3.2051 \(\int (a+b x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx\)

Optimal. Leaf size=158 \[ -\frac{2 b^4 (d+e x)^{15/2} (b d-a e)}{3 e^6}+\frac{20 b^3 (d+e x)^{13/2} (b d-a e)^2}{13 e^6}-\frac{20 b^2 (d+e x)^{11/2} (b d-a e)^3}{11 e^6}+\frac{10 b (d+e x)^{9/2} (b d-a e)^4}{9 e^6}-\frac{2 (d+e x)^{7/2} (b d-a e)^5}{7 e^6}+\frac{2 b^5 (d+e x)^{17/2}}{17 e^6} \]

[Out]

(-2*(b*d - a*e)^5*(d + e*x)^(7/2))/(7*e^6) + (10*b*(b*d - a*e)^4*(d + e*x)^(9/2)
)/(9*e^6) - (20*b^2*(b*d - a*e)^3*(d + e*x)^(11/2))/(11*e^6) + (20*b^3*(b*d - a*
e)^2*(d + e*x)^(13/2))/(13*e^6) - (2*b^4*(b*d - a*e)*(d + e*x)^(15/2))/(3*e^6) +
(2*b^5*(d + e*x)^(17/2))/(17*e^6)

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Rubi [A]  time = 0.143462, 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 \[ -\frac{2 b^4 (d+e x)^{15/2} (b d-a e)}{3 e^6}+\frac{20 b^3 (d+e x)^{13/2} (b d-a e)^2}{13 e^6}-\frac{20 b^2 (d+e x)^{11/2} (b d-a e)^3}{11 e^6}+\frac{10 b (d+e x)^{9/2} (b d-a e)^4}{9 e^6}-\frac{2 (d+e x)^{7/2} (b d-a e)^5}{7 e^6}+\frac{2 b^5 (d+e x)^{17/2}}{17 e^6} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(b*d - a*e)^5*(d + e*x)^(7/2))/(7*e^6) + (10*b*(b*d - a*e)^4*(d + e*x)^(9/2)
)/(9*e^6) - (20*b^2*(b*d - a*e)^3*(d + e*x)^(11/2))/(11*e^6) + (20*b^3*(b*d - a*
e)^2*(d + e*x)^(13/2))/(13*e^6) - (2*b^4*(b*d - a*e)*(d + e*x)^(15/2))/(3*e^6) +
(2*b^5*(d + e*x)^(17/2))/(17*e^6)

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Rubi in Sympy [A]  time = 73.8808, size = 146, normalized size = 0.92 \[ \frac{2 b^{5} \left (d + e x\right )^{\frac{17}{2}}}{17 e^{6}} + \frac{2 b^{4} \left (d + e x\right )^{\frac{15}{2}} \left (a e - b d\right )}{3 e^{6}} + \frac{20 b^{3} \left (d + e x\right )^{\frac{13}{2}} \left (a e - b d\right )^{2}}{13 e^{6}} + \frac{20 b^{2} \left (d + e x\right )^{\frac{11}{2}} \left (a e - b d\right )^{3}}{11 e^{6}} + \frac{10 b \left (d + e x\right )^{\frac{9}{2}} \left (a e - b d\right )^{4}}{9 e^{6}} + \frac{2 \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{5}}{7 e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(e*x+d)**(5/2)*(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

2*b**5*(d + e*x)**(17/2)/(17*e**6) + 2*b**4*(d + e*x)**(15/2)*(a*e - b*d)/(3*e**
6) + 20*b**3*(d + e*x)**(13/2)*(a*e - b*d)**2/(13*e**6) + 20*b**2*(d + e*x)**(11
/2)*(a*e - b*d)**3/(11*e**6) + 10*b*(d + e*x)**(9/2)*(a*e - b*d)**4/(9*e**6) + 2
*(d + e*x)**(7/2)*(a*e - b*d)**5/(7*e**6)

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Mathematica [A]  time = 0.302574, size = 217, normalized size = 1.37 \[ \frac{2 (d+e x)^{7/2} \left (21879 a^5 e^5+12155 a^4 b e^4 (7 e x-2 d)+2210 a^3 b^2 e^3 \left (8 d^2-28 d e x+63 e^2 x^2\right )+510 a^2 b^3 e^2 \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )+17 a b^4 e \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )+b^5 \left (-256 d^5+896 d^4 e x-2016 d^3 e^2 x^2+3696 d^2 e^3 x^3-6006 d e^4 x^4+9009 e^5 x^5\right )\right )}{153153 e^6} \]

Antiderivative was successfully verified.

