*a^2*b^5*e^7)*x^5 + 2145*
(3*b^7*d^3*e^4 + 152*a*b^6*d^2*e^5 + 422*a^2*b^5*d*e^6 + 158*a^3*b^4*e^7)*x^4 - 715*(2*b^7*d^4*e^3 - 44*a*b^6*
d^3*e^4 - 846*a^2*b^5*d^2*e^5 - 1124*a^3*b^4*d*e^6 - 193*a^4*b^3*e^7)*x^3 + 65*(8*b^7*d^5*e^2 - 106*a*b^6*d^4*
e^3 + 938*a^2*b^5*d^3*e^4 + 8368*a^3*b^4*d^2*e^5 + 5089*a^4*b^3*d*e^6 + 256*a^5*b^2*e^7)*x^2 - 5*(48*b^7*d^6*e
- 496*a*b^6*d^5*e^2 + 2618*a^2*b^5*d^4*e^3 - 11620*a^3*b^4*d^3*e^4 - 45677*a^4*b^3*d^2*e^5 - 8192*a^5*b^2*d*e
^6 + 256*a^6*b*e^7)*x)*sqrt(e*x + d))/(a^5*b^8*d^11 - 8*a^6*b^7*d^10*e + 28*a^7*b^6*d^9*e^2 - 56*a^8*b^5*d^8*e
^3 + 70*a^9*b^4*d^7*e^4 - 56*a^10*b^3*d^6*e^5 + 28*a^11*b^2*d^5*e^6 - 8*a^12*b*d^4*e^7 + a^13*d^3*e^8 + (b^13*
d^8*e^3 - 8*a*b^12*d^7*e^4 + 28*a^2*b^11*d^6*e^5 - 56*a^3*b^10*d^5*e^6 + 70*a^4*b^9*d^4*e^7 - 56*a^5*b^8*d^3*e
^8 + 28*a^6*b^7*d^2*e^9 - 8*a^7*b^6*d*e^10 + a^8*b^5*e^11)*x^8 + (3*b^13*d^9*e^2 - 19*a*b^12*d^8*e^3 + 44*a^2*
b^11*d^7*e^4 - 28*a^3*b^10*d^6*e^5 - 70*a^4*b^9*d^5*e^6 + 182*a^5*b^8*d^4*e^7 - 196*a^6*b^7*d^3*e^8 + 116*a^7*
b^6*d^2*e^9 - 37*a^8*b^5*d*e^10 + 5*a^9*b^4*e^11)*x^7 + (3*b^13*d^10*e - 9*a*b^12*d^9*e^2 - 26*a^2*b^11*d^8*e^
3 + 172*a^3*b^10*d^7*e^4 - 350*a^4*b^9*d^6*e^5 + 322*a^5*b^8*d^5*e^6 - 56*a^6*b^7*d^4*e^7 - 164*a^7*b^6*d^3*e^
8 + 163*a^8*b^5*d^2*e^9 - 65*a^9*b^4*d*e^10 + 10*a^10*b^3*e^11)*x^6 + (b^13*d^11 + 7*a*b^12*d^10*e - 62*a^2*b^
11*d^9*e^2 + 134*a^3*b^10*d^8*e^3 - 10*a^4*b^9*d^7*e^4 - 406*a^5*b^8*d^6*e^5 + 728*a^6*b^7*d^5*e^6 - 568*a^7*b
^6*d^4*e^7 + 161*a^8*b^5*d^3*e^8 + 55*a^9*b^4*d^2*e^9 - 50*a^10*b^3*d*e^10 + 10*a^11*b^2*e^11)*x^5 + 5*(a*b^12
*d^11 - 2*a^2*b^11*d^10*e - 14*a^3*b^10*d^9*e^2 + 65*a^4*b^9*d^8*e^3 - 106*a^5*b^8*d^7*e^4 + 56*a^6*b^7*d^6*e^
5 + 56*a^7*b^6*d^5*e^6 - 106*a^8*b^5*d^4*e^7 + 65*a^9*b^4*d^3*e^8 - 14*a^10*b^3*d^2*e^9 - 2*a^11*b^2*d*e^10 +
a^12*b*e^11)*x^4 + (10*a^2*b^11*d^11 - 50*a^3*b^10*d^10*e + 55*a^4*b^9*d^9*e^2 + 161*a^5*b^8*d^8*e^3 - 568*a^6
