### 3.1583 $$\int \frac{(a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^9} \, dx$$

Optimal. Leaf size=149 $\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{168 (d+e x)^6 (b d-a e)^3}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{28 (d+e x)^7 (b d-a e)^2}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{8 (d+e x)^8 (b d-a e)}$

[Out]

((a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*(b*d - a*e)*(d + e*x)^8) + (b*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x +
b^2*x^2])/(28*(b*d - a*e)^2*(d + e*x)^7) + (b^2*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(168*(b*d - a*e)^3*
(d + e*x)^6)

________________________________________________________________________________________

Rubi [A]  time = 0.0510391, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 28, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.107, Rules used = {646, 45, 37} $\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{168 (d+e x)^6 (b d-a e)^3}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{28 (d+e x)^7 (b d-a e)^2}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{8 (d+e x)^8 (b d-a e)}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^9,x]

[Out]

((a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*(b*d - a*e)*(d + e*x)^8) + (b*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x +
b^2*x^2])/(28*(b*d - a*e)^2*(d + e*x)^7) + (b^2*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(168*(b*d - a*e)^3*
(d + e*x)^6)

Rule 646

Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(a + b*x + c*x^2)^Fra
cPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b,
c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p] && NeQ[2*c*d - b*e, 0]

Rule 45

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*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 37

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] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^9} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{\left (a b+b^2 x\right )^5}{(d+e x)^9} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{(a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{8 (b d-a e) (d+e x)^8}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{\left (a b+b^2 x\right )^5}{(d+e x)^8} \, dx}{4 b^3 (b d-a e) \left (a b+b^2 x\right )}\\ &=\frac{(a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{8 (b d-a e) (d+e x)^8}+\frac{b (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{28 (b d-a e)^2 (d+e x)^7}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{\left (a b+b^2 x\right )^5}{(d+e x)^7} \, dx}{28 b^2 (b d-a e)^2 \left (a b+b^2 x\right )}\\ &=\frac{(a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{8 (b d-a e) (d+e x)^8}+\frac{b (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{28 (b d-a e)^2 (d+e x)^7}+\frac{b^2 (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{168 (b d-a e)^3 (d+e x)^6}\\ \end{align*}

Mathematica [A]  time = 0.0803123, size = 223, normalized size = 1.5 $-\frac{\sqrt{(a+b x)^2} \left (6 a^2 b^3 e^2 \left (8 d^2 e x+d^3+28 d e^2 x^2+56 e^3 x^3\right )+10 a^3 b^2 e^3 \left (d^2+8 d e x+28 e^2 x^2\right )+15 a^4 b e^4 (d+8 e x)+21 a^5 e^5+3 a b^4 e \left (28 d^2 e^2 x^2+8 d^3 e x+d^4+56 d e^3 x^3+70 e^4 x^4\right )+b^5 \left (28 d^3 e^2 x^2+56 d^2 e^3 x^3+8 d^4 e x+d^5+70 d e^4 x^4+56 e^5 x^5\right )\right )}{168 e^6 (a+b x) (d+e x)^8}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^9,x]

[Out]

-(Sqrt[(a + b*x)^2]*(21*a^5*e^5 + 15*a^4*b*e^4*(d + 8*e*x) + 10*a^3*b^2*e^3*(d^2 + 8*d*e*x + 28*e^2*x^2) + 6*a
^2*b^3*e^2*(d^3 + 8*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*x^3) + 3*a*b^4*e*(d^4 + 8*d^3*e*x + 28*d^2*e^2*x^2 + 56*d*
e^3*x^3 + 70*e^4*x^4) + b^5*(d^5 + 8*d^4*e*x + 28*d^3*e^2*x^2 + 56*d^2*e^3*x^3 + 70*d*e^4*x^4 + 56*e^5*x^5)))/
(168*e^6*(a + b*x)*(d + e*x)^8)