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

[Out]

(2*(d + e*x)^(7/2)*(21879*a^5*e^5 + 12155*a^4*b*e^4*(-2*d + 7*e*x) + 2210*a^3*b^
2*e^3*(8*d^2 - 28*d*e*x + 63*e^2*x^2) + 510*a^2*b^3*e^2*(-16*d^3 + 56*d^2*e*x -
126*d*e^2*x^2 + 231*e^3*x^3) + 17*a*b^4*e*(128*d^4 - 448*d^3*e*x + 1008*d^2*e^2*
x^2 - 1848*d*e^3*x^3 + 3003*e^4*x^4) + b^5*(-256*d^5 + 896*d^4*e*x - 2016*d^3*e^
2*x^2 + 3696*d^2*e^3*x^3 - 6006*d*e^4*x^4 + 9009*e^5*x^5)))/(153153*e^6)

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Maple [B]  time = 0.011, size = 273, normalized size = 1.7 \[{\frac{18018\,{x}^{5}{b}^{5}{e}^{5}+102102\,{x}^{4}a{b}^{4}{e}^{5}-12012\,{x}^{4}{b}^{5}d{e}^{4}+235620\,{x}^{3}{a}^{2}{b}^{3}{e}^{5}-62832\,{x}^{3}a{b}^{4}d{e}^{4}+7392\,{x}^{3}{b}^{5}{d}^{2}{e}^{3}+278460\,{x}^{2}{a}^{3}{b}^{2}{e}^{5}-128520\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}+34272\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}-4032\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}+170170\,x{a}^{4}b{e}^{5}-123760\,x{a}^{3}{b}^{2}d{e}^{4}+57120\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}-15232\,xa{b}^{4}{d}^{3}{e}^{2}+1792\,x{b}^{5}{d}^{4}e+43758\,{a}^{5}{e}^{5}-48620\,{a}^{4}bd{e}^{4}+35360\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}-16320\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+4352\,a{b}^{4}{d}^{4}e-512\,{b}^{5}{d}^{5}}{153153\,{e}^{6}} \left ( ex+d \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/153153*(e*x+d)^(7/2)*(9009*b^5*e^5*x^5+51051*a*b^4*e^5*x^4-6006*b^5*d*e^4*x^4+
117810*a^2*b^3*e^5*x^3-31416*a*b^4*d*e^4*x^3+3696*b^5*d^2*e^3*x^3+139230*a^3*b^2
*e^5*x^2-64260*a^2*b^3*d*e^4*x^2+17136*a*b^4*d^2*e^3*x^2-2016*b^5*d^3*e^2*x^2+85
085*a^4*b*e^5*x-61880*a^3*b^2*d*e^4*x+28560*a^2*b^3*d^2*e^3*x-7616*a*b^4*d^3*e^2
*x+896*b^5*d^4*e*x+21879*a^5*e^5-24310*a^4*b*d*e^4+17680*a^3*b^2*d^2*e^3-8160*a^
2*b^3*d^3*e^2+2176*a*b^4*d^4*e-256*b^5*d^5)/e^6

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Maxima [A]  time = 0.738439, size = 350, normalized size = 2.22 \[ \frac{2 \,{\left (9009 \,{\left (e x + d\right )}^{\frac{17}{2}} b^{5} - 51051 \,{\left (b^{5} d - a b^{4} e\right )}{\left (e x + d\right )}^{\frac{15}{2}} + 117810 \,{\left (b^{5} d^{2} - 2 \, a b^{4} d e + a^{2} b^{3} e^{2}\right )}{\left (e x + d\right )}^{\frac{13}{2}} - 139230 \,{\left (b^{5} d^{3} - 3 \, a b^{4} d^{2} e + 3 \, a^{2} b^{3} d e^{2} - a^{3} b^{2} e^{3}\right )}{\left (e x + d\right )}^{\frac{11}{2}} + 85085 \,{\left (b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}\right )}{\left (e x + d\right )}^{\frac{9}{2}} - 21879 \,{\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )}{\left (e x + d\right )}^{\frac{7}{2}}\right )}}{153153 \, e^{6}} \]

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)^(5/2),x, algorithm="maxima")

[Out]

2/153153*(9009*(e*x + d)^(17/2)*b^5 - 51051*(b^5*d - a*b^4*e)*(e*x + d)^(15/2) +
117810*(b^5*d^2 - 2*a*b^4*d*e + a^2*b^3*e^2)*(e*x + d)^(13/2) - 139230*(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)^(11/2) + 85085*(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)
^(9/2) - 21879*(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)^(7/2))/e^6

