3.1628 \(\int (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx\)

Optimal. Leaf size=187 \[ -\frac{4 b^5 (d+e x)^{15/2} (b d-a e)}{5 e^7}+\frac{30 b^4 (d+e x)^{13/2} (b d-a e)^2}{13 e^7}-\frac{40 b^3 (d+e x)^{11/2} (b d-a e)^3}{11 e^7}+\frac{10 b^2 (d+e x)^{9/2} (b d-a e)^4}{3 e^7}-\frac{12 b (d+e x)^{7/2} (b d-a e)^5}{7 e^7}+\frac{2 (d+e x)^{5/2} (b d-a e)^6}{5 e^7}+\frac{2 b^6 (d+e x)^{17/2}}{17 e^7} \]

[Out]

(2*(b*d - a*e)^6*(d + e*x)^(5/2))/(5*e^7) - (12*b*(b*d - a*e)^5*(d + e*x)^(7/2))
/(7*e^7) + (10*b^2*(b*d - a*e)^4*(d + e*x)^(9/2))/(3*e^7) - (40*b^3*(b*d - a*e)^
3*(d + e*x)^(11/2))/(11*e^7) + (30*b^4*(b*d - a*e)^2*(d + e*x)^(13/2))/(13*e^7)
- (4*b^5*(b*d - a*e)*(d + e*x)^(15/2))/(5*e^7) + (2*b^6*(d + e*x)^(17/2))/(17*e^
7)

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Rubi [A]  time = 0.173292, antiderivative size = 187, 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 \[ -\frac{4 b^5 (d+e x)^{15/2} (b d-a e)}{5 e^7}+\frac{30 b^4 (d+e x)^{13/2} (b d-a e)^2}{13 e^7}-\frac{40 b^3 (d+e x)^{11/2} (b d-a e)^3}{11 e^7}+\frac{10 b^2 (d+e x)^{9/2} (b d-a e)^4}{3 e^7}-\frac{12 b (d+e x)^{7/2} (b d-a e)^5}{7 e^7}+\frac{2 (d+e x)^{5/2} (b d-a e)^6}{5 e^7}+\frac{2 b^6 (d+e x)^{17/2}}{17 e^7} \]

Antiderivative was successfully verified.

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

[Out]

(2*(b*d - a*e)^6*(d + e*x)^(5/2))/(5*e^7) - (12*b*(b*d - a*e)^5*(d + e*x)^(7/2))
/(7*e^7) + (10*b^2*(b*d - a*e)^4*(d + e*x)^(9/2))/(3*e^7) - (40*b^3*(b*d - a*e)^
3*(d + e*x)^(11/2))/(11*e^7) + (30*b^4*(b*d - a*e)^2*(d + e*x)^(13/2))/(13*e^7)
- (4*b^5*(b*d - a*e)*(d + e*x)^(15/2))/(5*e^7) + (2*b^6*(d + e*x)^(17/2))/(17*e^
7)

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Rubi in Sympy [A]  time = 81.3885, size = 173, normalized size = 0.93 \[ \frac{2 b^{6} \left (d + e x\right )^{\frac{17}{2}}}{17 e^{7}} + \frac{4 b^{5} \left (d + e x\right )^{\frac{15}{2}} \left (a e - b d\right )}{5 e^{7}} + \frac{30 b^{4} \left (d + e x\right )^{\frac{13}{2}} \left (a e - b d\right )^{2}}{13 e^{7}} + \frac{40 b^{3} \left (d + e x\right )^{\frac{11}{2}} \left (a e - b d\right )^{3}}{11 e^{7}} + \frac{10 b^{2} \left (d + e x\right )^{\frac{9}{2}} \left (a e - b d\right )^{4}}{3 e^{7}} + \frac{12 b \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{5}}{7 e^{7}} + \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{6}}{5 e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b**6*(d + e*x)**(17/2)/(17*e**7) + 4*b**5*(d + e*x)**(15/2)*(a*e - b*d)/(5*e**
7) + 30*b**4*(d + e*x)**(13/2)*(a*e - b*d)**2/(13*e**7) + 40*b**3*(d + e*x)**(11
/2)*(a*e - b*d)**3/(11*e**7) + 10*b**2*(d + e*x)**(9/2)*(a*e - b*d)**4/(3*e**7)
+ 12*b*(d + e*x)**(7/2)*(a*e - b*d)**5/(7*e**7) + 2*(d + e*x)**(5/2)*(a*e - b*d)
**6/(5*e**7)

