3.1700 \(\int \frac{(a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^{11/2}} \, dx\)

Optimal. Leaf size=314 \[ \frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x}}{e^6 (a+b x)}+\frac{10 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^6 (a+b x) \sqrt{d+e x}}-\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{3 e^6 (a+b x) (d+e x)^{3/2}}+\frac{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^6 (a+b x) (d+e x)^{5/2}}-\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{7 e^6 (a+b x) (d+e x)^{7/2}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{9 e^6 (a+b x) (d+e x)^{9/2}} \]

[Out]

(2*(b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^6*(a + b*x)*(d + e*x)^(9/2)) - (10*b*(b*d - a*e)^4*Sqrt[a
^2 + 2*a*b*x + b^2*x^2])/(7*e^6*(a + b*x)*(d + e*x)^(7/2)) + (4*b^2*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2
])/(e^6*(a + b*x)*(d + e*x)^(5/2)) - (20*b^3*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^6*(a + b*x)*(d
+ e*x)^(3/2)) + (10*b^4*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x)*Sqrt[d + e*x]) + (2*b^5*Sqrt
[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x))

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Rubi [A]  time = 0.0963662, antiderivative size = 314, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {646, 43} \[ \frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x}}{e^6 (a+b x)}+\frac{10 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^6 (a+b x) \sqrt{d+e x}}-\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{3 e^6 (a+b x) (d+e x)^{3/2}}+\frac{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^6 (a+b x) (d+e x)^{5/2}}-\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{7 e^6 (a+b x) (d+e x)^{7/2}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{9 e^6 (a+b x) (d+e x)^{9/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^(11/2),x]

[Out]

(2*(b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^6*(a + b*x)*(d + e*x)^(9/2)) - (10*b*(b*d - a*e)^4*Sqrt[a
^2 + 2*a*b*x + b^2*x^2])/(7*e^6*(a + b*x)*(d + e*x)^(7/2)) + (4*b^2*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2
])/(e^6*(a + b*x)*(d + e*x)^(5/2)) - (20*b^3*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^6*(a + b*x)*(d
+ e*x)^(3/2)) + (10*b^4*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x)*Sqrt[d + e*x]) + (2*b^5*Sqrt
[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x))

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)^{11/2}} \, 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)^{11/2}} \, 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)^{11/2}}+\frac{5 b^6 (b d-a e)^4}{e^5 (d+e x)^{9/2}}-\frac{10 b^7 (b d-a e)^3}{e^5 (d+e x)^{7/2}}+\frac{10 b^8 (b d-a e)^2}{e^5 (d+e x)^{5/2}}-\frac{5 b^9 (b d-a e)}{e^5 (d+e x)^{3/2}}+\frac{b^{10}}{e^5 \sqrt{d+e x}}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{2 (b d-a e)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{9 e^6 (a+b x) (d+e x)^{9/2}}-\frac{10 b (b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}{7 e^6 (a+b x) (d+e x)^{7/2}}+\frac{4 b^2 (b d-a e)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{e^6 (a+b x) (d+e x)^{5/2}}-\frac{20 b^3 (b d-a e)^2 \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^6 (a+b x) (d+e x)^{3/2}}+\frac{10 b^4 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}{e^6 (a+b x) \sqrt{d+e x}}+\frac{2 b^5 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}\\ \end{align*}

Mathematica [A]  time = 0.119564, size = 235, normalized size = 0.75 \[ -\frac{2 \sqrt{(a+b x)^2} \left (2 a^2 b^3 e^2 \left (72 d^2 e x+16 d^3+126 d e^2 x^2+105 e^3 x^3\right )+2 a^3 b^2 e^3 \left (8 d^2+36 d e x+63 e^2 x^2\right )+5 a^4 b e^4 (2 d+9 e x)+7 a^5 e^5+a b^4 e \left (1008 d^2 e^2 x^2+576 d^3 e x+128 d^4+840 d e^3 x^3+315 e^4 x^4\right )+b^5 \left (-\left (2016 d^3 e^2 x^2+1680 d^2 e^3 x^3+1152 d^4 e x+256 d^5+630 d e^4 x^4+63 e^5 x^5\right )\right )\right )}{63 e^6 (a+b x) (d+e x)^{9/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^(11/2),x]

[Out]

