3.1399 \(\int (a+b x)^5 (c+d x)^{5/2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.154099, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ -\frac{2 b^4 (c+d x)^{15/2} (b c-a d)}{3 d^6}+\frac{20 b^3 (c+d x)^{13/2} (b c-a d)^2}{13 d^6}-\frac{20 b^2 (c+d x)^{11/2} (b c-a d)^3}{11 d^6}+\frac{10 b (c+d x)^{9/2} (b c-a d)^4}{9 d^6}-\frac{2 (c+d x)^{7/2} (b c-a d)^5}{7 d^6}+\frac{2 b^5 (c+d x)^{17/2}}{17 d^6} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 39.0692, size = 146, normalized size = 0.92 \[ \frac{2 b^{5} \left (c + d x\right )^{\frac{17}{2}}}{17 d^{6}} + \frac{2 b^{4} \left (c + d x\right )^{\frac{15}{2}} \left (a d - b c\right )}{3 d^{6}} + \frac{20 b^{3} \left (c + d x\right )^{\frac{13}{2}} \left (a d - b c\right )^{2}}{13 d^{6}} + \frac{20 b^{2} \left (c + d x\right )^{\frac{11}{2}} \left (a d - b c\right )^{3}}{11 d^{6}} + \frac{10 b \left (c + d x\right )^{\frac{9}{2}} \left (a d - b c\right )^{4}}{9 d^{6}} + \frac{2 \left (c + d x\right )^{\frac{7}{2}} \left (a d - b c\right )^{5}}{7 d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.292918, size = 217, normalized size = 1.37 \[ \frac{2 (c+d x)^{7/2} \left (21879 a^5 d^5+12155 a^4 b d^4 (7 d x-2 c)+2210 a^3 b^2 d^3 \left (8 c^2-28 c d x+63 d^2 x^2\right )+510 a^2 b^3 d^2 \left (-16 c^3+56 c^2 d x-126 c d^2 x^2+231 d^3 x^3\right )+17 a b^4 d \left (128 c^4-448 c^3 d x+1008 c^2 d^2 x^2-1848 c d^3 x^3+3003 d^4 x^4\right )+b^5 \left (-256 c^5+896 c^4 d x-2016 c^3 d^2 x^2+3696 c^2 d^3 x^3-6006 c d^4 x^4+9009 d^5 x^5\right )\right )}{153153 d^6} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.01, size = 273, normalized size = 1.7 \[{\frac{18018\,{b}^{5}{x}^{5}{d}^{5}+102102\,a{b}^{4}{d}^{5}{x}^{4}-12012\,{b}^{5}c{d}^{4}{x}^{4}+235620\,{a}^{2}{b}^{3}{d}^{5}{x}^{3}-62832\,a{b}^{4}c{d}^{4}{x}^{3}+7392\,{b}^{5}{c}^{2}{d}^{3}{x}^{3}+278460\,{a}^{3}{b}^{2}{d}^{5}{x}^{2}-128520\,{a}^{2}{b}^{3}c{d}^{4}{x}^{2}+34272\,a{b}^{4}{c}^{2}{d}^{3}{x}^{2}-4032\,{b}^{5}{c}^{3}{d}^{2}{x}^{2}+170170\,{a}^{4}b{d}^{5}x-123760\,{a}^{3}{b}^{2}c{d}^{4}x+57120\,{a}^{2}{b}^{3}{c}^{2}{d}^{3}x-15232\,a{b}^{4}{c}^{3}{d}^{2}x+1792\,{b}^{5}{c}^{4}dx+43758\,{a}^{5}{d}^{5}-48620\,{a}^{4}bc{d}^{4}+35360\,{a}^{3}{b}^{2}{c}^{2}{d}^{3}-16320\,{a}^{2}{b}^{3}{c}^{3}{d}^{2}+4352\,a{b}^{4}{c}^{4}d-512\,{b}^{5}{c}^{5}}{153153\,{d}^{6}} \left ( dx+c \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^5*(d*x+c)^(5/2),x)

[Out]