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Mathematica [A]  time = 0.349477, size = 291, normalized size = 1.56 \[ \frac{2 (d+e x)^{5/2} \left (51051 a^6 e^6+43758 a^5 b e^5 (5 e x-2 d)+12155 a^4 b^2 e^4 \left (8 d^2-20 d e x+35 e^2 x^2\right )+4420 a^3 b^3 e^3 \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )+255 a^2 b^4 e^2 \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )+34 a b^5 e \left (-256 d^5+640 d^4 e x-1120 d^3 e^2 x^2+1680 d^2 e^3 x^3-2310 d e^4 x^4+3003 e^5 x^5\right )+b^6 \left (1024 d^6-2560 d^5 e x+4480 d^4 e^2 x^2-6720 d^3 e^3 x^3+9240 d^2 e^4 x^4-12012 d e^5 x^5+15015 e^6 x^6\right )\right )}{255255 e^7} \]

Antiderivative was successfully verified.

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

[Out]

(2*(d + e*x)^(5/2)*(51051*a^6*e^6 + 43758*a^5*b*e^5*(-2*d + 5*e*x) + 12155*a^4*b
^2*e^4*(8*d^2 - 20*d*e*x + 35*e^2*x^2) + 4420*a^3*b^3*e^3*(-16*d^3 + 40*d^2*e*x
- 70*d*e^2*x^2 + 105*e^3*x^3) + 255*a^2*b^4*e^2*(128*d^4 - 320*d^3*e*x + 560*d^2
*e^2*x^2 - 840*d*e^3*x^3 + 1155*e^4*x^4) + 34*a*b^5*e*(-256*d^5 + 640*d^4*e*x -
1120*d^3*e^2*x^2 + 1680*d^2*e^3*x^3 - 2310*d*e^4*x^4 + 3003*e^5*x^5) + b^6*(1024
*d^6 - 2560*d^5*e*x + 4480*d^4*e^2*x^2 - 6720*d^3*e^3*x^3 + 9240*d^2*e^4*x^4 - 1
2012*d*e^5*x^5 + 15015*e^6*x^6)))/(255255*e^7)

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Maple [B]  time = 0.014, size = 377, normalized size = 2. \[{\frac{30030\,{x}^{6}{b}^{6}{e}^{6}+204204\,{x}^{5}a{b}^{5}{e}^{6}-24024\,{x}^{5}{b}^{6}d{e}^{5}+589050\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}-157080\,{x}^{4}a{b}^{5}d{e}^{5}+18480\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}+928200\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}-428400\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}+114240\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}-13440\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}+850850\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}-618800\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}+285600\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}-76160\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}+8960\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}+437580\,x{a}^{5}b{e}^{6}-486200\,x{a}^{4}{b}^{2}d{e}^{5}+353600\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}-163200\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}+43520\,xa{b}^{5}{d}^{4}{e}^{2}-5120\,x{b}^{6}{d}^{5}e+102102\,{a}^{6}{e}^{6}-175032\,{a}^{5}bd{e}^{5}+194480\,{b}^{2}{a}^{4}{d}^{2}{e}^{4}-141440\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}+65280\,{d}^{4}{e}^{2}{a}^{2}{b}^{4}-17408\,{d}^{5}a{b}^{5}e+2048\,{b}^{6}{d}^{6}}{255255\,{e}^{7}} \left ( ex+d \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/255255*(e*x+d)^(5/2)*(15015*b^6*e^6*x^6+102102*a*b^5*e^6*x^5-12012*b^6*d*e^5*x
^5+294525*a^2*b^4*e^6*x^4-78540*a*b^5*d*e^5*x^4+9240*b^6*d^2*e^4*x^4+464100*a^3*
b^3*e^6*x^3-214200*a^2*b^4*d*e^5*x^3+57120*a*b^5*d^2*e^4*x^3-6720*b^6*d^3*e^3*x^
3+425425*a^4*b^2*e^6*x^2-309400*a^3*b^3*d*e^5*x^2+142800*a^2*b^4*d^2*e^4*x^2-380
80*a*b^5*d^3*e^3*x^2+4480*b^6*d^4*e^2*x^2+218790*a^5*b*e^6*x-243100*a^4*b^2*d*e^
5*x+176800*a^3*b^3*d^2*e^4*x-81600*a^2*b^4*d^3*e^3*x+21760*a*b^5*d^4*e^2*x-2560*
b^6*d^5*e*x+51051*a^6*e^6-87516*a^5*b*d*e^5+97240*a^4*b^2*d^2*e^4-70720*a^3*b^3*
d^3*e^3+32640*a^2*b^4*d^4*e^2-8704*a*b^5*d^5*e+1024*b^6*d^6)/e^7

