### 3.1674 $$\int \frac{1}{(d+e x)^{5/2} (a^2+2 a b x+b^2 x^2)^3} \, dx$$

Optimal. Leaf size=266 $\frac{3003 b^{3/2} e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 (b d-a e)^{15/2}}-\frac{3003 b e^5}{128 \sqrt{d+e x} (b d-a e)^7}-\frac{1001 e^5}{128 (d+e x)^{3/2} (b d-a e)^6}-\frac{3003 e^4}{640 (a+b x) (d+e x)^{3/2} (b d-a e)^5}+\frac{429 e^3}{320 (a+b x)^2 (d+e x)^{3/2} (b d-a e)^4}-\frac{143 e^2}{240 (a+b x)^3 (d+e x)^{3/2} (b d-a e)^3}+\frac{13 e}{40 (a+b x)^4 (d+e x)^{3/2} (b d-a e)^2}-\frac{1}{5 (a+b x)^5 (d+e x)^{3/2} (b d-a e)}$

[Out]

(-1001*e^5)/(128*(b*d - a*e)^6*(d + e*x)^(3/2)) - 1/(5*(b*d - a*e)*(a + b*x)^5*(d + e*x)^(3/2)) + (13*e)/(40*(
b*d - a*e)^2*(a + b*x)^4*(d + e*x)^(3/2)) - (143*e^2)/(240*(b*d - a*e)^3*(a + b*x)^3*(d + e*x)^(3/2)) + (429*e
^3)/(320*(b*d - a*e)^4*(a + b*x)^2*(d + e*x)^(3/2)) - (3003*e^4)/(640*(b*d - a*e)^5*(a + b*x)*(d + e*x)^(3/2))
- (3003*b*e^5)/(128*(b*d - a*e)^7*Sqrt[d + e*x]) + (3003*b^(3/2)*e^5*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d
- a*e]])/(128*(b*d - a*e)^(15/2))

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Rubi [A]  time = 0.243315, antiderivative size = 266, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 4, integrand size = 28, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.143, Rules used = {27, 51, 63, 208} $\frac{3003 b^{3/2} e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 (b d-a e)^{15/2}}-\frac{3003 b e^5}{128 \sqrt{d+e x} (b d-a e)^7}-\frac{1001 e^5}{128 (d+e x)^{3/2} (b d-a e)^6}-\frac{3003 e^4}{640 (a+b x) (d+e x)^{3/2} (b d-a e)^5}+\frac{429 e^3}{320 (a+b x)^2 (d+e x)^{3/2} (b d-a e)^4}-\frac{143 e^2}{240 (a+b x)^3 (d+e x)^{3/2} (b d-a e)^3}+\frac{13 e}{40 (a+b x)^4 (d+e x)^{3/2} (b d-a e)^2}-\frac{1}{5 (a+b x)^5 (d+e x)^{3/2} (b d-a e)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-1001*e^5)/(128*(b*d - a*e)^6*(d + e*x)^(3/2)) - 1/(5*(b*d - a*e)*(a + b*x)^5*(d + e*x)^(3/2)) + (13*e)/(40*(
b*d - a*e)^2*(a + b*x)^4*(d + e*x)^(3/2)) - (143*e^2)/(240*(b*d - a*e)^3*(a + b*x)^3*(d + e*x)^(3/2)) + (429*e
^3)/(320*(b*d - a*e)^4*(a + b*x)^2*(d + e*x)^(3/2)) - (3003*e^4)/(640*(b*d - a*e)^5*(a + b*x)*(d + e*x)^(3/2))
- (3003*b*e^5)/(128*(b*d - a*e)^7*Sqrt[d + e*x]) + (3003*b^(3/2)*e^5*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d
