3.2050 \(\int (a+b x) (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx\)

Optimal. Leaf size=158 \[ -\frac{10 b^4 (d+e x)^{17/2} (b d-a e)}{17 e^6}+\frac{4 b^3 (d+e x)^{15/2} (b d-a e)^2}{3 e^6}-\frac{20 b^2 (d+e x)^{13/2} (b d-a e)^3}{13 e^6}+\frac{10 b (d+e x)^{11/2} (b d-a e)^4}{11 e^6}-\frac{2 (d+e x)^{9/2} (b d-a e)^5}{9 e^6}+\frac{2 b^5 (d+e x)^{19/2}}{19 e^6} \]

[Out]

(-2*(b*d - a*e)^5*(d + e*x)^(9/2))/(9*e^6) + (10*b*(b*d - a*e)^4*(d + e*x)^(11/2
))/(11*e^6) - (20*b^2*(b*d - a*e)^3*(d + e*x)^(13/2))/(13*e^6) + (4*b^3*(b*d - a
*e)^2*(d + e*x)^(15/2))/(3*e^6) - (10*b^4*(b*d - a*e)*(d + e*x)^(17/2))/(17*e^6)
+ (2*b^5*(d + e*x)^(19/2))/(19*e^6)

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Rubi [A]  time = 0.171867, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061 \[ -\frac{10 b^4 (d+e x)^{17/2} (b d-a e)}{17 e^6}+\frac{4 b^3 (d+e x)^{15/2} (b d-a e)^2}{3 e^6}-\frac{20 b^2 (d+e x)^{13/2} (b d-a e)^3}{13 e^6}+\frac{10 b (d+e x)^{11/2} (b d-a e)^4}{11 e^6}-\frac{2 (d+e x)^{9/2} (b d-a e)^5}{9 e^6}+\frac{2 b^5 (d+e x)^{19/2}}{19 e^6} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)*(d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

(-2*(b*d - a*e)^5*(d + e*x)^(9/2))/(9*e^6) + (10*b*(b*d - a*e)^4*(d + e*x)^(11/2
))/(11*e^6) - (20*b^2*(b*d - a*e)^3*(d + e*x)^(13/2))/(13*e^6) + (4*b^3*(b*d - a
*e)^2*(d + e*x)^(15/2))/(3*e^6) - (10*b^4*(b*d - a*e)*(d + e*x)^(17/2))/(17*e^6)
+ (2*b^5*(d + e*x)^(19/2))/(19*e^6)

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Rubi in Sympy [A]  time = 74.7228, size = 146, normalized size = 0.92 \[ \frac{2 b^{5} \left (d + e x\right )^{\frac{19}{2}}}{19 e^{6}} + \frac{10 b^{4} \left (d + e x\right )^{\frac{17}{2}} \left (a e - b d\right )}{17 e^{6}} + \frac{4 b^{3} \left (d + e x\right )^{\frac{15}{2}} \left (a e - b d\right )^{2}}{3 e^{6}} + \frac{20 b^{2} \left (d + e x\right )^{\frac{13}{2}} \left (a e - b d\right )^{3}}{13 e^{6}} + \frac{10 b \left (d + e x\right )^{\frac{11}{2}} \left (a e - b d\right )^{4}}{11 e^{6}} + \frac{2 \left (d + e x\right )^{\frac{9}{2}} \left (a e - b d\right )^{5}}{9 e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(e*x+d)**(7/2)*(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

2*b**5*(d + e*x)**(19/2)/(19*e**6) + 10*b**4*(d + e*x)**(17/2)*(a*e - b*d)/(17*e
**6) + 4*b**3*(d + e*x)**(15/2)*(a*e - b*d)**2/(3*e**6) + 20*b**2*(d + e*x)**(13
/2)*(a*e - b*d)**3/(13*e**6) + 10*b*(d + e*x)**(11/2)*(a*e - b*d)**4/(11*e**6) +
2*(d + e*x)**(9/2)*(a*e - b*d)**5/(9*e**6)