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Maxima [A]  time = 0.732202, size = 473, normalized size = 2.53 \[ \frac{2 \,{\left (15015 \,{\left (e x + d\right )}^{\frac{17}{2}} b^{6} - 102102 \,{\left (b^{6} d - a b^{5} e\right )}{\left (e x + d\right )}^{\frac{15}{2}} + 294525 \,{\left (b^{6} d^{2} - 2 \, a b^{5} d e + a^{2} b^{4} e^{2}\right )}{\left (e x + d\right )}^{\frac{13}{2}} - 464100 \,{\left (b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}\right )}{\left (e x + d\right )}^{\frac{11}{2}} + 425425 \,{\left (b^{6} d^{4} - 4 \, a b^{5} d^{3} e + 6 \, a^{2} b^{4} d^{2} e^{2} - 4 \, a^{3} b^{3} d e^{3} + a^{4} b^{2} e^{4}\right )}{\left (e x + d\right )}^{\frac{9}{2}} - 218790 \,{\left (b^{6} d^{5} - 5 \, a b^{5} d^{4} e + 10 \, a^{2} b^{4} d^{3} e^{2} - 10 \, a^{3} b^{3} d^{2} e^{3} + 5 \, a^{4} b^{2} d e^{4} - a^{5} b e^{5}\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 51051 \,{\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )}{\left (e x + d\right )}^{\frac{5}{2}}\right )}}{255255 \, e^{7}} \]

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="maxima")

[Out]

2/255255*(15015*(e*x + d)^(17/2)*b^6 - 102102*(b^6*d - a*b^5*e)*(e*x + d)^(15/2)
+ 294525*(b^6*d^2 - 2*a*b^5*d*e + a^2*b^4*e^2)*(e*x + d)^(13/2) - 464100*(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)^(11/2) + 425425*(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)^(9/2) - 218790*(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)*(e*x + d)^(7/2) + 51051*(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*x + d)^(5/2))/e^7

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Fricas [A]  time = 0.207697, size = 730, normalized size = 3.9 \[ \frac{2 \,{\left (15015 \, b^{6} e^{8} x^{8} + 1024 \, b^{6} d^{8} - 8704 \, a b^{5} d^{7} e + 32640 \, a^{2} b^{4} d^{6} e^{2} - 70720 \, a^{3} b^{3} d^{5} e^{3} + 97240 \, a^{4} b^{2} d^{4} e^{4} - 87516 \, a^{5} b d^{3} e^{5} + 51051 \, a^{6} d^{2} e^{6} + 6006 \,{\left (3 \, b^{6} d e^{7} + 17 \, a b^{5} e^{8}\right )} x^{7} + 231 \,{\left (b^{6} d^{2} e^{6} + 544 \, a b^{5} d e^{7} + 1275 \, a^{2} b^{4} e^{8}\right )} x^{6} - 42 \,{\left (6 \, b^{6} d^{3} e^{5} - 51 \, a b^{5} d^{2} e^{6} - 8925 \, a^{2} b^{4} d e^{7} - 11050 \, a^{3} b^{3} e^{8}\right )} x^{5} + 35 \,{\left (8 \, b^{6} d^{4} e^{4} - 68 \, a b^{5} d^{3} e^{5} + 255 \, a^{2} b^{4} d^{2} e^{6} + 17680 \, a^{3} b^{3} d e^{7} + 12155 \, a^{4} b^{2} e^{8}\right )} x^{4} - 10 \,{\left (32 \, b^{6} d^{5} e^{3} - 272 \, a b^{5} d^{4} e^{4} + 1020 \, a^{2} b^{4} d^{3} e^{5} - 2210 \, a^{3} b^{3} d^{2} e^{6} - 60775 \, a^{4} b^{2} d e^{7} - 21879 \, a^{5} b e^{8}\right )} x^{3} + 3 \,{\left (128 \, b^{6} d^{6} e^{2} - 1088 \, a b^{5} d^{5} e^{3} + 4080 \, a^{2} b^{4} d^{4} e^{4} - 8840 \, a^{3} b^{3} d^{3} e^{5} + 12155 \, a^{4} b^{2} d^{2} e^{6} + 116688 \, a^{5} b d e^{7} + 17017 \, a^{6} e^{8}\right )} x^{2} - 2 \,{\left (256 \, b^{6} d^{7} e - 2176 \, a b^{5} d^{6} e^{2} + 8160 \, a^{2} b^{4} d^{5} e^{3} - 17680 \, a^{3} b^{3} d^{4} e^{4} + 24310 \, a^{4} b^{2} d^{3} e^{5} - 21879 \, a^{5} b d^{2} e^{6} - 51051 \, a^{6} d e^{7}\right )} x\right )} \sqrt{e x + d}}{255255 \, e^{7}} \]