- a*e]])/(128*(b*d - a*e)^(15/2))

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)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac{1}{(a+b x)^6 (d+e x)^{5/2}} \, dx\\ &=-\frac{1}{5 (b d-a e) (a+b x)^5 (d+e x)^{3/2}}-\frac{(13 e) \int \frac{1}{(a+b x)^5 (d+e x)^{5/2}} \, dx}{10 (b d-a e)}\\ &=-\frac{1}{5 (b d-a e) (a+b x)^5 (d+e x)^{3/2}}+\frac{13 e}{40 (b d-a e)^2 (a+b x)^4 (d+e x)^{3/2}}+\frac{\left (143 e^2\right ) \int \frac{1}{(a+b x)^4 (d+e x)^{5/2}} \, dx}{80 (b d-a e)^2}\\ &=-\frac{1}{5 (b d-a e) (a+b x)^5 (d+e x)^{3/2}}+\frac{13 e}{40 (b d-a e)^2 (a+b x)^4 (d+e x)^{3/2}}-\frac{143 e^2}{240 (b d-a e)^3 (a+b x)^3 (d+e x)^{3/2}}-\frac{\left (429 e^3\right ) \int \frac{1}{(a+b x)^3 (d+e x)^{5/2}} \, dx}{160 (b d-a e)^3}\\ &=-\frac{1}{5 (b d-a e) (a+b x)^5 (d+e x)^{3/2}}+\frac{13 e}{40 (b d-a e)^2 (a+b x)^4 (d+e x)^{3/2}}-\frac{143 e^2}{240 (b d-a e)^3 (a+b x)^3 (d+e x)^{3/2}}+\frac{429 e^3}{320 (b d-a e)^4 (a+b x)^2 (d+e x)^{3/2}}+\frac{\left (3003 e^4\right ) \int \frac{1}{(a+b x)^2 (d+e x)^{5/2}} \, dx}{640 (b d-a e)^4}\\ &=-\frac{1}{5 (b d-a e) (a+b x)^5 (d+e x)^{3/2}}+\frac{13 e}{40 (b d-a e)^2 (a+b x)^4 (d+e x)^{3/2}}-\frac{143 e^2}{240 (b d-a e)^3 (a+b x)^3 (d+e x)^{3/2}}+\frac{429 e^3}{320 (b d-a e)^4 (a+b x)^2 (d+e x)^{3/2}}-\frac{3003 e^4}{640 (b d-a e)^5 (a+b x) (d+e x)^{3/2}}-\frac{\left (3003 e^5\right ) \int \frac{1}{(a+b x) (d+e x)^{5/2}} \, dx}{256 (b d-a e)^5}\\ &=-\frac{1001 e^5}{128 (b d-a e)^6 (d+e x)^{3/2}}-\frac{1}{5 (b d-a e) (a+b x)^5 (d+e x)^{3/2}}+\frac{13 e}{40 (b d-a e)^2 (a+b x)^4 (d+e x)^{3/2}}-\frac{143 e^2}{240 (b d-a e)^3 (a+b x)^3 (d+e x)^{3/2}}+\frac{429 e^3}{320 (b d-a e)^4 (a+b x)^2 (d+e x)^{3/2}}-\frac{3003 e^4}{640 (b d-a e)^5 (a+b x) (d+e x)^{3/2}}-\frac{\left (3003 b e^5\right ) \int \frac{1}{(a+b x) (d+e x)^{3/2}} \, dx}{256 (b d-a e)^6}\\ &=-\frac{1001 e^5}{128 (b d-a e)^6 (d+e x)^{3/2}}-\frac{1}{5 (b d-a e) (a+b x)^5 (d+e x)^{3/2}}+\frac{13 e}{40 (b d-a e)^2 (a+b x)^4 (d+e x)^{3/2}}-\frac{143 e^2}{240 (b d-a e)^3 (a+b x)^3 (d+e x)^{3/2}}+\frac{429 e^3}{320 (b d-a e)^4 (a+b x)^2 (d+e x)^{3/2}}-\frac{3003 e^4}{640 (b d-a e)^5 (a+b x) (d+e x)^{3/2}}-\frac{3003 b e^5}{128 (b d-a e)^7 \sqrt{d+e x}}-\frac{\left (3003 b^2 e^5\right ) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{256 (b d-a e)^7}\\ &=-\frac{1001 e^5}{128 (b d-a e)^6 (d+e x)^{3/2}}-\frac{1}{5 (b d-a e) (a+b x)^5 (d+e x)^{3/2}}+\frac{13 e}{40 (b d-a e)^2 (a+b x)^4 (d+e x)^{3/2}}-\frac{143 e^2}{240 (b d-a e)^3 (a+b x)^3 (d+e x)^{3/2}}+\frac{429 e^3}{320 (b d-a e)^4 (a+b x)^2 (d+e x)^{3/2}}-\frac{3003 e^4}{640 (b d-a e)^5 (a+b x) (d+e x)^{3/2}}-\frac{3003 b e^5}{128 (b d-a e)^7 \sqrt{d+e x}}-\frac{\left (3003 b^2 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)^7}\\ &=-\frac{1001 e^5}{128 (b d-a e)^6 (d+e x)^{3/2}}-\frac{1}{5 (b d-a e) (a+b x)^5 (d+e x)^{3/2}}+\frac{13 e}{40 (b d-a e)^2 (a+b x)^4 (d+e x)^{3/2}}-\frac{143 e^2}{240 (b d-a e)^3 (a+b x)^3 (d+e x)^{3/2}}+\frac{429 e^3}{320 (b d-a e)^4 (a+b x)^2 (d+e x)^{3/2}}-\frac{3003 e^4}{640 (b d-a e)^5 (a+b x) (d+e x)^{3/2}}-\frac{3003 b e^5}{128 (b d-a e)^7 \sqrt{d+e x}}+\frac{3003 b^{3/2} e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 (b d-a e)^{15/2}}\\ \end{align*}

Mathematica [C]  time = 0.0207705, size = 52, normalized size = 0.2 $-\frac{2 e^5 \, _2F_1\left (-\frac{3}{2},6;-\frac{1}{2};-\frac{b (d+e x)}{a e-b d}\right )}{3 (d+e x)^{3/2} (a e-b d)^6}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-2*e^5*Hypergeometric2F1[-3/2, 6, -1/2, -((b*(d + e*x))/(-(b*d) + a*e))])/(3*(-(b*d) + a*e)^6*(d + e*x)^(3/2)
)

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Maple [B]  time = 0.217, size = 668, normalized size = 2.5 \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)^(5/2)/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

-2/3*e^5/(a*e-b*d)^6/(e*x+d)^(3/2)+12*e^5/(a*e-b*d)^7*b/(e*x+d)^(1/2)+1467/128*e^5/(a*e-b*d)^7*b^6/(b*e*x+a*e)
^5*(e*x+d)^(9/2)+9629/192*e^6/(a*e-b*d)^7*b^5/(b*e*x+a*e)^5*(e*x+d)^(7/2)*a-9629/192*e^5/(a*e-b*d)^7*b^6/(b*e*
x+a*e)^5*(e*x+d)^(7/2)*d+1253/15*e^7/(a*e-b*d)^7*b^4/(b*e*x+a*e)^5*(e*x+d)^(5/2)*a^2-2506/15*e^6/(a*e-b*d)^7*b
^5/(b*e*x+a*e)^5*(e*x+d)^(5/2)*a*d+1253/15*e^5/(a*e-b*d)^7*b^6/(b*e*x+a*e)^5*(e*x+d)^(5/2)*d^2+12131/192*e^8/(
a*e-b*d)^7*b^3/(b*e*x+a*e)^5*(e*x+d)^(3/2)*a^3-12131/64*e^7/(a*e-b*d)^7*b^4/(b*e*x+a*e)^5*(e*x+d)^(3/2)*a^2*d+
12131/64*e^6/(a*e-b*d)^7*b^5/(b*e*x+a*e)^5*(e*x+d)^(3/2)*a*d^2-12131/192*e^5/(a*e-b*d)^7*b^6/(b*e*x+a*e)^5*(e*
