3.1701 \(\int \frac{(a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^{13/2}} \, dx\)

Optimal. Leaf size=316 \[ -\frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}{e^6 (a+b x) \sqrt{d+e x}}+\frac{10 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{3 e^6 (a+b x) (d+e x)^{3/2}}-\frac{4 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^6 (a+b x) (d+e x)^{5/2}}+\frac{20 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{7 e^6 (a+b x) (d+e x)^{7/2}}-\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{9 e^6 (a+b x) (d+e x)^{9/2}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{11 e^6 (a+b x) (d+e x)^{11/2}} \]

[Out]

(2*(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)^(11/2)) - (10*b*(b*d - a*e)^4*Sqrt
[a^2 + 2*a*b*x + b^2*x^2])/(9*e^6*(a + b*x)*(d + e*x)^(9/2)) + (20*b^2*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*
x^2])/(7*e^6*(a + b*x)*(d + e*x)^(7/2)) - (4*b^3*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x)*(
d + e*x)^(5/2)) + (10*b^4*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^6*(a + b*x)*(d + e*x)^(3/2)) - (2*b^
5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x)*Sqrt[d + e*x])

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Rubi [A]  time = 0.0957733, antiderivative size = 316, 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} \[ -\frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}{e^6 (a+b x) \sqrt{d+e x}}+\frac{10 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{3 e^6 (a+b x) (d+e x)^{3/2}}-\frac{4 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^6 (a+b x) (d+e x)^{5/2}}+\frac{20 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{7 e^6 (a+b x) (d+e x)^{7/2}}-\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{9 e^6 (a+b x) (d+e x)^{9/2}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{11 e^6 (a+b x) (d+e x)^{11/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^(13/2),x]

[Out]

(2*(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)^(11/2)) - (10*b*(b*d - a*e)^4*Sqrt
[a^2 + 2*a*b*x + b^2*x^2])/(9*e^6*(a + b*x)*(d + e*x)^(9/2)) + (20*b^2*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*
x^2])/(7*e^6*(a + b*x)*(d + e*x)^(7/2)) - (4*b^3*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x)*(
d + e*x)^(5/2)) + (10*b^4*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^6*(a + b*x)*(d + e*x)^(3/2)) - (2*b^
5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x)*Sqrt[d + e*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)^{13/2}} \, 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)^{13/2}} \, 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)^{13/2}}+\frac{5 b^6 (b d-a e)^4}{e^5 (d+e x)^{11/2}}-\frac{10 b^7 (b d-a e)^3}{e^5 (d+e x)^{9/2}}+\frac{10 b^8 (b d-a e)^2}{e^5 (d+e x)^{7/2}}-\frac{5 b^9 (b d-a e)}{e^5 (d+e x)^{5/2}}+\frac{b^{10}}{e^5 (d+e x)^{3/2}}\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}}{11 e^6 (a+b x) (d+e x)^{11/2}}-\frac{10 b (b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}{9 e^6 (a+b x) (d+e x)^{9/2}}+\frac{20 b^2 (b d-a e)^3 \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^3 (b d-a e)^2 \sqrt{a^2+2 a b x+b^2 x^2}}{e^6 (a+b x) (d+e x)^{5/2}}+\frac{10 b^4 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^6 (a+b x) (d+e x)^{3/2}}-\frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}{e^6 (a+b x) \sqrt{d+e x}}\\ \end{align*}

Mathematica [A]  time = 0.117079, size = 234, normalized size = 0.74 \[ -\frac{2 \sqrt{(a+b x)^2} \left (6 a^2 b^3 e^2 \left (88 d^2 e x+16 d^3+198 d e^2 x^2+231 e^3 x^3\right )+10 a^3 b^2 e^3 \left (8 d^2+44 d e x+99 e^2 x^2\right )+35 a^4 b e^4 (2 d+11 e x)+63 a^5 e^5+a b^4 e \left (1584 d^2 e^2 x^2+704 d^3 e x+128 d^4+1848 d e^3 x^3+1155 e^4 x^4\right )+b^5 \left (3168 d^3 e^2 x^2+3696 d^2 e^3 x^3+1408 d^4 e x+256 d^5+2310 d e^4 x^4+693 e^5 x^5\right )\right )}{693 e^6 (a+b x) (d+e x)^{11/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^(13/2),x]