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Mathematica [A]  time = 0.344362, size = 217, normalized size = 1.37 \[ \frac{2 (d+e x)^{9/2} \left (46189 a^5 e^5+20995 a^4 b e^4 (9 e x-2 d)+3230 a^3 b^2 e^3 \left (8 d^2-36 d e x+99 e^2 x^2\right )+646 a^2 b^3 e^2 \left (-16 d^3+72 d^2 e x-198 d e^2 x^2+429 e^3 x^3\right )+19 a b^4 e \left (128 d^4-576 d^3 e x+1584 d^2 e^2 x^2-3432 d e^3 x^3+6435 e^4 x^4\right )+b^5 \left (-256 d^5+1152 d^4 e x-3168 d^3 e^2 x^2+6864 d^2 e^3 x^3-12870 d e^4 x^4+21879 e^5 x^5\right )\right )}{415701 e^6} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)*(d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

(2*(d + e*x)^(9/2)*(46189*a^5*e^5 + 20995*a^4*b*e^4*(-2*d + 9*e*x) + 3230*a^3*b^
2*e^3*(8*d^2 - 36*d*e*x + 99*e^2*x^2) + 646*a^2*b^3*e^2*(-16*d^3 + 72*d^2*e*x -
198*d*e^2*x^2 + 429*e^3*x^3) + 19*a*b^4*e*(128*d^4 - 576*d^3*e*x + 1584*d^2*e^2*
x^2 - 3432*d*e^3*x^3 + 6435*e^4*x^4) + b^5*(-256*d^5 + 1152*d^4*e*x - 3168*d^3*e
^2*x^2 + 6864*d^2*e^3*x^3 - 12870*d*e^4*x^4 + 21879*e^5*x^5)))/(415701*e^6)

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Maple [B]  time = 0.011, size = 273, normalized size = 1.7 \[{\frac{43758\,{x}^{5}{b}^{5}{e}^{5}+244530\,{x}^{4}a{b}^{4}{e}^{5}-25740\,{x}^{4}{b}^{5}d{e}^{4}+554268\,{x}^{3}{a}^{2}{b}^{3}{e}^{5}-130416\,{x}^{3}a{b}^{4}d{e}^{4}+13728\,{x}^{3}{b}^{5}{d}^{2}{e}^{3}+639540\,{x}^{2}{a}^{3}{b}^{2}{e}^{5}-255816\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}+60192\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}-6336\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}+377910\,x{a}^{4}b{e}^{5}-232560\,x{a}^{3}{b}^{2}d{e}^{4}+93024\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}-21888\,xa{b}^{4}{d}^{3}{e}^{2}+2304\,x{b}^{5}{d}^{4}e+92378\,{a}^{5}{e}^{5}-83980\,{a}^{4}bd{e}^{4}+51680\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}-20672\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+4864\,a{b}^{4}{d}^{4}e-512\,{b}^{5}{d}^{5}}{415701\,{e}^{6}} \left ( ex+d \right ) ^{{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(e*x+d)^(7/2)*(b^2*x^2+2*a*b*x+a^2)^2,x)

[Out]

2/415701*(e*x+d)^(9/2)*(21879*b^5*e^5*x^5+122265*a*b^4*e^5*x^4-12870*b^5*d*e^4*x
^4+277134*a^2*b^3*e^5*x^3-65208*a*b^4*d*e^4*x^3+6864*b^5*d^2*e^3*x^3+319770*a^3*
b^2*e^5*x^2-127908*a^2*b^3*d*e^4*x^2+30096*a*b^4*d^2*e^3*x^2-3168*b^5*d^3*e^2*x^
2+188955*a^4*b*e^5*x-116280*a^3*b^2*d*e^4*x+46512*a^2*b^3*d^2*e^3*x-10944*a*b^4*
d^3*e^2*x+1152*b^5*d^4*e*x+46189*a^5*e^5-41990*a^4*b*d*e^4+25840*a^3*b^2*d^2*e^3
-10336*a^2*b^3*d^3*e^2+2432*a*b^4*d^4*e-256*b^5*d^5)/e^6