*b^7*d^7*e^4 + 728*a^7*b^6*d^6*e^5 - 406*a^8*b^5*d^5*e^6 - 10*a^9*b^4*d^4*e^7 + 134*a^10*b^3*d^3*e^8 - 62*a^11
*b^2*d^2*e^9 + 7*a^12*b*d*e^10 + a^13*e^11)*x^3 + (10*a^3*b^10*d^11 - 65*a^4*b^9*d^10*e + 163*a^5*b^8*d^9*e^2
- 164*a^6*b^7*d^8*e^3 - 56*a^7*b^6*d^7*e^4 + 322*a^8*b^5*d^6*e^5 - 350*a^9*b^4*d^5*e^6 + 172*a^10*b^3*d^4*e^7
- 26*a^11*b^2*d^3*e^8 - 9*a^12*b*d^2*e^9 + 3*a^13*d*e^10)*x^2 + (5*a^4*b^9*d^11 - 37*a^5*b^8*d^10*e + 116*a^6*
b^7*d^9*e^2 - 196*a^7*b^6*d^8*e^3 + 182*a^8*b^5*d^7*e^4 - 70*a^9*b^4*d^6*e^5 - 28*a^10*b^3*d^5*e^6 + 44*a^11*b
^2*d^4*e^7 - 19*a^12*b*d^3*e^8 + 3*a^13*d^2*e^9)*x), 1/640*(45045*(b^7*e^8*x^8 + a^5*b^2*d^3*e^5 + (3*b^7*d*e^
7 + 5*a*b^6*e^8)*x^7 + (3*b^7*d^2*e^6 + 15*a*b^6*d*e^7 + 10*a^2*b^5*e^8)*x^6 + (b^7*d^3*e^5 + 15*a*b^6*d^2*e^6
+ 30*a^2*b^5*d*e^7 + 10*a^3*b^4*e^8)*x^5 + 5*(a*b^6*d^3*e^5 + 6*a^2*b^5*d^2*e^6 + 6*a^3*b^4*d*e^7 + a^4*b^3*e
^8)*x^4 + (10*a^2*b^5*d^3*e^5 + 30*a^3*b^4*d^2*e^6 + 15*a^4*b^3*d*e^7 + a^5*b^2*e^8)*x^3 + (10*a^3*b^4*d^3*e^5
+ 15*a^4*b^3*d^2*e^6 + 3*a^5*b^2*d*e^7)*x^2 + (5*a^4*b^3*d^3*e^5 + 3*a^5*b^2*d^2*e^6)*x)*sqrt(-b/(b*d - a*e))
*arctan(-(b*d - a*e)*sqrt(e*x + d)*sqrt(-b/(b*d - a*e))/(b*e*x + b*d)) - (45045*b^7*e^7*x^7 + 128*b^7*d^7 - 11
36*a*b^6*d^6*e + 4648*a^2*b^5*d^5*e^2 - 12110*a^3*b^4*d^4*e^3 + 26635*a^4*b^3*d^3*e^4 + 29696*a^5*b^2*d^2*e^5
- 3072*a^6*b*d*e^6 + 256*a^7*e^7 + 105105*(b^7*d*e^6 + 2*a*b^6*e^7)*x^6 + 3003*(23*b^7*d^2*e^5 + 164*a*b^6*d*e
^6 + 128*a^2*b^5*e^7)*x^5 + 2145*(3*b^7*d^3*e^4 + 152*a*b^6*d^2*e^5 + 422*a^2*b^5*d*e^6 + 158*a^3*b^4*e^7)*x^4
- 715*(2*b^7*d^4*e^3 - 44*a*b^6*d^3*e^4 - 846*a^2*b^5*d^2*e^5 - 1124*a^3*b^4*d*e^6 - 193*a^4*b^3*e^7)*x^3 + 6
5*(8*b^7*d^5*e^2 - 106*a*b^6*d^4*e^3 + 938*a^2*b^5*d^3*e^4 + 8368*a^3*b^4*d^2*e^5 + 5089*a^4*b^3*d*e^6 + 256*a
^5*b^2*e^7)*x^2 - 5*(48*b^7*d^6*e - 496*a*b^6*d^5*e^2 + 2618*a^2*b^5*d^4*e^3 - 11620*a^3*b^4*d^3*e^4 - 45677*a
