### 3.1586 $$\int \frac{(a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^{12}} \, dx$$

Optimal. Leaf size=308 $-\frac{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 e^6 (a+b x) (d+e x)^6}+\frac{5 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{7 e^6 (a+b x) (d+e x)^7}-\frac{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{4 e^6 (a+b x) (d+e x)^8}+\frac{10 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{9 e^6 (a+b x) (d+e x)^9}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{2 e^6 (a+b x) (d+e x)^{10}}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{11 e^6 (a+b x) (d+e x)^{11}}$

[Out]

((b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^6*(a + b*x)*(d + e*x)^11) - (b*(b*d - a*e)^4*Sqrt[a^2 + 2*
a*b*x + b^2*x^2])/(2*e^6*(a + b*x)*(d + e*x)^10) + (10*b^2*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^6
*(a + b*x)*(d + e*x)^9) - (5*b^3*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^6*(a + b*x)*(d + e*x)^8) +
(5*b^4*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^6*(a + b*x)*(d + e*x)^7) - (b^5*Sqrt[a^2 + 2*a*b*x + b^
2*x^2])/(6*e^6*(a + b*x)*(d + e*x)^6)

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Rubi [A]  time = 0.138008, antiderivative size = 308, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.071, Rules used = {646, 43} $-\frac{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 e^6 (a+b x) (d+e x)^6}+\frac{5 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{7 e^6 (a+b x) (d+e x)^7}-\frac{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{4 e^6 (a+b x) (d+e x)^8}+\frac{10 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{9 e^6 (a+b x) (d+e x)^9}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{2 e^6 (a+b x) (d+e x)^{10}}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{11 e^6 (a+b x) (d+e x)^{11}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^12,x]

[Out]

((b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^6*(a + b*x)*(d + e*x)^11) - (b*(b*d - a*e)^4*Sqrt[a^2 + 2*
a*b*x + b^2*x^2])/(2*e^6*(a + b*x)*(d + e*x)^10) + (10*b^2*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^6
*(a + b*x)*(d + e*x)^9) - (5*b^3*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^6*(a + b*x)*(d + e*x)^8) +
(5*b^4*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^6*(a + b*x)*(d + e*x)^7) - (b^5*Sqrt[a^2 + 2*a*b*x + b^
2*x^2])/(6*e^6*(a + b*x)*(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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{12}} \, 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)^{12}} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (-\frac{b^5 (b d-a e)^5}{e^5 (d+e x)^{12}}+\frac{5 b^6 (b d-a e)^4}{e^5 (d+e x)^{11}}-\frac{10 b^7 (b d-a e)^3}{e^5 (d+e x)^{10}}+\frac{10 b^8 (b d-a e)^2}{e^5 (d+e x)^9}-\frac{5 b^9 (b d-a e)}{e^5 (d+e x)^8}+\frac{b^{10}}{e^5 (d+e x)^7}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{(b d-a e)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{11 e^6 (a+b x) (d+e x)^{11}}-\frac{b (b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^6 (a+b x) (d+e x)^{10}}+\frac{10 b^2 (b d-a e)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{9 e^6 (a+b x) (d+e x)^9}-\frac{5 b^3 (b d-a e)^2 \sqrt{a^2+2 a b x+b^2 x^2}}{4 e^6 (a+b x) (d+e x)^8}+\frac{5 b^4 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}{7 e^6 (a+b x) (d+e x)^7}-\frac{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 e^6 (a+b x) (d+e x)^6}\\ \end{align*}

Mathematica [A]  time = 0.0794436, size = 223, normalized size = 0.72 $-\frac{\sqrt{(a+b x)^2} \left (21 a^2 b^3 e^2 \left (11 d^2 e x+d^3+55 d e^2 x^2+165 e^3 x^3\right )+56 a^3 b^2 e^3 \left (d^2+11 d e x+55 e^2 x^2\right )+126 a^4 b e^4 (d+11 e x)+252 a^5 e^5+6 a b^4 e \left (55 d^2 e^2 x^2+11 d^3 e x+d^4+165 d e^3 x^3+330 e^4 x^4\right )+b^5 \left (55 d^3 e^2 x^2+165 d^2 e^3 x^3+11 d^4 e x+d^5+330 d e^4 x^4+462 e^5 x^5\right )\right )}{2772 e^6 (a+b x) (d+e x)^{11}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^12,x]

[Out]

-(Sqrt[(a + b*x)^2]*(252*a^5*e^5 + 126*a^4*b*e^4*(d + 11*e*x) + 56*a^3*b^2*e^3*(d^2 + 11*d*e*x + 55*e^2*x^2) +
21*a^2*b^3*e^2*(d^3 + 11*d^2*e*x + 55*d*e^2*x^2 + 165*e^3*x^3) + 6*a*b^4*e*(d^4 + 11*d^3*e*x + 55*d^2*e^2*x^2
+ 165*d*e^3*x^3 + 330*e^4*x^4) + b^5*(d^5 + 11*d^4*e*x + 55*d^3*e^2*x^2 + 165*d^2*e^3*x^3 + 330*d*e^4*x^4 + 4
62*e^5*x^5)))/(2772*e^6*(a + b*x)*(d + e*x)^11)