(-2*Sqrt[(a + b*x)^2]*(7*a^5*e^5 + 5*a^4*b*e^4*(2*d + 9*e*x) + 2*a^3*b^2*e^3*(8*d^2 + 36*d*e*x + 63*e^2*x^2) +
2*a^2*b^3*e^2*(16*d^3 + 72*d^2*e*x + 126*d*e^2*x^2 + 105*e^3*x^3) + a*b^4*e*(128*d^4 + 576*d^3*e*x + 1008*d^2
*e^2*x^2 + 840*d*e^3*x^3 + 315*e^4*x^4) - b^5*(256*d^5 + 1152*d^4*e*x + 2016*d^3*e^2*x^2 + 1680*d^2*e^3*x^3 +
630*d*e^4*x^4 + 63*e^5*x^5)))/(63*e^6*(a + b*x)*(d + e*x)^(9/2))

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Maple [A]  time = 0.154, size = 289, normalized size = 0.9 \begin{align*} -{\frac{-126\,{x}^{5}{b}^{5}{e}^{5}+630\,{x}^{4}a{b}^{4}{e}^{5}-1260\,{x}^{4}{b}^{5}d{e}^{4}+420\,{x}^{3}{a}^{2}{b}^{3}{e}^{5}+1680\,{x}^{3}a{b}^{4}d{e}^{4}-3360\,{x}^{3}{b}^{5}{d}^{2}{e}^{3}+252\,{x}^{2}{a}^{3}{b}^{2}{e}^{5}+504\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}+2016\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}-4032\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}+90\,x{a}^{4}b{e}^{5}+144\,x{a}^{3}{b}^{2}d{e}^{4}+288\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}+1152\,xa{b}^{4}{d}^{3}{e}^{2}-2304\,x{b}^{5}{d}^{4}e+14\,{a}^{5}{e}^{5}+20\,d{e}^{4}{a}^{4}b+32\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}+64\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+256\,a{b}^{4}{d}^{4}e-512\,{b}^{5}{d}^{5}}{63\, \left ( bx+a \right ) ^{5}{e}^{6}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}} \left ( ex+d \right ) ^{-{\frac{9}{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)^(11/2),x)

[Out]

-2/63/(e*x+d)^(9/2)*(-63*b^5*e^5*x^5+315*a*b^4*e^5*x^4-630*b^5*d*e^4*x^4+210*a^2*b^3*e^5*x^3+840*a*b^4*d*e^4*x
^3-1680*b^5*d^2*e^3*x^3+126*a^3*b^2*e^5*x^2+252*a^2*b^3*d*e^4*x^2+1008*a*b^4*d^2*e^3*x^2-2016*b^5*d^3*e^2*x^2+
45*a^4*b*e^5*x+72*a^3*b^2*d*e^4*x+144*a^2*b^3*d^2*e^3*x+576*a*b^4*d^3*e^2*x-1152*b^5*d^4*e*x+7*a^5*e^5+10*a^4*
b*d*e^4+16*a^3*b^2*d^2*e^3+32*a^2*b^3*d^3*e^2+128*a*b^4*d^4*e-256*b^5*d^5)*((b*x+a)^2)^(5/2)/e^6/(b*x+a)^5

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Maxima [A]  time = 1.13801, size = 412, normalized size = 1.31 \begin{align*} \frac{2 \,{\left (63 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} - 128 \, a b^{4} d^{4} e - 32 \, a^{2} b^{3} d^{3} e^{2} - 16 \, a^{3} b^{2} d^{2} e^{3} - 10 \, a^{4} b d e^{4} - 7 \, a^{5} e^{5} + 315 \,{\left (2 \, b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} + 210 \,{\left (8 \, b^{5} d^{2} e^{3} - 4 \, a b^{4} d e^{4} - a^{2} b^{3} e^{5}\right )} x^{3} + 126 \,{\left (16 \, b^{5} d^{3} e^{2} - 8 \, a b^{4} d^{2} e^{3} - 2 \, a^{2} b^{3} d e^{4} - a^{3} b^{2} e^{5}\right )} x^{2} + 9 \,{\left (128 \, b^{5} d^{4} e - 64 \, a b^{4} d^{3} e^{2} - 16 \, a^{2} b^{3} d^{2} e^{3} - 8 \, a^{3} b^{2} d e^{4} - 5 \, a^{4} b e^{5}\right )} x\right )}}{63 \,{\left (e^{10} x^{4} + 4 \, d e^{9} x^{3} + 6 \, d^{2} e^{8} x^{2} + 4 \, d^{3} e^{7} x + d^{4} e^{6}\right )} \sqrt{e x + d}} \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)^(11/2),x, algorithm="maxima")