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

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Maxima [A]  time = 1.3791, size = 350, normalized size = 2.22 \[ \frac{2 \,{\left (9009 \,{\left (d x + c\right )}^{\frac{17}{2}} b^{5} - 51051 \,{\left (b^{5} c - a b^{4} d\right )}{\left (d x + c\right )}^{\frac{15}{2}} + 117810 \,{\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )}{\left (d x + c\right )}^{\frac{13}{2}} - 139230 \,{\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )}{\left (d x + c\right )}^{\frac{11}{2}} + 85085 \,{\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )}{\left (d x + c\right )}^{\frac{9}{2}} - 21879 \,{\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )}{\left (d x + c\right )}^{\frac{7}{2}}\right )}}{153153 \, d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5*(d*x + c)^(5/2),x, algorithm="maxima")

[Out]

2/153153*(9009*(d*x + c)^(17/2)*b^5 - 51051*(b^5*c - a*b^4*d)*(d*x + c)^(15/2) +
117810*(b^5*c^2 - 2*a*b^4*c*d + a^2*b^3*d^2)*(d*x + c)^(13/2) - 139230*(b^5*c^3
- 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*(d*x + c)^(11/2) + 85085*(b^5*
c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*(d*x + c)
^(9/2) - 21879*(b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^
3 + 5*a^4*b*c*d^4 - a^5*d^5)*(d*x + c)^(7/2))/d^6

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Fricas [A]  time = 0.224659, size = 671, normalized size = 4.25 \[ \frac{2 \,{\left (9009 \, b^{5} d^{8} x^{8} - 256 \, b^{5} c^{8} + 2176 \, a b^{4} c^{7} d - 8160 \, a^{2} b^{3} c^{6} d^{2} + 17680 \, a^{3} b^{2} c^{5} d^{3} - 24310 \, a^{4} b c^{4} d^{4} + 21879 \, a^{5} c^{3} d^{5} + 3003 \,{\left (7 \, b^{5} c d^{7} + 17 \, a b^{4} d^{8}\right )} x^{7} + 231 \,{\left (55 \, b^{5} c^{2} d^{6} + 527 \, a b^{4} c d^{7} + 510 \, a^{2} b^{3} d^{8}\right )} x^{6} + 63 \,{\left (b^{5} c^{3} d^{5} + 1207 \, a b^{4} c^{2} d^{6} + 4590 \, a^{2} b^{3} c d^{7} + 2210 \, a^{3} b^{2} d^{8}\right )} x^{5} - 35 \,{\left (2 \, b^{5} c^{4} d^{4} - 17 \, a b^{4} c^{3} d^{5} - 5406 \, a^{2} b^{3} c^{2} d^{6} - 10166 \, a^{3} b^{2} c d^{7} - 2431 \, a^{4} b d^{8}\right )} x^{4} +{\left (80 \, b^{5} c^{5} d^{3} - 680 \, a b^{4} c^{4} d^{4} + 2550 \, a^{2} b^{3} c^{3} d^{5} + 249730 \, a^{3} b^{2} c^{2} d^{6} + 230945 \, a^{4} b c d^{7} + 21879 \, a^{5} d^{8}\right )} x^{3} - 3 \,{\left (32 \, b^{5} c^{6} d^{2} - 272 \, a b^{4} c^{5} d^{3} + 1020 \, a^{2} b^{3} c^{4} d^{4} - 2210 \, a^{3} b^{2} c^{3} d^{5} - 60775 \, a^{4} b c^{2} d^{6} - 21879 \, a^{5} c d^{7}\right )} x^{2} +{\left (128 \, b^{5} c^{7} d - 1088 \, a b^{4} c^{6} d^{2} + 4080 \, a^{2} b^{3} c^{5} d^{3} - 8840 \, a^{3} b^{2} c^{4} d^{4} + 12155 \, a^{4} b c^{3} d^{5} + 65637 \, a^{5} c^{2} d^{6}\right )} x\right )} \sqrt{d x + c}}{153153 \, d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5*(d*x + c)^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 7.4485, 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)**5*(d*x+c)**(5/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.242158, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5*(d*x + c)^(5/2),x, algorithm="giac")

[Out]

Done