[Out]

(-2*Sqrt[(a + b*x)^2]*(63*a^5*e^5 + 35*a^4*b*e^4*(2*d + 11*e*x) + 10*a^3*b^2*e^3*(8*d^2 + 44*d*e*x + 99*e^2*x^
2) + 6*a^2*b^3*e^2*(16*d^3 + 88*d^2*e*x + 198*d*e^2*x^2 + 231*e^3*x^3) + a*b^4*e*(128*d^4 + 704*d^3*e*x + 1584
*d^2*e^2*x^2 + 1848*d*e^3*x^3 + 1155*e^4*x^4) + b^5*(256*d^5 + 1408*d^4*e*x + 3168*d^3*e^2*x^2 + 3696*d^2*e^3*
x^3 + 2310*d*e^4*x^4 + 693*e^5*x^5)))/(693*e^6*(a + b*x)*(d + e*x)^(11/2))

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Maple [A]  time = 0.155, size = 289, normalized size = 0.9 \begin{align*} -{\frac{1386\,{x}^{5}{b}^{5}{e}^{5}+2310\,{x}^{4}a{b}^{4}{e}^{5}+4620\,{x}^{4}{b}^{5}d{e}^{4}+2772\,{x}^{3}{a}^{2}{b}^{3}{e}^{5}+3696\,{x}^{3}a{b}^{4}d{e}^{4}+7392\,{x}^{3}{b}^{5}{d}^{2}{e}^{3}+1980\,{x}^{2}{a}^{3}{b}^{2}{e}^{5}+2376\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}+3168\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}+6336\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}+770\,x{a}^{4}b{e}^{5}+880\,x{a}^{3}{b}^{2}d{e}^{4}+1056\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}+1408\,xa{b}^{4}{d}^{3}{e}^{2}+2816\,x{b}^{5}{d}^{4}e+126\,{a}^{5}{e}^{5}+140\,d{e}^{4}{a}^{4}b+160\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}+192\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+256\,a{b}^{4}{d}^{4}e+512\,{b}^{5}{d}^{5}}{693\, \left ( bx+a \right ) ^{5}{e}^{6}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}} \left ( ex+d \right ) ^{-{\frac{11}{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)^(13/2),x)

[Out]

-2/693/(e*x+d)^(11/2)*(693*b^5*e^5*x^5+1155*a*b^4*e^5*x^4+2310*b^5*d*e^4*x^4+1386*a^2*b^3*e^5*x^3+1848*a*b^4*d
*e^4*x^3+3696*b^5*d^2*e^3*x^3+990*a^3*b^2*e^5*x^2+1188*a^2*b^3*d*e^4*x^2+1584*a*b^4*d^2*e^3*x^2+3168*b^5*d^3*e
^2*x^2+385*a^4*b*e^5*x+440*a^3*b^2*d*e^4*x+528*a^2*b^3*d^2*e^3*x+704*a*b^4*d^3*e^2*x+1408*b^5*d^4*e*x+63*a^5*e
^5+70*a^4*b*d*e^4+80*a^3*b^2*d^2*e^3+96*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 = 1.13069, size = 425, normalized size = 1.34 \begin{align*} -\frac{2 \,{\left (693 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} + 128 \, a b^{4} d^{4} e + 96 \, a^{2} b^{3} d^{3} e^{2} + 80 \, a^{3} b^{2} d^{2} e^{3} + 70 \, a^{4} b d e^{4} + 63 \, a^{5} e^{5} + 1155 \,{\left (2 \, b^{5} d e^{4} + a b^{4} e^{5}\right )} x^{4} + 462 \,{\left (8 \, b^{5} d^{2} e^{3} + 4 \, a b^{4} d e^{4} + 3 \, a^{2} b^{3} e^{5}\right )} x^{3} + 198 \,{\left (16 \, b^{5} d^{3} e^{2} + 8 \, a b^{4} d^{2} e^{3} + 6 \, a^{2} b^{3} d e^{4} + 5 \, a^{3} b^{2} e^{5}\right )} x^{2} + 11 \,{\left (128 \, b^{5} d^{4} e + 64 \, a b^{4} d^{3} e^{2} + 48 \, a^{2} b^{3} d^{2} e^{3} + 40 \, a^{3} b^{2} d e^{4} + 35 \, a^{4} b e^{5}\right )} x\right )}}{693 \,{\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 )} \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)^(13/2),x, algorithm="maxima")

