3.1631 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=179 \[ -\frac{4 b^5 (d+e x)^{9/2} (b d-a e)}{3 e^7}+\frac{30 b^4 (d+e x)^{7/2} (b d-a e)^2}{7 e^7}-\frac{8 b^3 (d+e x)^{5/2} (b d-a e)^3}{e^7}+\frac{10 b^2 (d+e x)^{3/2} (b d-a e)^4}{e^7}-\frac{12 b \sqrt{d+e x} (b d-a e)^5}{e^7}-\frac{2 (b d-a e)^6}{e^7 \sqrt{d+e x}}+\frac{2 b^6 (d+e x)^{11/2}}{11 e^7} \]

[Out]

(-2*(b*d - a*e)^6)/(e^7*Sqrt[d + e*x]) - (12*b*(b*d - a*e)^5*Sqrt[d + e*x])/e^7
+ (10*b^2*(b*d - a*e)^4*(d + e*x)^(3/2))/e^7 - (8*b^3*(b*d - a*e)^3*(d + e*x)^(5
/2))/e^7 + (30*b^4*(b*d - a*e)^2*(d + e*x)^(7/2))/(7*e^7) - (4*b^5*(b*d - a*e)*(
d + e*x)^(9/2))/(3*e^7) + (2*b^6*(d + e*x)^(11/2))/(11*e^7)

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Rubi [A]  time = 0.17501, antiderivative size = 179, 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 \[ -\frac{4 b^5 (d+e x)^{9/2} (b d-a e)}{3 e^7}+\frac{30 b^4 (d+e x)^{7/2} (b d-a e)^2}{7 e^7}-\frac{8 b^3 (d+e x)^{5/2} (b d-a e)^3}{e^7}+\frac{10 b^2 (d+e x)^{3/2} (b d-a e)^4}{e^7}-\frac{12 b \sqrt{d+e x} (b d-a e)^5}{e^7}-\frac{2 (b d-a e)^6}{e^7 \sqrt{d+e x}}+\frac{2 b^6 (d+e x)^{11/2}}{11 e^7} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(b*d - a*e)^6)/(e^7*Sqrt[d + e*x]) - (12*b*(b*d - a*e)^5*Sqrt[d + e*x])/e^7
+ (10*b^2*(b*d - a*e)^4*(d + e*x)^(3/2))/e^7 - (8*b^3*(b*d - a*e)^3*(d + e*x)^(5
/2))/e^7 + (30*b^4*(b*d - a*e)^2*(d + e*x)^(7/2))/(7*e^7) - (4*b^5*(b*d - a*e)*(
d + e*x)^(9/2))/(3*e^7) + (2*b^6*(d + e*x)^(11/2))/(11*e^7)

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Rubi in Sympy [A]  time = 78.8285, size = 167, normalized size = 0.93 \[ \frac{2 b^{6} \left (d + e x\right )^{\frac{11}{2}}}{11 e^{7}} + \frac{4 b^{5} \left (d + e x\right )^{\frac{9}{2}} \left (a e - b d\right )}{3 e^{7}} + \frac{30 b^{4} \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{2}}{7 e^{7}} + \frac{8 b^{3} \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{3}}{e^{7}} + \frac{10 b^{2} \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{4}}{e^{7}} + \frac{12 b \sqrt{d + e x} \left (a e - b d\right )^{5}}{e^{7}} - \frac{2 \left (a e - b d\right )^{6}}{e^{7} \sqrt{d + e x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**2+2*a*b*x+a**2)**3/(e*x+d)**(3/2),x)

[Out]

2*b**6*(d + e*x)**(11/2)/(11*e**7) + 4*b**5*(d + e*x)**(9/2)*(a*e - b*d)/(3*e**7
) + 30*b**4*(d + e*x)**(7/2)*(a*e - b*d)**2/(7*e**7) + 8*b**3*(d + e*x)**(5/2)*(
a*e - b*d)**3/e**7 + 10*b**2*(d + e*x)**(3/2)*(a*e - b*d)**4/e**7 + 12*b*sqrt(d
+ e*x)*(a*e - b*d)**5/e**7 - 2*(a*e - b*d)**6/(e**7*sqrt(d + e*x))