x+d)^(3/2)*d^3+2373/128*e^9/(a*e-b*d)^7*b^2/(b*e*x+a*e)^5*(e*x+d)^(1/2)*a^4-2373/32*e^8/(a*e-b*d)^7*b^3/(b*e*x
+a*e)^5*(e*x+d)^(1/2)*a^3*d+7119/64*e^7/(a*e-b*d)^7*b^4/(b*e*x+a*e)^5*(e*x+d)^(1/2)*d^2*a^2-2373/32*e^6/(a*e-b
*d)^7*b^5/(b*e*x+a*e)^5*(e*x+d)^(1/2)*a*d^3+2373/128*e^5/(a*e-b*d)^7*b^6/(b*e*x+a*e)^5*(e*x+d)^(1/2)*d^4+3003/
128*e^5/(a*e-b*d)^7*b^2/((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)^(5/2)/(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 = 2.50026, size = 6792, normalized size = 25.53 \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)^(5/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

[-1/3840*(45045*(b^6*e^7*x^7 + a^5*b*d^2*e^5 + (2*b^6*d*e^6 + 5*a*b^5*e^7)*x^6 + (b^6*d^2*e^5 + 10*a*b^5*d*e^6
+ 10*a^2*b^4*e^7)*x^5 + 5*(a*b^5*d^2*e^5 + 4*a^2*b^4*d*e^6 + 2*a^3*b^3*e^7)*x^4 + 5*(2*a^2*b^4*d^2*e^5 + 4*a^
3*b^3*d*e^6 + a^4*b^2*e^7)*x^3 + (10*a^3*b^3*d^2*e^5 + 10*a^4*b^2*d*e^6 + a^5*b*e^7)*x^2 + (5*a^4*b^2*d^2*e^5
+ 2*a^5*b*d*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^6*e^6*x^6 + 384*b^6*d^6 - 2928*a*b^5*d^5*e + 10024*a^2*b^4*d^4*e^2 - 21070*a^3*b
^3*d^3*e^3 + 35595*a^4*b^2*d^2*e^4 + 24320*a^5*b*d*e^5 - 1280*a^6*e^6 + 30030*(2*b^6*d*e^5 + 7*a*b^5*e^6)*x^5
+ 3003*(3*b^6*d^2*e^4 + 94*a*b^5*d*e^5 + 128*a^2*b^4*e^6)*x^4 - 858*(3*b^6*d^3*e^3 - 51*a*b^5*d^2*e^4 - 607*a^
2*b^4*d*e^5 - 395*a^3*b^3*e^6)*x^3 + 143*(8*b^6*d^4*e^2 - 86*a*b^5*d^3*e^3 + 588*a^2*b^4*d^2*e^4 + 3250*a^3*b^
3*d*e^5 + 965*a^4*b^2*e^6)*x^2 - 26*(24*b^6*d^5*e - 208*a*b^5*d^4*e^2 + 889*a^2*b^4*d^3*e^3 - 3045*a^3*b^3*d^2
*e^4 - 7415*a^4*b^2*d*e^5 - 640*a^5*b*e^6)*x)*sqrt(e*x + d))/(a^5*b^7*d^9 - 7*a^6*b^6*d^8*e + 21*a^7*b^5*d^7*e
^2 - 35*a^8*b^4*d^6*e^3 + 35*a^9*b^3*d^5*e^4 - 21*a^10*b^2*d^4*e^5 + 7*a^11*b*d^3*e^6 - a^12*d^2*e^7 + (b^12*d
^7*e^2 - 7*a*b^11*d^6*e^3 + 21*a^2*b^10*d^5*e^4 - 35*a^3*b^9*d^4*e^5 + 35*a^4*b^8*d^3*e^6 - 21*a^5*b^7*d^2*e^7
+ 7*a^6*b^6*d*e^8 - a^7*b^5*e^9)*x^7 + (2*b^12*d^8*e - 9*a*b^11*d^7*e^2 + 7*a^2*b^10*d^6*e^3 + 35*a^3*b^9*d^5
*e^4 - 105*a^4*b^8*d^4*e^5 + 133*a^5*b^7*d^3*e^6 - 91*a^6*b^6*d^2*e^7 + 33*a^7*b^5*d*e^8 - 5*a^8*b^4*e^9)*x^6
+ (b^12*d^9 + 3*a*b^11*d^8*e - 39*a^2*b^10*d^7*e^2 + 105*a^3*b^9*d^6*e^3 - 105*a^4*b^8*d^5*e^4 - 21*a^5*b^7*d^