[Out]

-2/693*(693*b^5*e^5*x^5 + 256*b^5*d^5 + 128*a*b^4*d^4*e + 96*a^2*b^3*d^3*e^2 + 80*a^3*b^2*d^2*e^3 + 70*a^4*b*d
*e^4 + 63*a^5*e^5 + 1155*(2*b^5*d*e^4 + a*b^4*e^5)*x^4 + 462*(8*b^5*d^2*e^3 + 4*a*b^4*d*e^4 + 3*a^2*b^3*e^5)*x
^3 + 198*(16*b^5*d^3*e^2 + 8*a*b^4*d^2*e^3 + 6*a^2*b^3*d*e^4 + 5*a^3*b^2*e^5)*x^2 + 11*(128*b^5*d^4*e + 64*a*b
^4*d^3*e^2 + 48*a^2*b^3*d^2*e^3 + 40*a^3*b^2*d*e^4 + 35*a^4*b*e^5)*x)/((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)*sqrt(e*x + d))

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Fricas [A]  time = 1.57252, size = 698, normalized size = 2.21 \begin{align*} -\frac{2 \,{\left (693 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} + 128 \, a b^{4} d^{4} e + 96 \, a^{2} b^{3} d^{3} e^{2} + 80 \, a^{3} b^{2} d^{2} e^{3} + 70 \, a^{4} b d e^{4} + 63 \, a^{5} e^{5} + 1155 \,{\left (2 \, b^{5} d e^{4} + a b^{4} e^{5}\right )} x^{4} + 462 \,{\left (8 \, b^{5} d^{2} e^{3} + 4 \, a b^{4} d e^{4} + 3 \, a^{2} b^{3} e^{5}\right )} x^{3} + 198 \,{\left (16 \, b^{5} d^{3} e^{2} + 8 \, a b^{4} d^{2} e^{3} + 6 \, a^{2} b^{3} d e^{4} + 5 \, a^{3} b^{2} e^{5}\right )} x^{2} + 11 \,{\left (128 \, b^{5} d^{4} e + 64 \, a b^{4} d^{3} e^{2} + 48 \, a^{2} b^{3} d^{2} e^{3} + 40 \, a^{3} b^{2} d e^{4} + 35 \, a^{4} b e^{5}\right )} x\right )} \sqrt{e x + d}}{693 \,{\left (e^{12} x^{6} + 6 \, d e^{11} x^{5} + 15 \, d^{2} e^{10} x^{4} + 20 \, d^{3} e^{9} x^{3} + 15 \, d^{4} e^{8} x^{2} + 6 \, d^{5} e^{7} x + d^{6} 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)^(13/2),x, algorithm="fricas")

[Out]

-2/693*(693*b^5*e^5*x^5 + 256*b^5*d^5 + 128*a*b^4*d^4*e + 96*a^2*b^3*d^3*e^2 + 80*a^3*b^2*d^2*e^3 + 70*a^4*b*d
*e^4 + 63*a^5*e^5 + 1155*(2*b^5*d*e^4 + a*b^4*e^5)*x^4 + 462*(8*b^5*d^2*e^3 + 4*a*b^4*d*e^4 + 3*a^2*b^3*e^5)*x
^3 + 198*(16*b^5*d^3*e^2 + 8*a*b^4*d^2*e^3 + 6*a^2*b^3*d*e^4 + 5*a^3*b^2*e^5)*x^2 + 11*(128*b^5*d^4*e + 64*a*b
^4*d^3*e^2 + 48*a^2*b^3*d^2*e^3 + 40*a^3*b^2*d*e^4 + 35*a^4*b*e^5)*x)*sqrt(e*x + d)/(e^12*x^6 + 6*d*e^11*x^5 +
15*d^2*e^10*x^4 + 20*d^3*e^9*x^3 + 15*d^4*e^8*x^2 + 6*d^5*e^7*x + d^6*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)**(13/2),x)

[Out]