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Mathematica [A]  time = 0.317456, size = 288, normalized size = 1.61 \[ \frac{2 \left (-231 a^6 e^6+1386 a^5 b e^5 (2 d+e x)+1155 a^4 b^2 e^4 \left (-8 d^2-4 d e x+e^2 x^2\right )+924 a^3 b^3 e^3 \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )+99 a^2 b^4 e^2 \left (-128 d^4-64 d^3 e x+16 d^2 e^2 x^2-8 d e^3 x^3+5 e^4 x^4\right )+22 a b^5 e \left (256 d^5+128 d^4 e x-32 d^3 e^2 x^2+16 d^2 e^3 x^3-10 d e^4 x^4+7 e^5 x^5\right )+b^6 \left (-1024 d^6-512 d^5 e x+128 d^4 e^2 x^2-64 d^3 e^3 x^3+40 d^2 e^4 x^4-28 d e^5 x^5+21 e^6 x^6\right )\right )}{231 e^7 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-231*a^6*e^6 + 1386*a^5*b*e^5*(2*d + e*x) + 1155*a^4*b^2*e^4*(-8*d^2 - 4*d*e
*x + e^2*x^2) + 924*a^3*b^3*e^3*(16*d^3 + 8*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3) + 9
9*a^2*b^4*e^2*(-128*d^4 - 64*d^3*e*x + 16*d^2*e^2*x^2 - 8*d*e^3*x^3 + 5*e^4*x^4)
+ 22*a*b^5*e*(256*d^5 + 128*d^4*e*x - 32*d^3*e^2*x^2 + 16*d^2*e^3*x^3 - 10*d*e^
4*x^4 + 7*e^5*x^5) + b^6*(-1024*d^6 - 512*d^5*e*x + 128*d^4*e^2*x^2 - 64*d^3*e^3
*x^3 + 40*d^2*e^4*x^4 - 28*d*e^5*x^5 + 21*e^6*x^6)))/(231*e^7*Sqrt[d + e*x])

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Maple [B]  time = 0.013, size = 377, normalized size = 2.1 \[ -{\frac{-42\,{x}^{6}{b}^{6}{e}^{6}-308\,{x}^{5}a{b}^{5}{e}^{6}+56\,{x}^{5}{b}^{6}d{e}^{5}-990\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}+440\,{x}^{4}a{b}^{5}d{e}^{5}-80\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}-1848\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}+1584\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}-704\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}+128\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}-2310\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}+3696\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}-3168\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}+1408\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}-256\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}-2772\,x{a}^{5}b{e}^{6}+9240\,x{a}^{4}{b}^{2}d{e}^{5}-14784\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}+12672\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}-5632\,xa{b}^{5}{d}^{4}{e}^{2}+1024\,x{b}^{6}{d}^{5}e+462\,{a}^{6}{e}^{6}-5544\,{a}^{5}bd{e}^{5}+18480\,{b}^{2}{a}^{4}{d}^{2}{e}^{4}-29568\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}+25344\,{d}^{4}{e}^{2}{a}^{2}{b}^{4}-11264\,{d}^{5}a{b}^{5}e+2048\,{b}^{6}{d}^{6}}{231\,{e}^{7}}{\frac{1}{\sqrt{ex+d}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^(3/2),x)

[Out]

-2/231*(-21*b^6*e^6*x^6-154*a*b^5*e^6*x^5+28*b^6*d*e^5*x^5-495*a^2*b^4*e^6*x^4+2
20*a*b^5*d*e^5*x^4-40*b^6*d^2*e^4*x^4-924*a^3*b^3*e^6*x^3+792*a^2*b^4*d*e^5*x^3-
352*a*b^5*d^2*e^4*x^3+64*b^6*d^3*e^3*x^3-1155*a^4*b^2*e^6*x^2+1848*a^3*b^3*d*e^5
*x^2-1584*a^2*b^4*d^2*e^4*x^2+704*a*b^5*d^3*e^3*x^2-128*b^6*d^4*e^2*x^2-1386*a^5
*b*e^6*x+4620*a^4*b^2*d*e^5*x-7392*a^3*b^3*d^2*e^4*x+6336*a^2*b^4*d^3*e^3*x-2816
*a*b^5*d^4*e^2*x+512*b^6*d^5*e*x+231*a^6*e^6-2772*a^5*b*d*e^5+9240*a^4*b^2*d^2*e
^4-14784*a^3*b^3*d^3*e^3+12672*a^2*b^4*d^4*e^2-5632*a*b^5*d^5*e+1024*b^6*d^6)/(e
*x+d)^(1/2)/e^7