4*e^5 + 147*a^6*b^6*d^3*e^6 - 141*a^7*b^5*d^2*e^7 + 60*a^8*b^4*d*e^8 - 10*a^9*b^3*e^9)*x^5 + 5*(a*b^11*d^9 - 3
*a^2*b^10*d^8*e - 5*a^3*b^9*d^7*e^2 + 35*a^4*b^8*d^6*e^3 - 63*a^5*b^7*d^5*e^4 + 49*a^6*b^6*d^4*e^5 - 7*a^7*b^5
*d^3*e^6 - 15*a^8*b^4*d^2*e^7 + 10*a^9*b^3*d*e^8 - 2*a^10*b^2*e^9)*x^4 + 5*(2*a^2*b^10*d^9 - 10*a^3*b^9*d^8*e
+ 15*a^4*b^8*d^7*e^2 + 7*a^5*b^7*d^6*e^3 - 49*a^6*b^6*d^5*e^4 + 63*a^7*b^5*d^4*e^5 - 35*a^8*b^4*d^3*e^6 + 5*a^
9*b^3*d^2*e^7 + 3*a^10*b^2*d*e^8 - a^11*b*e^9)*x^3 + (10*a^3*b^9*d^9 - 60*a^4*b^8*d^8*e + 141*a^5*b^7*d^7*e^2
- 147*a^6*b^6*d^6*e^3 + 21*a^7*b^5*d^5*e^4 + 105*a^8*b^4*d^4*e^5 - 105*a^9*b^3*d^3*e^6 + 39*a^10*b^2*d^2*e^7 -
3*a^11*b*d*e^8 - a^12*e^9)*x^2 + (5*a^4*b^8*d^9 - 33*a^5*b^7*d^8*e + 91*a^6*b^6*d^7*e^2 - 133*a^7*b^5*d^6*e^3
+ 105*a^8*b^4*d^5*e^4 - 35*a^9*b^3*d^4*e^5 - 7*a^10*b^2*d^3*e^6 + 9*a^11*b*d^2*e^7 - 2*a^12*d*e^8)*x), 1/1920
*(45045*(b^6*e^7*x^7 + a^5*b*d^2*e^5 + (2*b^6*d*e^6 + 5*a*b^5*e^7)*x^6 + (b^6*d^2*e^5 + 10*a*b^5*d*e^6 + 10*a^
2*b^4*e^7)*x^5 + 5*(a*b^5*d^2*e^5 + 4*a^2*b^4*d*e^6 + 2*a^3*b^3*e^7)*x^4 + 5*(2*a^2*b^4*d^2*e^5 + 4*a^3*b^3*d*
e^6 + a^4*b^2*e^7)*x^3 + (10*a^3*b^3*d^2*e^5 + 10*a^4*b^2*d*e^6 + a^5*b*e^7)*x^2 + (5*a^4*b^2*d^2*e^5 + 2*a^5*
b*d*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)) - (4504
5*b^6*e^6*x^6 + 384*b^6*d^6 - 2928*a*b^5*d^5*e + 10024*a^2*b^4*d^4*e^2 - 21070*a^3*b^3*d^3*e^3 + 35595*a^4*b^2
*d^2*e^4 + 24320*a^5*b*d*e^5 - 1280*a^6*e^6 + 30030*(2*b^6*d*e^5 + 7*a*b^5*e^6)*x^5 + 3003*(3*b^6*d^2*e^4 + 94
*a*b^5*d*e^5 + 128*a^2*b^4*e^6)*x^4 - 858*(3*b^6*d^3*e^3 - 51*a*b^5*d^2*e^4 - 607*a^2*b^4*d*e^5 - 395*a^3*b^3*
e^6)*x^3 + 143*(8*b^6*d^4*e^2 - 86*a*b^5*d^3*e^3 + 588*a^2*b^4*d^2*e^4 + 3250*a^3*b^3*d*e^5 + 965*a^4*b^2*e^6)
*x^2 - 26*(24*b^6*d^5*e - 208*a*b^5*d^4*e^2 + 889*a^2*b^4*d^3*e^3 - 3045*a^3*b^3*d^2*e^4 - 7415*a^4*b^2*d*e^5
- 640*a^5*b*e^6)*x)*sqrt(e*x + d))/(a^5*b^7*d^9 - 7*a^6*b^6*d^8*e + 21*a^7*b^5*d^7*e^2 - 35*a^8*b^4*d^6*e^3 +
35*a^9*b^3*d^5*e^4 - 21*a^10*b^2*d^4*e^5 + 7*a^11*b*d^3*e^6 - a^12*d^2*e^7 + (b^12*d^7*e^2 - 7*a*b^11*d^6*e^3
+ 21*a^2*b^10*d^5*e^4 - 35*a^3*b^9*d^4*e^5 + 35*a^4*b^8*d^3*e^6 - 21*a^5*b^7*d^2*e^7 + 7*a^6*b^6*d*e^8 - a^7*b