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Maple [A]  time = 0.156, size = 288, normalized size = 0.9 \begin{align*} -{\frac{462\,{x}^{5}{b}^{5}{e}^{5}+1980\,{x}^{4}a{b}^{4}{e}^{5}+330\,{x}^{4}{b}^{5}d{e}^{4}+3465\,{x}^{3}{a}^{2}{b}^{3}{e}^{5}+990\,{x}^{3}a{b}^{4}d{e}^{4}+165\,{x}^{3}{b}^{5}{d}^{2}{e}^{3}+3080\,{x}^{2}{a}^{3}{b}^{2}{e}^{5}+1155\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}+330\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}+55\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}+1386\,x{a}^{4}b{e}^{5}+616\,x{a}^{3}{b}^{2}d{e}^{4}+231\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}+66\,xa{b}^{4}{d}^{3}{e}^{2}+11\,x{b}^{5}{d}^{4}e+252\,{a}^{5}{e}^{5}+126\,d{e}^{4}{a}^{4}b+56\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}+21\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+6\,a{b}^{4}{d}^{4}e+{b}^{5}{d}^{5}}{2772\,{e}^{6} \left ( ex+d \right ) ^{11} \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)^12,x)

[Out]

-1/2772/e^6*(462*b^5*e^5*x^5+1980*a*b^4*e^5*x^4+330*b^5*d*e^4*x^4+3465*a^2*b^3*e^5*x^3+990*a*b^4*d*e^4*x^3+165
*b^5*d^2*e^3*x^3+3080*a^3*b^2*e^5*x^2+1155*a^2*b^3*d*e^4*x^2+330*a*b^4*d^2*e^3*x^2+55*b^5*d^3*e^2*x^2+1386*a^4
*b*e^5*x+616*a^3*b^2*d*e^4*x+231*a^2*b^3*d^2*e^3*x+66*a*b^4*d^3*e^2*x+11*b^5*d^4*e*x+252*a^5*e^5+126*a^4*b*d*e
^4+56*a^3*b^2*d^2*e^3+21*a^2*b^3*d^3*e^2+6*a*b^4*d^4*e+b^5*d^5)*((b*x+a)^2)^(5/2)/(e*x+d)^11/(b*x+a)^5

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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((b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^12,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.61667, size = 802, normalized size = 2.6 \begin{align*} -\frac{462 \, b^{5} e^{5} x^{5} + b^{5} d^{5} + 6 \, a b^{4} d^{4} e + 21 \, a^{2} b^{3} d^{3} e^{2} + 56 \, a^{3} b^{2} d^{2} e^{3} + 126 \, a^{4} b d e^{4} + 252 \, a^{5} e^{5} + 330 \,{\left (b^{5} d e^{4} + 6 \, a b^{4} e^{5}\right )} x^{4} + 165 \,{\left (b^{5} d^{2} e^{3} + 6 \, a b^{4} d e^{4} + 21 \, a^{2} b^{3} e^{5}\right )} x^{3} + 55 \,{\left (b^{5} d^{3} e^{2} + 6 \, a b^{4} d^{2} e^{3} + 21 \, a^{2} b^{3} d e^{4} + 56 \, a^{3} b^{2} e^{5}\right )} x^{2} + 11 \,{\left (b^{5} d^{4} e + 6 \, a b^{4} d^{3} e^{2} + 21 \, a^{2} b^{3} d^{2} e^{3} + 56 \, a^{3} b^{2} d e^{4} + 126 \, a^{4} b e^{5}\right )} x}{2772 \,{\left (e^{17} x^{11} + 11 \, d e^{16} x^{10} + 55 \, d^{2} e^{15} x^{9} + 165 \, d^{3} e^{14} x^{8} + 330 \, d^{4} e^{13} x^{7} + 462 \, d^{5} e^{12} x^{6} + 462 \, d^{6} e^{11} x^{5} + 330 \, d^{7} e^{10} x^{4} + 165 \, d^{8} e^{9} x^{3} + 55 \, d^{9} e^{8} x^{2} + 11 \, d^{10} e^{7} x + d^{11} 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)^12,x, algorithm="fricas")

[Out]