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Fricas [A]  time = 0.285046, size = 671, normalized size = 4.25 \[ \frac{2 \,{\left (9009 \, b^{5} e^{8} x^{8} - 256 \, b^{5} d^{8} + 2176 \, a b^{4} d^{7} e - 8160 \, a^{2} b^{3} d^{6} e^{2} + 17680 \, a^{3} b^{2} d^{5} e^{3} - 24310 \, a^{4} b d^{4} e^{4} + 21879 \, a^{5} d^{3} e^{5} + 3003 \,{\left (7 \, b^{5} d e^{7} + 17 \, a b^{4} e^{8}\right )} x^{7} + 231 \,{\left (55 \, b^{5} d^{2} e^{6} + 527 \, a b^{4} d e^{7} + 510 \, a^{2} b^{3} e^{8}\right )} x^{6} + 63 \,{\left (b^{5} d^{3} e^{5} + 1207 \, a b^{4} d^{2} e^{6} + 4590 \, a^{2} b^{3} d e^{7} + 2210 \, a^{3} b^{2} e^{8}\right )} x^{5} - 35 \,{\left (2 \, b^{5} d^{4} e^{4} - 17 \, a b^{4} d^{3} e^{5} - 5406 \, a^{2} b^{3} d^{2} e^{6} - 10166 \, a^{3} b^{2} d e^{7} - 2431 \, a^{4} b e^{8}\right )} x^{4} +{\left (80 \, b^{5} d^{5} e^{3} - 680 \, a b^{4} d^{4} e^{4} + 2550 \, a^{2} b^{3} d^{3} e^{5} + 249730 \, a^{3} b^{2} d^{2} e^{6} + 230945 \, a^{4} b d e^{7} + 21879 \, a^{5} e^{8}\right )} x^{3} - 3 \,{\left (32 \, b^{5} d^{6} e^{2} - 272 \, a b^{4} d^{5} e^{3} + 1020 \, a^{2} b^{3} d^{4} e^{4} - 2210 \, a^{3} b^{2} d^{3} e^{5} - 60775 \, a^{4} b d^{2} e^{6} - 21879 \, a^{5} d e^{7}\right )} x^{2} +{\left (128 \, b^{5} d^{7} e - 1088 \, a b^{4} d^{6} e^{2} + 4080 \, a^{2} b^{3} d^{5} e^{3} - 8840 \, a^{3} b^{2} d^{4} e^{4} + 12155 \, a^{4} b d^{3} e^{5} + 65637 \, a^{5} d^{2} e^{6}\right )} x\right )} \sqrt{e x + d}}{153153 \, e^{6}} \]

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)^(5/2),x, algorithm="fricas")

[Out]

2/153153*(9009*b^5*e^8*x^8 - 256*b^5*d^8 + 2176*a*b^4*d^7*e - 8160*a^2*b^3*d^6*e
^2 + 17680*a^3*b^2*d^5*e^3 - 24310*a^4*b*d^4*e^4 + 21879*a^5*d^3*e^5 + 3003*(7*b
^5*d*e^7 + 17*a*b^4*e^8)*x^7 + 231*(55*b^5*d^2*e^6 + 527*a*b^4*d*e^7 + 510*a^2*b
^3*e^8)*x^6 + 63*(b^5*d^3*e^5 + 1207*a*b^4*d^2*e^6 + 4590*a^2*b^3*d*e^7 + 2210*a
^3*b^2*e^8)*x^5 - 35*(2*b^5*d^4*e^4 - 17*a*b^4*d^3*e^5 - 5406*a^2*b^3*d^2*e^6 -
10166*a^3*b^2*d*e^7 - 2431*a^4*b*e^8)*x^4 + (80*b^5*d^5*e^3 - 680*a*b^4*d^4*e^4
+ 2550*a^2*b^3*d^3*e^5 + 249730*a^3*b^2*d^2*e^6 + 230945*a^4*b*d*e^7 + 21879*a^5
*e^8)*x^3 - 3*(32*b^5*d^6*e^2 - 272*a*b^4*d^5*e^3 + 1020*a^2*b^3*d^4*e^4 - 2210*
a^3*b^2*d^3*e^5 - 60775*a^4*b*d^2*e^6 - 21879*a^5*d*e^7)*x^2 + (128*b^5*d^7*e -
1088*a*b^4*d^6*e^2 + 4080*a^2*b^3*d^5*e^3 - 8840*a^3*b^2*d^4*e^4 + 12155*a^4*b*d
^3*e^5 + 65637*a^5*d^2*e^6)*x)*sqrt(e*x + d)/e^6

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Sympy [A]  time = 17.6205, size = 1292, normalized size = 8.18 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*(e*x+d)**(5/2)*(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.31671, 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)^(5/2),x, algorithm="giac")

[Out]

Done