[Out]

2/63*(63*b^5*e^5*x^5 + 256*b^5*d^5 - 128*a*b^4*d^4*e - 32*a^2*b^3*d^3*e^2 - 16*a^3*b^2*d^2*e^3 - 10*a^4*b*d*e^
4 - 7*a^5*e^5 + 315*(2*b^5*d*e^4 - a*b^4*e^5)*x^4 + 210*(8*b^5*d^2*e^3 - 4*a*b^4*d*e^4 - a^2*b^3*e^5)*x^3 + 12
6*(16*b^5*d^3*e^2 - 8*a*b^4*d^2*e^3 - 2*a^2*b^3*d*e^4 - a^3*b^2*e^5)*x^2 + 9*(128*b^5*d^4*e - 64*a*b^4*d^3*e^2
- 16*a^2*b^3*d^2*e^3 - 8*a^3*b^2*d*e^4 - 5*a^4*b*e^5)*x)/((e^10*x^4 + 4*d*e^9*x^3 + 6*d^2*e^8*x^2 + 4*d^3*e^7
*x + d^4*e^6)*sqrt(e*x + d))

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Fricas [A]  time = 1.65967, size = 657, normalized size = 2.09 \begin{align*} \frac{2 \,{\left (63 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} - 128 \, a b^{4} d^{4} e - 32 \, a^{2} b^{3} d^{3} e^{2} - 16 \, a^{3} b^{2} d^{2} e^{3} - 10 \, a^{4} b d e^{4} - 7 \, a^{5} e^{5} + 315 \,{\left (2 \, b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} + 210 \,{\left (8 \, b^{5} d^{2} e^{3} - 4 \, a b^{4} d e^{4} - a^{2} b^{3} e^{5}\right )} x^{3} + 126 \,{\left (16 \, b^{5} d^{3} e^{2} - 8 \, a b^{4} d^{2} e^{3} - 2 \, a^{2} b^{3} d e^{4} - a^{3} b^{2} e^{5}\right )} x^{2} + 9 \,{\left (128 \, b^{5} d^{4} e - 64 \, a b^{4} d^{3} e^{2} - 16 \, a^{2} b^{3} d^{2} e^{3} - 8 \, a^{3} b^{2} d e^{4} - 5 \, a^{4} b e^{5}\right )} x\right )} \sqrt{e x + d}}{63 \,{\left (e^{11} x^{5} + 5 \, d e^{10} x^{4} + 10 \, d^{2} e^{9} x^{3} + 10 \, d^{3} e^{8} x^{2} + 5 \, d^{4} e^{7} x + d^{5} 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)^(11/2),x, algorithm="fricas")

[Out]

2/63*(63*b^5*e^5*x^5 + 256*b^5*d^5 - 128*a*b^4*d^4*e - 32*a^2*b^3*d^3*e^2 - 16*a^3*b^2*d^2*e^3 - 10*a^4*b*d*e^
4 - 7*a^5*e^5 + 315*(2*b^5*d*e^4 - a*b^4*e^5)*x^4 + 210*(8*b^5*d^2*e^3 - 4*a*b^4*d*e^4 - a^2*b^3*e^5)*x^3 + 12
6*(16*b^5*d^3*e^2 - 8*a*b^4*d^2*e^3 - 2*a^2*b^3*d*e^4 - a^3*b^2*e^5)*x^2 + 9*(128*b^5*d^4*e - 64*a*b^4*d^3*e^2
- 16*a^2*b^3*d^2*e^3 - 8*a^3*b^2*d*e^4 - 5*a^4*b*e^5)*x)*sqrt(e*x + d)/(e^11*x^5 + 5*d*e^10*x^4 + 10*d^2*e^9*
x^3 + 10*d^3*e^8*x^2 + 5*d^4*e^7*x + d^5*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)**(11/2),x)

[Out]