________________________________________________________________________________________

Maple [B]  time = 0.158, size = 288, normalized size = 1.9 \begin{align*} -{\frac{56\,{x}^{5}{b}^{5}{e}^{5}+210\,{x}^{4}a{b}^{4}{e}^{5}+70\,{x}^{4}{b}^{5}d{e}^{4}+336\,{x}^{3}{a}^{2}{b}^{3}{e}^{5}+168\,{x}^{3}a{b}^{4}d{e}^{4}+56\,{x}^{3}{b}^{5}{d}^{2}{e}^{3}+280\,{x}^{2}{a}^{3}{b}^{2}{e}^{5}+168\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}+84\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}+28\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}+120\,x{a}^{4}b{e}^{5}+80\,x{a}^{3}{b}^{2}d{e}^{4}+48\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}+24\,xa{b}^{4}{d}^{3}{e}^{2}+8\,x{b}^{5}{d}^{4}e+21\,{a}^{5}{e}^{5}+15\,d{e}^{4}{a}^{4}b+10\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}+6\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+3\,a{b}^{4}{d}^{4}e+{b}^{5}{d}^{5}}{168\,{e}^{6} \left ( ex+d \right ) ^{8} \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^9,x)

[Out]

-1/168/e^6*(56*b^5*e^5*x^5+210*a*b^4*e^5*x^4+70*b^5*d*e^4*x^4+336*a^2*b^3*e^5*x^3+168*a*b^4*d*e^4*x^3+56*b^5*d
^2*e^3*x^3+280*a^3*b^2*e^5*x^2+168*a^2*b^3*d*e^4*x^2+84*a*b^4*d^2*e^3*x^2+28*b^5*d^3*e^2*x^2+120*a^4*b*e^5*x+8
0*a^3*b^2*d*e^4*x+48*a^2*b^3*d^2*e^3*x+24*a*b^4*d^3*e^2*x+8*b^5*d^4*e*x+21*a^5*e^5+15*a^4*b*d*e^4+10*a^3*b^2*d
^2*e^3+6*a^2*b^3*d^3*e^2+3*a*b^4*d^4*e+b^5*d^5)*((b*x+a)^2)^(5/2)/(e*x+d)^8/(b*x+a)^5

________________________________________________________________________________________

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((b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^9,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 1.62102, size = 697, normalized size = 4.68 \begin{align*} -\frac{56 \, b^{5} e^{5} x^{5} + b^{5} d^{5} + 3 \, a b^{4} d^{4} e + 6 \, a^{2} b^{3} d^{3} e^{2} + 10 \, a^{3} b^{2} d^{2} e^{3} + 15 \, a^{4} b d e^{4} + 21 \, a^{5} e^{5} + 70 \,{\left (b^{5} d e^{4} + 3 \, a b^{4} e^{5}\right )} x^{4} + 56 \,{\left (b^{5} d^{2} e^{3} + 3 \, a b^{4} d e^{4} + 6 \, a^{2} b^{3} e^{5}\right )} x^{3} + 28 \,{\left (b^{5} d^{3} e^{2} + 3 \, a b^{4} d^{2} e^{3} + 6 \, a^{2} b^{3} d e^{4} + 10 \, a^{3} b^{2} e^{5}\right )} x^{2} + 8 \,{\left (b^{5} d^{4} e + 3 \, a b^{4} d^{3} e^{2} + 6 \, a^{2} b^{3} d^{2} e^{3} + 10 \, a^{3} b^{2} d e^{4} + 15 \, a^{4} b e^{5}\right )} x}{168 \,{\left (e^{14} x^{8} + 8 \, d e^{13} x^{7} + 28 \, d^{2} e^{12} x^{6} + 56 \, d^{3} e^{11} x^{5} + 70 \, d^{4} e^{10} x^{4} + 56 \, d^{5} e^{9} x^{3} + 28 \, d^{6} e^{8} x^{2} + 8 \, d^{7} e^{7} x + d^{8} e^{6}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^9,x, algorithm="fricas")

[Out]

-1/168*(56*b^5*e^5*x^5 + b^5*d^5 + 3*a*b^4*d^4*e + 6*a^2*b^3*d^3*e^2 + 10*a^3*b^2*d^2*e^3 + 15*a^4*b*d*e^4 + 2
1*a^5*e^5 + 70*(b^5*d*e^4 + 3*a*b^4*e^5)*x^4 + 56*(b^5*d^2*e^3 + 3*a*b^4*d*e^4 + 6*a^2*b^3*e^5)*x^3 + 28*(b^5*
d^3*e^2 + 3*a*b^4*d^2*e^3 + 6*a^2*b^3*d*e^4 + 10*a^3*b^2*e^5)*x^2 + 8*(b^5*d^4*e + 3*a*b^4*d^3*e^2 + 6*a^2*b^3
*d^2*e^3 + 10*a^3*b^2*d*e^4 + 15*a^4*b*e^5)*x)/(e^14*x^8 + 8*d*e^13*x^7 + 28*d^2*e^12*x^6 + 56*d^3*e^11*x^5 +
70*d^4*e^10*x^4 + 56*d^5*e^9*x^3 + 28*d^6*e^8*x^2 + 8*d^7*e^7*x + d^8*e^6)