-1/2772*(462*b^5*e^5*x^5 + b^5*d^5 + 6*a*b^4*d^4*e + 21*a^2*b^3*d^3*e^2 + 56*a^3*b^2*d^2*e^3 + 126*a^4*b*d*e^4
+ 252*a^5*e^5 + 330*(b^5*d*e^4 + 6*a*b^4*e^5)*x^4 + 165*(b^5*d^2*e^3 + 6*a*b^4*d*e^4 + 21*a^2*b^3*e^5)*x^3 +
55*(b^5*d^3*e^2 + 6*a*b^4*d^2*e^3 + 21*a^2*b^3*d*e^4 + 56*a^3*b^2*e^5)*x^2 + 11*(b^5*d^4*e + 6*a*b^4*d^3*e^2 +
21*a^2*b^3*d^2*e^3 + 56*a^3*b^2*d*e^4 + 126*a^4*b*e^5)*x)/(e^17*x^11 + 11*d*e^16*x^10 + 55*d^2*e^15*x^9 + 165
*d^3*e^14*x^8 + 330*d^4*e^13*x^7 + 462*d^5*e^12*x^6 + 462*d^6*e^11*x^5 + 330*d^7*e^10*x^4 + 165*d^8*e^9*x^3 +
55*d^9*e^8*x^2 + 11*d^10*e^7*x + d^11*e^6)

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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((b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**12,x)

[Out]

Timed out

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Giac [A]  time = 1.17248, size = 514, normalized size = 1.67 \begin{align*} -\frac{{\left (462 \, b^{5} x^{5} e^{5} \mathrm{sgn}\left (b x + a\right ) + 330 \, b^{5} d x^{4} e^{4} \mathrm{sgn}\left (b x + a\right ) + 165 \, b^{5} d^{2} x^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) + 55 \, b^{5} d^{3} x^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 11 \, b^{5} d^{4} x e \mathrm{sgn}\left (b x + a\right ) + b^{5} d^{5} \mathrm{sgn}\left (b x + a\right ) + 1980 \, a b^{4} x^{4} e^{5} \mathrm{sgn}\left (b x + a\right ) + 990 \, a b^{4} d x^{3} e^{4} \mathrm{sgn}\left (b x + a\right ) + 330 \, a b^{4} d^{2} x^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) + 66 \, a b^{4} d^{3} x e^{2} \mathrm{sgn}\left (b x + a\right ) + 6 \, a b^{4} d^{4} e \mathrm{sgn}\left (b x + a\right ) + 3465 \, a^{2} b^{3} x^{3} e^{5} \mathrm{sgn}\left (b x + a\right ) + 1155 \, a^{2} b^{3} d x^{2} e^{4} \mathrm{sgn}\left (b x + a\right ) + 231 \, a^{2} b^{3} d^{2} x e^{3} \mathrm{sgn}\left (b x + a\right ) + 21 \, a^{2} b^{3} d^{3} e^{2} \mathrm{sgn}\left (b x + a\right ) + 3080 \, a^{3} b^{2} x^{2} e^{5} \mathrm{sgn}\left (b x + a\right ) + 616 \, a^{3} b^{2} d x e^{4} \mathrm{sgn}\left (b x + a\right ) + 56 \, a^{3} b^{2} d^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) + 1386 \, a^{4} b x e^{5} \mathrm{sgn}\left (b x + a\right ) + 126 \, a^{4} b d e^{4} \mathrm{sgn}\left (b x + a\right ) + 252 \, a^{5} e^{5} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-6\right )}}{2772 \,{\left (x e + d\right )}^{11}} \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)^12,x, algorithm="giac")

[Out]

-1/2772*(462*b^5*x^5*e^5*sgn(b*x + a) + 330*b^5*d*x^4*e^4*sgn(b*x + a) + 165*b^5*d^2*x^3*e^3*sgn(b*x + a) + 55
*b^5*d^3*x^2*e^2*sgn(b*x + a) + 11*b^5*d^4*x*e*sgn(b*x + a) + b^5*d^5*sgn(b*x + a) + 1980*a*b^4*x^4*e^5*sgn(b*
x + a) + 990*a*b^4*d*x^3*e^4*sgn(b*x + a) + 330*a*b^4*d^2*x^2*e^3*sgn(b*x + a) + 66*a*b^4*d^3*x*e^2*sgn(b*x +
a) + 6*a*b^4*d^4*e*sgn(b*x + a) + 3465*a^2*b^3*x^3*e^5*sgn(b*x + a) + 1155*a^2*b^3*d*x^2*e^4*sgn(b*x + a) + 23
1*a^2*b^3*d^2*x*e^3*sgn(b*x + a) + 21*a^2*b^3*d^3*e^2*sgn(b*x + a) + 3080*a^3*b^2*x^2*e^5*sgn(b*x + a) + 616*a
^3*b^2*d*x*e^4*sgn(b*x + a) + 56*a^3*b^2*d^2*e^3*sgn(b*x + a) + 1386*a^4*b*x*e^5*sgn(b*x + a) + 126*a^4*b*d*e^
4*sgn(b*x + a) + 252*a^5*e^5*sgn(b*x + a))*e^(-6)/(x*e + d)^11