Timed out

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Giac [A]  time = 1.18664, size = 608, normalized size = 1.94 \begin{align*} 2 \, \sqrt{x e + d} b^{5} e^{\left (-6\right )} \mathrm{sgn}\left (b x + a\right ) + \frac{2 \,{\left (315 \,{\left (x e + d\right )}^{4} b^{5} d \mathrm{sgn}\left (b x + a\right ) - 210 \,{\left (x e + d\right )}^{3} b^{5} d^{2} \mathrm{sgn}\left (b x + a\right ) + 126 \,{\left (x e + d\right )}^{2} b^{5} d^{3} \mathrm{sgn}\left (b x + a\right ) - 45 \,{\left (x e + d\right )} b^{5} d^{4} \mathrm{sgn}\left (b x + a\right ) + 7 \, b^{5} d^{5} \mathrm{sgn}\left (b x + a\right ) - 315 \,{\left (x e + d\right )}^{4} a b^{4} e \mathrm{sgn}\left (b x + a\right ) + 420 \,{\left (x e + d\right )}^{3} a b^{4} d e \mathrm{sgn}\left (b x + a\right ) - 378 \,{\left (x e + d\right )}^{2} a b^{4} d^{2} e \mathrm{sgn}\left (b x + a\right ) + 180 \,{\left (x e + d\right )} a b^{4} d^{3} e \mathrm{sgn}\left (b x + a\right ) - 35 \, a b^{4} d^{4} e \mathrm{sgn}\left (b x + a\right ) - 210 \,{\left (x e + d\right )}^{3} a^{2} b^{3} e^{2} \mathrm{sgn}\left (b x + a\right ) + 378 \,{\left (x e + d\right )}^{2} a^{2} b^{3} d e^{2} \mathrm{sgn}\left (b x + a\right ) - 270 \,{\left (x e + d\right )} a^{2} b^{3} d^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 70 \, a^{2} b^{3} d^{3} e^{2} \mathrm{sgn}\left (b x + a\right ) - 126 \,{\left (x e + d\right )}^{2} a^{3} b^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) + 180 \,{\left (x e + d\right )} a^{3} b^{2} d e^{3} \mathrm{sgn}\left (b x + a\right ) - 70 \, a^{3} b^{2} d^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) - 45 \,{\left (x e + d\right )} a^{4} b e^{4} \mathrm{sgn}\left (b x + a\right ) + 35 \, a^{4} b d e^{4} \mathrm{sgn}\left (b x + a\right ) - 7 \, a^{5} e^{5} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-6\right )}}{63 \,{\left (x e + d\right )}^{\frac{9}{2}}} \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)^(11/2),x, algorithm="giac")

[Out]

2*sqrt(x*e + d)*b^5*e^(-6)*sgn(b*x + a) + 2/63*(315*(x*e + d)^4*b^5*d*sgn(b*x + a) - 210*(x*e + d)^3*b^5*d^2*s
gn(b*x + a) + 126*(x*e + d)^2*b^5*d^3*sgn(b*x + a) - 45*(x*e + d)*b^5*d^4*sgn(b*x + a) + 7*b^5*d^5*sgn(b*x + a
) - 315*(x*e + d)^4*a*b^4*e*sgn(b*x + a) + 420*(x*e + d)^3*a*b^4*d*e*sgn(b*x + a) - 378*(x*e + d)^2*a*b^4*d^2*
e*sgn(b*x + a) + 180*(x*e + d)*a*b^4*d^3*e*sgn(b*x + a) - 35*a*b^4*d^4*e*sgn(b*x + a) - 210*(x*e + d)^3*a^2*b^
3*e^2*sgn(b*x + a) + 378*(x*e + d)^2*a^2*b^3*d*e^2*sgn(b*x + a) - 270*(x*e + d)*a^2*b^3*d^2*e^2*sgn(b*x + a) +
70*a^2*b^3*d^3*e^2*sgn(b*x + a) - 126*(x*e + d)^2*a^3*b^2*e^3*sgn(b*x + a) + 180*(x*e + d)*a^3*b^2*d*e^3*sgn(
b*x + a) - 70*a^3*b^2*d^2*e^3*sgn(b*x + a) - 45*(x*e + d)*a^4*b*e^4*sgn(b*x + a) + 35*a^4*b*d*e^4*sgn(b*x + a)
- 7*a^5*e^5*sgn(b*x + a))*e^(-6)/(x*e + d)^(9/2)