^4*b^3*d^2*e^5 - 8192*a^5*b^2*d*e^6 + 256*a^6*b*e^7)*x)*sqrt(e*x + d))/(a^5*b^8*d^11 - 8*a^6*b^7*d^10*e + 28*a
^7*b^6*d^9*e^2 - 56*a^8*b^5*d^8*e^3 + 70*a^9*b^4*d^7*e^4 - 56*a^10*b^3*d^6*e^5 + 28*a^11*b^2*d^5*e^6 - 8*a^12*
b*d^4*e^7 + a^13*d^3*e^8 + (b^13*d^8*e^3 - 8*a*b^12*d^7*e^4 + 28*a^2*b^11*d^6*e^5 - 56*a^3*b^10*d^5*e^6 + 70*a
^4*b^9*d^4*e^7 - 56*a^5*b^8*d^3*e^8 + 28*a^6*b^7*d^2*e^9 - 8*a^7*b^6*d*e^10 + a^8*b^5*e^11)*x^8 + (3*b^13*d^9*
e^2 - 19*a*b^12*d^8*e^3 + 44*a^2*b^11*d^7*e^4 - 28*a^3*b^10*d^6*e^5 - 70*a^4*b^9*d^5*e^6 + 182*a^5*b^8*d^4*e^7
- 196*a^6*b^7*d^3*e^8 + 116*a^7*b^6*d^2*e^9 - 37*a^8*b^5*d*e^10 + 5*a^9*b^4*e^11)*x^7 + (3*b^13*d^10*e - 9*a*
b^12*d^9*e^2 - 26*a^2*b^11*d^8*e^3 + 172*a^3*b^10*d^7*e^4 - 350*a^4*b^9*d^6*e^5 + 322*a^5*b^8*d^5*e^6 - 56*a^6
*b^7*d^4*e^7 - 164*a^7*b^6*d^3*e^8 + 163*a^8*b^5*d^2*e^9 - 65*a^9*b^4*d*e^10 + 10*a^10*b^3*e^11)*x^6 + (b^13*d
^11 + 7*a*b^12*d^10*e - 62*a^2*b^11*d^9*e^2 + 134*a^3*b^10*d^8*e^3 - 10*a^4*b^9*d^7*e^4 - 406*a^5*b^8*d^6*e^5
+ 728*a^6*b^7*d^5*e^6 - 568*a^7*b^6*d^4*e^7 + 161*a^8*b^5*d^3*e^8 + 55*a^9*b^4*d^2*e^9 - 50*a^10*b^3*d*e^10 +
10*a^11*b^2*e^11)*x^5 + 5*(a*b^12*d^11 - 2*a^2*b^11*d^10*e - 14*a^3*b^10*d^9*e^2 + 65*a^4*b^9*d^8*e^3 - 106*a^
5*b^8*d^7*e^4 + 56*a^6*b^7*d^6*e^5 + 56*a^7*b^6*d^5*e^6 - 106*a^8*b^5*d^4*e^7 + 65*a^9*b^4*d^3*e^8 - 14*a^10*b
^3*d^2*e^9 - 2*a^11*b^2*d*e^10 + a^12*b*e^11)*x^4 + (10*a^2*b^11*d^11 - 50*a^3*b^10*d^10*e + 55*a^4*b^9*d^9*e^
2 + 161*a^5*b^8*d^8*e^3 - 568*a^6*b^7*d^7*e^4 + 728*a^7*b^6*d^6*e^5 - 406*a^8*b^5*d^5*e^6 - 10*a^9*b^4*d^4*e^7
+ 134*a^10*b^3*d^3*e^8 - 62*a^11*b^2*d^2*e^9 + 7*a^12*b*d*e^10 + a^13*e^11)*x^3 + (10*a^3*b^10*d^11 - 65*a^4*
b^9*d^10*e + 163*a^5*b^8*d^9*e^2 - 164*a^6*b^7*d^8*e^3 - 56*a^7*b^6*d^7*e^4 + 322*a^8*b^5*d^6*e^5 - 350*a^9*b^
4*d^5*e^6 + 172*a^10*b^3*d^4*e^7 - 26*a^11*b^2*d^3*e^8 - 9*a^12*b*d^2*e^9 + 3*a^13*d*e^10)*x^2 + (5*a^4*b^9*d^
11 - 37*a^5*b^8*d^10*e + 116*a^6*b^7*d^9*e^2 - 196*a^7*b^6*d^8*e^3 + 182*a^8*b^5*d^7*e^4 - 70*a^9*b^4*d^6*e^5