________________________________________________________________________________________

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((b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**9,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.17226, size = 514, normalized size = 3.45 \begin{align*} -\frac{{\left (56 \, b^{5} x^{5} e^{5} \mathrm{sgn}\left (b x + a\right ) + 70 \, b^{5} d x^{4} e^{4} \mathrm{sgn}\left (b x + a\right ) + 56 \, b^{5} d^{2} x^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) + 28 \, b^{5} d^{3} x^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 8 \, b^{5} d^{4} x e \mathrm{sgn}\left (b x + a\right ) + b^{5} d^{5} \mathrm{sgn}\left (b x + a\right ) + 210 \, a b^{4} x^{4} e^{5} \mathrm{sgn}\left (b x + a\right ) + 168 \, a b^{4} d x^{3} e^{4} \mathrm{sgn}\left (b x + a\right ) + 84 \, a b^{4} d^{2} x^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) + 24 \, a b^{4} d^{3} x e^{2} \mathrm{sgn}\left (b x + a\right ) + 3 \, a b^{4} d^{4} e \mathrm{sgn}\left (b x + a\right ) + 336 \, a^{2} b^{3} x^{3} e^{5} \mathrm{sgn}\left (b x + a\right ) + 168 \, a^{2} b^{3} d x^{2} e^{4} \mathrm{sgn}\left (b x + a\right ) + 48 \, a^{2} b^{3} d^{2} x e^{3} \mathrm{sgn}\left (b x + a\right ) + 6 \, a^{2} b^{3} d^{3} e^{2} \mathrm{sgn}\left (b x + a\right ) + 280 \, a^{3} b^{2} x^{2} e^{5} \mathrm{sgn}\left (b x + a\right ) + 80 \, a^{3} b^{2} d x e^{4} \mathrm{sgn}\left (b x + a\right ) + 10 \, a^{3} b^{2} d^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) + 120 \, a^{4} b x e^{5} \mathrm{sgn}\left (b x + a\right ) + 15 \, a^{4} b d e^{4} \mathrm{sgn}\left (b x + a\right ) + 21 \, a^{5} e^{5} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-6\right )}}{168 \,{\left (x e + d\right )}^{8}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^9,x, algorithm="giac")

[Out]

-1/168*(56*b^5*x^5*e^5*sgn(b*x + a) + 70*b^5*d*x^4*e^4*sgn(b*x + a) + 56*b^5*d^2*x^3*e^3*sgn(b*x + a) + 28*b^5
*d^3*x^2*e^2*sgn(b*x + a) + 8*b^5*d^4*x*e*sgn(b*x + a) + b^5*d^5*sgn(b*x + a) + 210*a*b^4*x^4*e^5*sgn(b*x + a)
+ 168*a*b^4*d*x^3*e^4*sgn(b*x + a) + 84*a*b^4*d^2*x^2*e^3*sgn(b*x + a) + 24*a*b^4*d^3*x*e^2*sgn(b*x + a) + 3*
a*b^4*d^4*e*sgn(b*x + a) + 336*a^2*b^3*x^3*e^5*sgn(b*x + a) + 168*a^2*b^3*d*x^2*e^4*sgn(b*x + a) + 48*a^2*b^3*
d^2*x*e^3*sgn(b*x + a) + 6*a^2*b^3*d^3*e^2*sgn(b*x + a) + 280*a^3*b^2*x^2*e^5*sgn(b*x + a) + 80*a^3*b^2*d*x*e^
4*sgn(b*x + a) + 10*a^3*b^2*d^2*e^3*sgn(b*x + a) + 120*a^4*b*x*e^5*sgn(b*x + a) + 15*a^4*b*d*e^4*sgn(b*x + a)
+ 21*a^5*e^5*sgn(b*x + a))*e^(-6)/(x*e + d)^8