^5*e^9)*x^7 + (2*b^12*d^8*e - 9*a*b^11*d^7*e^2 + 7*a^2*b^10*d^6*e^3 + 35*a^3*b^9*d^5*e^4 - 105*a^4*b^8*d^4*e^5
+ 133*a^5*b^7*d^3*e^6 - 91*a^6*b^6*d^2*e^7 + 33*a^7*b^5*d*e^8 - 5*a^8*b^4*e^9)*x^6 + (b^12*d^9 + 3*a*b^11*d^8
*e - 39*a^2*b^10*d^7*e^2 + 105*a^3*b^9*d^6*e^3 - 105*a^4*b^8*d^5*e^4 - 21*a^5*b^7*d^4*e^5 + 147*a^6*b^6*d^3*e^
6 - 141*a^7*b^5*d^2*e^7 + 60*a^8*b^4*d*e^8 - 10*a^9*b^3*e^9)*x^5 + 5*(a*b^11*d^9 - 3*a^2*b^10*d^8*e - 5*a^3*b^
9*d^7*e^2 + 35*a^4*b^8*d^6*e^3 - 63*a^5*b^7*d^5*e^4 + 49*a^6*b^6*d^4*e^5 - 7*a^7*b^5*d^3*e^6 - 15*a^8*b^4*d^2*
e^7 + 10*a^9*b^3*d*e^8 - 2*a^10*b^2*e^9)*x^4 + 5*(2*a^2*b^10*d^9 - 10*a^3*b^9*d^8*e + 15*a^4*b^8*d^7*e^2 + 7*a
^5*b^7*d^6*e^3 - 49*a^6*b^6*d^5*e^4 + 63*a^7*b^5*d^4*e^5 - 35*a^8*b^4*d^3*e^6 + 5*a^9*b^3*d^2*e^7 + 3*a^10*b^2
*d*e^8 - a^11*b*e^9)*x^3 + (10*a^3*b^9*d^9 - 60*a^4*b^8*d^8*e + 141*a^5*b^7*d^7*e^2 - 147*a^6*b^6*d^6*e^3 + 21
*a^7*b^5*d^5*e^4 + 105*a^8*b^4*d^4*e^5 - 105*a^9*b^3*d^3*e^6 + 39*a^10*b^2*d^2*e^7 - 3*a^11*b*d*e^8 - a^12*e^9
)*x^2 + (5*a^4*b^8*d^9 - 33*a^5*b^7*d^8*e + 91*a^6*b^6*d^7*e^2 - 133*a^7*b^5*d^6*e^3 + 105*a^8*b^4*d^5*e^4 - 3
5*a^9*b^3*d^4*e^5 - 7*a^10*b^2*d^3*e^6 + 9*a^11*b*d^2*e^7 - 2*a^12*d*e^8)*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)**(5/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.32092, size = 860, normalized size = 3.23 \begin{align*} -\frac{3003 \, b^{2} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{5}}{128 \,{\left (b^{7} d^{7} - 7 \, a b^{6} d^{6} e + 21 \, a^{2} b^{5} d^{5} e^{2} - 35 \, a^{3} b^{4} d^{4} e^{3} + 35 \, a^{4} b^{3} d^{3} e^{4} - 21 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} - a^{7} e^{7}\right )} \sqrt{-b^{2} d + a b e}} - \frac{2 \,{\left (18 \,{\left (x e + d\right )} b e^{5} + b d e^{5} - a e^{6}\right )}}{3 \,{\left (b^{7} d^{7} - 7 \, a b^{6} d^{6} e + 21 \, a^{2} b^{5} d^{5} e^{2} - 35 \, a^{3} b^{4} d^{4} e^{3} + 35 \, a^{4} b^{3} d^{3} e^{4} - 21 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} - a^{7} e^{7}\right )}{\left (x e + d\right )}^{\frac{3}{2}}} - \frac{22005 \,{\left (x e + d\right )}^{\frac{9}{2}} b^{6} e^{5} - 96290 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{6} d e^{5} + 160384 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{6} d^{2} e^{5} - 121310 