- 28*a^10*b^3*d^5*e^6 + 44*a^11*b^2*d^4*e^7 - 19*a^12*b*d^3*e^8 + 3*a^13*d^2*e^9)*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)**(7/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.21175, size = 1193, normalized size = 4.04 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

-9009/128*b^3*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^5/((b^8*d^8 - 8*a*b^7*d^7*e + 28*a^2*b^6*d^6*e^2
- 56*a^3*b^5*d^5*e^3 + 70*a^4*b^4*d^4*e^4 - 56*a^5*b^3*d^3*e^5 + 28*a^6*b^2*d^2*e^6 - 8*a^7*b*d*e^7 + a^8*e^8)
*sqrt(-b^2*d + a*b*e)) - 1/640*(45045*(x*e + d)^7*b^7*e^5 - 210210*(x*e + d)^6*b^7*d*e^5 + 384384*(x*e + d)^5*
b^7*d^2*e^5 - 338910*(x*e + d)^4*b^7*d^3*e^5 + 137995*(x*e + d)^3*b^7*d^4*e^5 - 16640*(x*e + d)^2*b^7*d^5*e^5
- 1280*(x*e + d)*b^7*d^6*e^5 - 256*b^7*d^7*e^5 + 210210*(x*e + d)^6*a*b^6*e^6 - 768768*(x*e + d)^5*a*b^6*d*e^6
+ 1016730*(x*e + d)^4*a*b^6*d^2*e^6 - 551980*(x*e + d)^3*a*b^6*d^3*e^6 + 83200*(x*e + d)^2*a*b^6*d^4*e^6 + 76
80*(x*e + d)*a*b^6*d^5*e^6 + 1792*a*b^6*d^6*e^6 + 384384*(x*e + d)^5*a^2*b^5*e^7 - 1016730*(x*e + d)^4*a^2*b^5
*d*e^7 + 827970*(x*e + d)^3*a^2*b^5*d^2*e^7 - 166400*(x*e + d)^2*a^2*b^5*d^3*e^7 - 19200*(x*e + d)*a^2*b^5*d^4
*e^7 - 5376*a^2*b^5*d^5*e^7 + 338910*(x*e + d)^4*a^3*b^4*e^8 - 551980*(x*e + d)^3*a^3*b^4*d*e^8 + 166400*(x*e
+ d)^2*a^3*b^4*d^2*e^8 + 25600*(x*e + d)*a^3*b^4*d^3*e^8 + 8960*a^3*b^4*d^4*e^8 + 137995*(x*e + d)^3*a^4*b^3*e
^9 - 83200*(x*e + d)^2*a^4*b^3*d*e^9 - 19200*(x*e + d)*a^4*b^3*d^2*e^9 - 8960*a^4*b^3*d^3*e^9 + 16640*(x*e + d
)^2*a^5*b^2*e^10 + 7680*(x*e + d)*a^5*b^2*d*e^10 + 5376*a^5*b^2*d^2*e^10 - 1280*(x*e + d)*a^6*b*e^11 - 1792*a^
6*b*d*e^11 + 256*a^7*e^12)/((b^8*d^8 - 8*a*b^7*d^7*e + 28*a^2*b^6*d^6*e^2 - 56*a^3*b^5*d^5*e^3 + 70*a^4*b^4*d^
4*e^4 - 56*a^5*b^3*d^3*e^5 + 28*a^6*b^2*d^2*e^6 - 8*a^7*b*d*e^7 + a^8*e^8)*((x*e + d)^(3/2)*b - sqrt(x*e + d)*
b*d + sqrt(x*e + d)*a*e)^5)