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Maxima [A]  time = 0.735215, size = 483, normalized size = 2.7 \[ \frac{2 \,{\left (\frac{21 \,{\left (e x + d\right )}^{\frac{11}{2}} b^{6} - 154 \,{\left (b^{6} d - a b^{5} e\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 495 \,{\left (b^{6} d^{2} - 2 \, a b^{5} d e + a^{2} b^{4} e^{2}\right )}{\left (e x + d\right )}^{\frac{7}{2}} - 924 \,{\left (b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 1155 \,{\left (b^{6} d^{4} - 4 \, a b^{5} d^{3} e + 6 \, a^{2} b^{4} d^{2} e^{2} - 4 \, a^{3} b^{3} d e^{3} + a^{4} b^{2} e^{4}\right )}{\left (e x + d\right )}^{\frac{3}{2}} - 1386 \,{\left (b^{6} d^{5} - 5 \, a b^{5} d^{4} e + 10 \, a^{2} b^{4} d^{3} e^{2} - 10 \, a^{3} b^{3} d^{2} e^{3} + 5 \, a^{4} b^{2} d e^{4} - a^{5} b e^{5}\right )} \sqrt{e x + d}}{e^{6}} - \frac{231 \,{\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )}}{\sqrt{e x + d} e^{6}}\right )}}{231 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^3/(e*x + d)^(3/2),x, algorithm="maxima")

[Out]

2/231*((21*(e*x + d)^(11/2)*b^6 - 154*(b^6*d - a*b^5*e)*(e*x + d)^(9/2) + 495*(b
^6*d^2 - 2*a*b^5*d*e + a^2*b^4*e^2)*(e*x + d)^(7/2) - 924*(b^6*d^3 - 3*a*b^5*d^2
*e + 3*a^2*b^4*d*e^2 - a^3*b^3*e^3)*(e*x + d)^(5/2) + 1155*(b^6*d^4 - 4*a*b^5*d^
3*e + 6*a^2*b^4*d^2*e^2 - 4*a^3*b^3*d*e^3 + a^4*b^2*e^4)*(e*x + d)^(3/2) - 1386*
(b^6*d^5 - 5*a*b^5*d^4*e + 10*a^2*b^4*d^3*e^2 - 10*a^3*b^3*d^2*e^3 + 5*a^4*b^2*d
*e^4 - a^5*b*e^5)*sqrt(e*x + d))/e^6 - 231*(b^6*d^6 - 6*a*b^5*d^5*e + 15*a^2*b^4
*d^4*e^2 - 20*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 - 6*a^5*b*d*e^5 + a^6*e^6)/(s
qrt(e*x + d)*e^6))/e

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Fricas [A]  time = 0.205388, size = 479, normalized size = 2.68 \[ \frac{2 \,{\left (21 \, b^{6} e^{6} x^{6} - 1024 \, b^{6} d^{6} + 5632 \, a b^{5} d^{5} e - 12672 \, a^{2} b^{4} d^{4} e^{2} + 14784 \, a^{3} b^{3} d^{3} e^{3} - 9240 \, a^{4} b^{2} d^{2} e^{4} + 2772 \, a^{5} b d e^{5} - 231 \, a^{6} e^{6} - 14 \,{\left (2 \, b^{6} d e^{5} - 11 \, a b^{5} e^{6}\right )} x^{5} + 5 \,{\left (8 \, b^{6} d^{2} e^{4} - 44 \, a b^{5} d e^{5} + 99 \, a^{2} b^{4} e^{6}\right )} x^{4} - 4 \,{\left (16 \, b^{6} d^{3} e^{3} - 88 \, a b^{5} d^{2} e^{4} + 198 \, a^{2} b^{4} d e^{5} - 231 \, a^{3} b^{3} e^{6}\right )} x^{3} +{\left (128 \, b^{6} d^{4} e^{2} - 704 \, a b^{5} d^{3} e^{3} + 1584 \, a^{2} b^{4} d^{2} e^{4} - 1848 \, a^{3} b^{3} d e^{5} + 1155 \, a^{4} b^{2} e^{6}\right )} x^{2} - 2 \,{\left (256 \, b^{6} d^{5} e - 1408 \, a b^{5} d^{4} e^{2} + 3168 \, a^{2} b^{4} d^{3} e^{3} - 3696 \, a^{3} b^{3} d^{2} e^{4} + 2310 \, a^{4} b^{2} d e^{5} - 693 \, a^{5} b e^{6}\right )} x\right )}}{231 \, \sqrt{e x + d} e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^3/(e*x + d)^(3/2),x, algorithm="fricas")

[Out]