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Maxima [A]  time = 0.727202, size = 350, normalized size = 2.22 \[ \frac{2 \,{\left (21879 \,{\left (e x + d\right )}^{\frac{19}{2}} b^{5} - 122265 \,{\left (b^{5} d - a b^{4} e\right )}{\left (e x + d\right )}^{\frac{17}{2}} + 277134 \,{\left (b^{5} d^{2} - 2 \, a b^{4} d e + a^{2} b^{3} e^{2}\right )}{\left (e x + d\right )}^{\frac{15}{2}} - 319770 \,{\left (b^{5} d^{3} - 3 \, a b^{4} d^{2} e + 3 \, a^{2} b^{3} d e^{2} - a^{3} b^{2} e^{3}\right )}{\left (e x + d\right )}^{\frac{13}{2}} + 188955 \,{\left (b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}\right )}{\left (e x + d\right )}^{\frac{11}{2}} - 46189 \,{\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )}{\left (e x + d\right )}^{\frac{9}{2}}\right )}}{415701 \, e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^2*(b*x + a)*(e*x + d)^(7/2),x, algorithm="maxima")

[Out]

2/415701*(21879*(e*x + d)^(19/2)*b^5 - 122265*(b^5*d - a*b^4*e)*(e*x + d)^(17/2)
+ 277134*(b^5*d^2 - 2*a*b^4*d*e + a^2*b^3*e^2)*(e*x + d)^(15/2) - 319770*(b^5*d
^3 - 3*a*b^4*d^2*e + 3*a^2*b^3*d*e^2 - a^3*b^2*e^3)*(e*x + d)^(13/2) + 188955*(b
^5*d^4 - 4*a*b^4*d^3*e + 6*a^2*b^3*d^2*e^2 - 4*a^3*b^2*d*e^3 + a^4*b*e^4)*(e*x +
d)^(11/2) - 46189*(b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^
2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5)*(e*x + d)^(9/2))/e^6

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Fricas [A]  time = 0.284201, size = 782, normalized size = 4.95 \[ \frac{2 \,{\left (21879 \, b^{5} e^{9} x^{9} - 256 \, b^{5} d^{9} + 2432 \, a b^{4} d^{8} e - 10336 \, a^{2} b^{3} d^{7} e^{2} + 25840 \, a^{3} b^{2} d^{6} e^{3} - 41990 \, a^{4} b d^{5} e^{4} + 46189 \, a^{5} d^{4} e^{5} + 1287 \,{\left (58 \, b^{5} d e^{8} + 95 \, a b^{4} e^{9}\right )} x^{8} + 858 \,{\left (101 \, b^{5} d^{2} e^{7} + 494 \, a b^{4} d e^{8} + 323 \, a^{2} b^{3} e^{9}\right )} x^{7} + 66 \,{\left (524 \, b^{5} d^{3} e^{6} + 7619 \, a b^{4} d^{2} e^{7} + 14858 \, a^{2} b^{3} d e^{8} + 4845 \, a^{3} b^{2} e^{9}\right )} x^{6} + 9 \,{\left (7 \, b^{5} d^{4} e^{5} + 23028 \, a b^{4} d^{3} e^{6} + 133076 \, a^{2} b^{3} d^{2} e^{7} + 129200 \, a^{3} b^{2} d e^{8} + 20995 \, a^{4} b e^{9}\right )} x^{5} -{\left (70 \, b^{5} d^{5} e^{4} - 665 \, a b^{4} d^{4} e^{5} - 516800 \, a^{2} b^{3} d^{3} e^{6} - 1479340 \, a^{3} b^{2} d^{2} e^{7} - 713830 \, a^{4} b d e^{8} - 46189 \, a^{5} e^{9}\right )} x^{4} + 2 \,{\left (40 \, b^{5} d^{6} e^{3} - 380 \, a b^{4} d^{5} e^{4} + 1615 \, a^{2} b^{3} d^{4} e^{5} + 342380 \, a^{3} b^{2} d^{3} e^{6} + 482885 \, a^{4} b d^{2} e^{7} + 92378 \, a^{5} d e^{8}\right )} x^{3} - 6 \,{\left (16 \, b^{5} d^{7} e^{2} - 152 \, a b^{4} d^{6} e^{3} + 646 \, a^{2} b^{3} d^{5} e^{4} - 1615 \, a^{3} b^{2} d^{4} e^{5} - 83980 \, a^{4} b d^{3} e^{6} - 46189 \, a^{5} d^{2} e^{7}\right )} x^{2} +{\left (128 \, b^{5} d^{8} e - 1216 \, a b^{4} d^{7} e^{2} + 5168 \, a^{2} b^{3} d^{6} e^{3} - 12920 \, a^{3} b^{2} d^{5} e^{4} + 20995 \, a^{4} b d^{4} e^{5} + 184756 \, a^{5} d^{3} e^{6}\right )} x\right )} \sqrt{e x + d}}{415701 \, e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^2*(b*x + a)*(e*x + d)^(7/2),x, algorithm="fricas")