Timed out

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Giac [A]  time = 1.21552, size = 603, normalized size = 1.91 \begin{align*} -\frac{2 \,{\left (693 \,{\left (x e + d\right )}^{5} b^{5} \mathrm{sgn}\left (b x + a\right ) - 1155 \,{\left (x e + d\right )}^{4} b^{5} d \mathrm{sgn}\left (b x + a\right ) + 1386 \,{\left (x e + d\right )}^{3} b^{5} d^{2} \mathrm{sgn}\left (b x + a\right ) - 990 \,{\left (x e + d\right )}^{2} b^{5} d^{3} \mathrm{sgn}\left (b x + a\right ) + 385 \,{\left (x e + d\right )} b^{5} d^{4} \mathrm{sgn}\left (b x + a\right ) - 63 \, b^{5} d^{5} \mathrm{sgn}\left (b x + a\right ) + 1155 \,{\left (x e + d\right )}^{4} a b^{4} e \mathrm{sgn}\left (b x + a\right ) - 2772 \,{\left (x e + d\right )}^{3} a b^{4} d e \mathrm{sgn}\left (b x + a\right ) + 2970 \,{\left (x e + d\right )}^{2} a b^{4} d^{2} e \mathrm{sgn}\left (b x + a\right ) - 1540 \,{\left (x e + d\right )} a b^{4} d^{3} e \mathrm{sgn}\left (b x + a\right ) + 315 \, a b^{4} d^{4} e \mathrm{sgn}\left (b x + a\right ) + 1386 \,{\left (x e + d\right )}^{3} a^{2} b^{3} e^{2} \mathrm{sgn}\left (b x + a\right ) - 2970 \,{\left (x e + d\right )}^{2} a^{2} b^{3} d e^{2} \mathrm{sgn}\left (b x + a\right ) + 2310 \,{\left (x e + d\right )} a^{2} b^{3} d^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) - 630 \, a^{2} b^{3} d^{3} e^{2} \mathrm{sgn}\left (b x + a\right ) + 990 \,{\left (x e + d\right )}^{2} a^{3} b^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) - 1540 \,{\left (x e + d\right )} a^{3} b^{2} d e^{3} \mathrm{sgn}\left (b x + a\right ) + 630 \, a^{3} b^{2} d^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) + 385 \,{\left (x e + d\right )} a^{4} b e^{4} \mathrm{sgn}\left (b x + a\right ) - 315 \, a^{4} b d e^{4} \mathrm{sgn}\left (b x + a\right ) + 63 \, a^{5} e^{5} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-6\right )}}{693 \,{\left (x e + d\right )}^{\frac{11}{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)^(13/2),x, algorithm="giac")

[Out]

-2/693*(693*(x*e + d)^5*b^5*sgn(b*x + a) - 1155*(x*e + d)^4*b^5*d*sgn(b*x + a) + 1386*(x*e + d)^3*b^5*d^2*sgn(
b*x + a) - 990*(x*e + d)^2*b^5*d^3*sgn(b*x + a) + 385*(x*e + d)*b^5*d^4*sgn(b*x + a) - 63*b^5*d^5*sgn(b*x + a)
+ 1155*(x*e + d)^4*a*b^4*e*sgn(b*x + a) - 2772*(x*e + d)^3*a*b^4*d*e*sgn(b*x + a) + 2970*(x*e + d)^2*a*b^4*d^
2*e*sgn(b*x + a) - 1540*(x*e + d)*a*b^4*d^3*e*sgn(b*x + a) + 315*a*b^4*d^4*e*sgn(b*x + a) + 1386*(x*e + d)^3*a
^2*b^3*e^2*sgn(b*x + a) - 2970*(x*e + d)^2*a^2*b^3*d*e^2*sgn(b*x + a) + 2310*(x*e + d)*a^2*b^3*d^2*e^2*sgn(b*x
+ a) - 630*a^2*b^3*d^3*e^2*sgn(b*x + a) + 990*(x*e + d)^2*a^3*b^2*e^3*sgn(b*x + a) - 1540*(x*e + d)*a^3*b^2*d
*e^3*sgn(b*x + a) + 630*a^3*b^2*d^2*e^3*sgn(b*x + a) + 385*(x*e + d)*a^4*b*e^4*sgn(b*x + a) - 315*a^4*b*d*e^4*
sgn(b*x + a) + 63*a^5*e^5*sgn(b*x + a))*e^(-6)/(x*e + d)^(11/2)