2/231*(21*b^6*e^6*x^6 - 1024*b^6*d^6 + 5632*a*b^5*d^5*e - 12672*a^2*b^4*d^4*e^2
+ 14784*a^3*b^3*d^3*e^3 - 9240*a^4*b^2*d^2*e^4 + 2772*a^5*b*d*e^5 - 231*a^6*e^6
- 14*(2*b^6*d*e^5 - 11*a*b^5*e^6)*x^5 + 5*(8*b^6*d^2*e^4 - 44*a*b^5*d*e^5 + 99*a
^2*b^4*e^6)*x^4 - 4*(16*b^6*d^3*e^3 - 88*a*b^5*d^2*e^4 + 198*a^2*b^4*d*e^5 - 231
*a^3*b^3*e^6)*x^3 + (128*b^6*d^4*e^2 - 704*a*b^5*d^3*e^3 + 1584*a^2*b^4*d^2*e^4
- 1848*a^3*b^3*d*e^5 + 1155*a^4*b^2*e^6)*x^2 - 2*(256*b^6*d^5*e - 1408*a*b^5*d^4
*e^2 + 3168*a^2*b^4*d^3*e^3 - 3696*a^3*b^3*d^2*e^4 + 2310*a^4*b^2*d*e^5 - 693*a^
5*b*e^6)*x)/(sqrt(e*x + d)*e^7)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x\right )^{6}}{\left (d + e x\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b**2*x**2+2*a*b*x+a**2)**3/(e*x+d)**(3/2),x)

[Out]

Integral((a + b*x)**6/(d + e*x)**(3/2), x)

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GIAC/XCAS [A]  time = 0.218279, size = 640, normalized size = 3.58 \[ \frac{2}{231} \,{\left (21 \,{\left (x e + d\right )}^{\frac{11}{2}} b^{6} e^{70} - 154 \,{\left (x e + d\right )}^{\frac{9}{2}} b^{6} d e^{70} + 495 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{6} d^{2} e^{70} - 924 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{6} d^{3} e^{70} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{6} d^{4} e^{70} - 1386 \, \sqrt{x e + d} b^{6} d^{5} e^{70} + 154 \,{\left (x e + d\right )}^{\frac{9}{2}} a b^{5} e^{71} - 990 \,{\left (x e + d\right )}^{\frac{7}{2}} a b^{5} d e^{71} + 2772 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{5} d^{2} e^{71} - 4620 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{5} d^{3} e^{71} + 6930 \, \sqrt{x e + d} a b^{5} d^{4} e^{71} + 495 \,{\left (x e + d\right )}^{\frac{7}{2}} a^{2} b^{4} e^{72} - 2772 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{2} b^{4} d e^{72} + 6930 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{4} d^{2} e^{72} - 13860 \, \sqrt{x e + d} a^{2} b^{4} d^{3} e^{72} + 924 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{3} b^{3} e^{73} - 4620 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3} b^{3} d e^{73} + 13860 \, \sqrt{x e + d} a^{3} b^{3} d^{2} e^{73} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{4} b^{2} e^{74} - 6930 \, \sqrt{x e + d} a^{4} b^{2} d e^{74} + 1386 \, \sqrt{x e + d} a^{5} b e^{75}\right )} e^{\left (-77\right )} - \frac{2 \,{\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} e^{\left (-7\right )}}{\sqrt{x e + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^3/(e*x + d)^(3/2),x, algorithm="giac")

[Out]

2/231*(21*(x*e + d)^(11/2)*b^6*e^70 - 154*(x*e + d)^(9/2)*b^6*d*e^70 + 495*(x*e
+ d)^(7/2)*b^6*d^2*e^70 - 924*(x*e + d)^(5/2)*b^6*d^3*e^70 + 1155*(x*e + d)^(3/2
)*b^6*d^4*e^70 - 1386*sqrt(x*e + d)*b^6*d^5*e^70 + 154*(x*e + d)^(9/2)*a*b^5*e^7
1 - 990*(x*e + d)^(7/2)*a*b^5*d*e^71 + 2772*(x*e + d)^(5/2)*a*b^5*d^2*e^71 - 462
0*(x*e + d)^(3/2)*a*b^5*d^3*e^71 + 6930*sqrt(x*e + d)*a*b^5*d^4*e^71 + 495*(x*e
+ d)^(7/2)*a^2*b^4*e^72 - 2772*(x*e + d)^(5/2)*a^2*b^4*d*e^72 + 6930*(x*e + d)^(
3/2)*a^2*b^4*d^2*e^72 - 13860*sqrt(x*e + d)*a^2*b^4*d^3*e^72 + 924*(x*e + d)^(5/
2)*a^3*b^3*e^73 - 4620*(x*e + d)^(3/2)*a^3*b^3*d*e^73 + 13860*sqrt(x*e + d)*a^3*
b^3*d^2*e^73 + 1155*(x*e + d)^(3/2)*a^4*b^2*e^74 - 6930*sqrt(x*e + d)*a^4*b^2*d*
e^74 + 1386*sqrt(x*e + d)*a^5*b*e^75)*e^(-77) - 2*(b^6*d^6 - 6*a*b^5*d^5*e + 15*
a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 - 6*a^5*b*d*e^5 + a^6*
e^6)*e^(-7)/sqrt(x*e + d)