[Out]

2/415701*(21879*b^5*e^9*x^9 - 256*b^5*d^9 + 2432*a*b^4*d^8*e - 10336*a^2*b^3*d^7
*e^2 + 25840*a^3*b^2*d^6*e^3 - 41990*a^4*b*d^5*e^4 + 46189*a^5*d^4*e^5 + 1287*(5
8*b^5*d*e^8 + 95*a*b^4*e^9)*x^8 + 858*(101*b^5*d^2*e^7 + 494*a*b^4*d*e^8 + 323*a
^2*b^3*e^9)*x^7 + 66*(524*b^5*d^3*e^6 + 7619*a*b^4*d^2*e^7 + 14858*a^2*b^3*d*e^8
+ 4845*a^3*b^2*e^9)*x^6 + 9*(7*b^5*d^4*e^5 + 23028*a*b^4*d^3*e^6 + 133076*a^2*b
^3*d^2*e^7 + 129200*a^3*b^2*d*e^8 + 20995*a^4*b*e^9)*x^5 - (70*b^5*d^5*e^4 - 665
*a*b^4*d^4*e^5 - 516800*a^2*b^3*d^3*e^6 - 1479340*a^3*b^2*d^2*e^7 - 713830*a^4*b
*d*e^8 - 46189*a^5*e^9)*x^4 + 2*(40*b^5*d^6*e^3 - 380*a*b^4*d^5*e^4 + 1615*a^2*b
^3*d^4*e^5 + 342380*a^3*b^2*d^3*e^6 + 482885*a^4*b*d^2*e^7 + 92378*a^5*d*e^8)*x^
3 - 6*(16*b^5*d^7*e^2 - 152*a*b^4*d^6*e^3 + 646*a^2*b^3*d^5*e^4 - 1615*a^3*b^2*d
^4*e^5 - 83980*a^4*b*d^3*e^6 - 46189*a^5*d^2*e^7)*x^2 + (128*b^5*d^8*e - 1216*a*
b^4*d^7*e^2 + 5168*a^2*b^3*d^6*e^3 - 12920*a^3*b^2*d^5*e^4 + 20995*a^4*b*d^4*e^5
+ 184756*a^5*d^3*e^6)*x)*sqrt(e*x + d)/e^6

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Sympy [A]  time = 46.9795, size = 1187, normalized size = 7.51 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*(e*x+d)**(7/2)*(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