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]

2/255255*(15015*b^6*e^8*x^8 + 1024*b^6*d^8 - 8704*a*b^5*d^7*e + 32640*a^2*b^4*d^
6*e^2 - 70720*a^3*b^3*d^5*e^3 + 97240*a^4*b^2*d^4*e^4 - 87516*a^5*b*d^3*e^5 + 51
051*a^6*d^2*e^6 + 6006*(3*b^6*d*e^7 + 17*a*b^5*e^8)*x^7 + 231*(b^6*d^2*e^6 + 544
*a*b^5*d*e^7 + 1275*a^2*b^4*e^8)*x^6 - 42*(6*b^6*d^3*e^5 - 51*a*b^5*d^2*e^6 - 89
25*a^2*b^4*d*e^7 - 11050*a^3*b^3*e^8)*x^5 + 35*(8*b^6*d^4*e^4 - 68*a*b^5*d^3*e^5
+ 255*a^2*b^4*d^2*e^6 + 17680*a^3*b^3*d*e^7 + 12155*a^4*b^2*e^8)*x^4 - 10*(32*b
^6*d^5*e^3 - 272*a*b^5*d^4*e^4 + 1020*a^2*b^4*d^3*e^5 - 2210*a^3*b^3*d^2*e^6 - 6
0775*a^4*b^2*d*e^7 - 21879*a^5*b*e^8)*x^3 + 3*(128*b^6*d^6*e^2 - 1088*a*b^5*d^5*
e^3 + 4080*a^2*b^4*d^4*e^4 - 8840*a^3*b^3*d^3*e^5 + 12155*a^4*b^2*d^2*e^6 + 1166
88*a^5*b*d*e^7 + 17017*a^6*e^8)*x^2 - 2*(256*b^6*d^7*e - 2176*a*b^5*d^6*e^2 + 81
60*a^2*b^4*d^5*e^3 - 17680*a^3*b^3*d^4*e^4 + 24310*a^4*b^2*d^3*e^5 - 21879*a^5*b
*d^2*e^6 - 51051*a^6*d*e^7)*x)*sqrt(e*x + d)/e^7

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Sympy [A]  time = 12.1903, size = 1000, normalized size = 5.35 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**6*d*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 2*a*
*6*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e + 12*a**5*b*d*(-d*(d + e*x)**(
3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 12*a**5*b*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d
+ e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 30*a**4*b**2*d*(d**2*(d + e*x)**(3/
2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 + 30*a**4*b**2*(-d**3*(
d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*
x)**(9/2)/9)/e**3 + 40*a**3*b**3*d*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)*
*(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**4 + 40*a**3*b**3*(d**
4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4
*d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**4 + 30*a**2*b**4*d*(d**4*(d + e
*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d +
e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**5 + 30*a**2*b**4*(-d**5*(d + e*x)**(3/2
)/3 + d**4*(d + e*x)**(5/2) - 10*d**3*(d + e*x)**(7/2)/7 + 10*d**2*(d + e*x)**(9
/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**5 + 12*a*b**5*d*(-d*
*5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 10*d**3*(d + e*x)**(7/2)/7 + 10*
d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**6
+ 12*a*b**5*(d**6*(d + e*x)**(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d +
e*x)**(7/2)/7 - 20*d**3*(d + e*x)**(9/2)/9 + 15*d**2*(d + e*x)**(11/2)/11 - 6*d*
(d + e*x)**(13/2)/13 + (d + e*x)**(15/2)/15)/e**6 + 2*b**6*d*(d**6*(d + e*x)**(3
/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d + e*x)**(7/2)/7 - 20*d**3*(d + e*
x)**(9/2)/9 + 15*d**2*(d + e*x)**(11/2)/11 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x
)**(15/2)/15)/e**7 + 2*b**6*(-d**7*(d + e*x)**(3/2)/3 + 7*d**6*(d + e*x)**(5/2)/
5 - 3*d**5*(d + e*x)**(7/2) + 35*d**4*(d + e*x)**(9/2)/9 - 35*d**3*(d + e*x)**(1
1/2)/11 + 21*d**2*(d + e*x)**(13/2)/13 - 7*d*(d + e*x)**(15/2)/15 + (d + e*x)**(
17/2)/17)/e**7

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.228883, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

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]

Done