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{6} d^{3} e^{5} + 35595 \, \sqrt{x e + d} b^{6} d^{4} e^{5} + 96290 \,{\left (x e + d\right )}^{\frac{7}{2}} a b^{5} e^{6} - 320768 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{5} d e^{6} + 363930 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{5} d^{2} e^{6} - 142380 \, \sqrt{x e + d} a b^{5} d^{3} e^{6} + 160384 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{2} b^{4} e^{7} - 363930 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{4} d e^{7} + 213570 \, \sqrt{x e + d} a^{2} b^{4} d^{2} e^{7} + 121310 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3} b^{3} e^{8} - 142380 \, \sqrt{x e + d} a^{3} b^{3} d e^{8} + 35595 \, \sqrt{x e + d} a^{4} b^{2} e^{9}}{1920 \,{\left (b^{7} d^{7} - 7 \, a b^{6} d^{6} e + 21 \, a^{2} b^{5} d^{5} e^{2} - 35 \, a^{3} b^{4} d^{4} e^{3} + 35 \, a^{4} b^{3} d^{3} e^{4} - 21 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} - a^{7} e^{7}\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(5/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

-3003/128*b^2*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^5/((b^7*d^7 - 7*a*b^6*d^6*e + 21*a^2*b^5*d^5*e^2
- 35*a^3*b^4*d^4*e^3 + 35*a^4*b^3*d^3*e^4 - 21*a^5*b^2*d^2*e^5 + 7*a^6*b*d*e^6 - a^7*e^7)*sqrt(-b^2*d + a*b*e)
) - 2/3*(18*(x*e + d)*b*e^5 + b*d*e^5 - a*e^6)/((b^7*d^7 - 7*a*b^6*d^6*e + 21*a^2*b^5*d^5*e^2 - 35*a^3*b^4*d^4
*e^3 + 35*a^4*b^3*d^3*e^4 - 21*a^5*b^2*d^2*e^5 + 7*a^6*b*d*e^6 - a^7*e^7)*(x*e + d)^(3/2)) - 1/1920*(22005*(x*
e + d)^(9/2)*b^6*e^5 - 96290*(x*e + d)^(7/2)*b^6*d*e^5 + 160384*(x*e + d)^(5/2)*b^6*d^2*e^5 - 121310*(x*e + d)
^(3/2)*b^6*d^3*e^5 + 35595*sqrt(x*e + d)*b^6*d^4*e^5 + 96290*(x*e + d)^(7/2)*a*b^5*e^6 - 320768*(x*e + d)^(5/2
)*a*b^5*d*e^6 + 363930*(x*e + d)^(3/2)*a*b^5*d^2*e^6 - 142380*sqrt(x*e + d)*a*b^5*d^3*e^6 + 160384*(x*e + d)^(
5/2)*a^2*b^4*e^7 - 363930*(x*e + d)^(3/2)*a^2*b^4*d*e^7 + 213570*sqrt(x*e + d)*a^2*b^4*d^2*e^7 + 121310*(x*e +
d)^(3/2)*a^3*b^3*e^8 - 142380*sqrt(x*e + d)*a^3*b^3*d*e^8 + 35595*sqrt(x*e + d)*a^4*b^2*e^9)/((b^7*d^7 - 7*a*
b^6*d^6*e + 21*a^2*b^5*d^5*e^2 - 35*a^3*b^4*d^4*e^3 + 35*a^4*b^3*d^3*e^4 - 21*a^5*b^2*d^2*e^5 + 7*a^6*b*d*e^6
- a^7*e^7)*((x*e + d)*b - b*d + a*e)^5)