Piecewise((2*a**5*d**4*sqrt(d + e*x)/(9*e) + 8*a**5*d**3*x*sqrt(d + e*x)/9 + 4*a
**5*d**2*e*x**2*sqrt(d + e*x)/3 + 8*a**5*d*e**2*x**3*sqrt(d + e*x)/9 + 2*a**5*e*
*3*x**4*sqrt(d + e*x)/9 - 20*a**4*b*d**5*sqrt(d + e*x)/(99*e**2) + 10*a**4*b*d**
4*x*sqrt(d + e*x)/(99*e) + 80*a**4*b*d**3*x**2*sqrt(d + e*x)/33 + 460*a**4*b*d**
2*e*x**3*sqrt(d + e*x)/99 + 340*a**4*b*d*e**2*x**4*sqrt(d + e*x)/99 + 10*a**4*b*
e**3*x**5*sqrt(d + e*x)/11 + 160*a**3*b**2*d**6*sqrt(d + e*x)/(1287*e**3) - 80*a
**3*b**2*d**5*x*sqrt(d + e*x)/(1287*e**2) + 20*a**3*b**2*d**4*x**2*sqrt(d + e*x)
/(429*e) + 4240*a**3*b**2*d**3*x**3*sqrt(d + e*x)/1287 + 9160*a**3*b**2*d**2*e*x
**4*sqrt(d + e*x)/1287 + 800*a**3*b**2*d*e**2*x**5*sqrt(d + e*x)/143 + 20*a**3*b
**2*e**3*x**6*sqrt(d + e*x)/13 - 64*a**2*b**3*d**7*sqrt(d + e*x)/(1287*e**4) + 3
2*a**2*b**3*d**6*x*sqrt(d + e*x)/(1287*e**3) - 8*a**2*b**3*d**5*x**2*sqrt(d + e*
x)/(429*e**2) + 20*a**2*b**3*d**4*x**3*sqrt(d + e*x)/(1287*e) + 3200*a**2*b**3*d
**3*x**4*sqrt(d + e*x)/1287 + 824*a**2*b**3*d**2*e*x**5*sqrt(d + e*x)/143 + 184*
a**2*b**3*d*e**2*x**6*sqrt(d + e*x)/39 + 4*a**2*b**3*e**3*x**7*sqrt(d + e*x)/3 +
256*a*b**4*d**8*sqrt(d + e*x)/(21879*e**5) - 128*a*b**4*d**7*x*sqrt(d + e*x)/(2
1879*e**4) + 32*a*b**4*d**6*x**2*sqrt(d + e*x)/(7293*e**3) - 80*a*b**4*d**5*x**3
*sqrt(d + e*x)/(21879*e**2) + 70*a*b**4*d**4*x**4*sqrt(d + e*x)/(21879*e) + 2424
*a*b**4*d**3*x**5*sqrt(d + e*x)/2431 + 1604*a*b**4*d**2*e*x**6*sqrt(d + e*x)/663
+ 104*a*b**4*d*e**2*x**7*sqrt(d + e*x)/51 + 10*a*b**4*e**3*x**8*sqrt(d + e*x)/1
7 - 512*b**5*d**9*sqrt(d + e*x)/(415701*e**6) + 256*b**5*d**8*x*sqrt(d + e*x)/(4
15701*e**5) - 64*b**5*d**7*x**2*sqrt(d + e*x)/(138567*e**4) + 160*b**5*d**6*x**3
*sqrt(d + e*x)/(415701*e**3) - 140*b**5*d**5*x**4*sqrt(d + e*x)/(415701*e**2) +
14*b**5*d**4*x**5*sqrt(d + e*x)/(46189*e) + 2096*b**5*d**3*x**6*sqrt(d + e*x)/12
597 + 404*b**5*d**2*e*x**7*sqrt(d + e*x)/969 + 116*b**5*d*e**2*x**8*sqrt(d + e*x
)/323 + 2*b**5*e**3*x**9*sqrt(d + e*x)/19, Ne(e, 0)), (d**(7/2)*(a**5*x + 5*a**4
*b*x**2/2 + 10*a**3*b**2*x**3/3 + 5*a**2*b**3*x**4/2 + a*b**4*x**5 + b**5*x**6/6
), True))

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GIAC/XCAS [A]  time = 0.3348, 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)^2*(b*x + a)*(e*x + d)^(7/2),x, algorithm="giac")

[Out]

Done