3.1387 \(\int (a+b x)^5 (c+d x)^{3/2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.15737, 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 \[ -\frac{10 b^4 (c+d x)^{13/2} (b c-a d)}{13 d^6}+\frac{20 b^3 (c+d x)^{11/2} (b c-a d)^2}{11 d^6}-\frac{20 b^2 (c+d x)^{9/2} (b c-a d)^3}{9 d^6}+\frac{10 b (c+d x)^{7/2} (b c-a d)^4}{7 d^6}-\frac{2 (c+d x)^{5/2} (b c-a d)^5}{5 d^6}+\frac{2 b^5 (c+d x)^{15/2}}{15 d^6} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^5*(c + d*x)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 42.4734, size = 146, normalized size = 0.92 \[ \frac{2 b^{5} \left (c + d x\right )^{\frac{15}{2}}}{15 d^{6}} + \frac{10 b^{4} \left (c + d x\right )^{\frac{13}{2}} \left (a d - b c\right )}{13 d^{6}} + \frac{20 b^{3} \left (c + d x\right )^{\frac{11}{2}} \left (a d - b c\right )^{2}}{11 d^{6}} + \frac{20 b^{2} \left (c + d x\right )^{\frac{9}{2}} \left (a d - b c\right )^{3}}{9 d^{6}} + \frac{10 b \left (c + d x\right )^{\frac{7}{2}} \left (a d - b c\right )^{4}}{7 d^{6}} + \frac{2 \left (c + d x\right )^{\frac{5}{2}} \left (a d - b c\right )^{5}}{5 d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**5*(d*x+c)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.251692, size = 217, normalized size = 1.37 \[ \frac{2 (c+d x)^{5/2} \left (9009 a^5 d^5+6435 a^4 b d^4 (5 d x-2 c)+1430 a^3 b^2 d^3 \left (8 c^2-20 c d x+35 d^2 x^2\right )+390 a^2 b^3 d^2 \left (-16 c^3+40 c^2 d x-70 c d^2 x^2+105 d^3 x^3\right )+15 a b^4 d \left (128 c^4-320 c^3 d x+560 c^2 d^2 x^2-840 c d^3 x^3+1155 d^4 x^4\right )+b^5 \left (-256 c^5+640 c^4 d x-1120 c^3 d^2 x^2+1680 c^2 d^3 x^3-2310 c d^4 x^4+3003 d^5 x^5\right )\right )}{45045 d^6} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^5*(c + d*x)^(3/2),x]

[Out]

(2*(c + d*x)^(5/2)*(9009*a^5*d^5 + 6435*a^4*b*d^4*(-2*c + 5*d*x) + 1430*a^3*b^2*
d^3*(8*c^2 - 20*c*d*x + 35*d^2*x^2) + 390*a^2*b^3*d^2*(-16*c^3 + 40*c^2*d*x - 70
*c*d^2*x^2 + 105*d^3*x^3) + 15*a*b^4*d*(128*c^4 - 320*c^3*d*x + 560*c^2*d^2*x^2
- 840*c*d^3*x^3 + 1155*d^4*x^4) + b^5*(-256*c^5 + 640*c^4*d*x - 1120*c^3*d^2*x^2
+ 1680*c^2*d^3*x^3 - 2310*c*d^4*x^4 + 3003*d^5*x^5)))/(45045*d^6)

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Maple [B]  time = 0.01, size = 273, normalized size = 1.7 \[{\frac{6006\,{b}^{5}{x}^{5}{d}^{5}+34650\,a{b}^{4}{d}^{5}{x}^{4}-4620\,{b}^{5}c{d}^{4}{x}^{4}+81900\,{a}^{2}{b}^{3}{d}^{5}{x}^{3}-25200\,a{b}^{4}c{d}^{4}{x}^{3}+3360\,{b}^{5}{c}^{2}{d}^{3}{x}^{3}+100100\,{a}^{3}{b}^{2}{d}^{5}{x}^{2}-54600\,{a}^{2}{b}^{3}c{d}^{4}{x}^{2}+16800\,a{b}^{4}{c}^{2}{d}^{3}{x}^{2}-2240\,{b}^{5}{c}^{3}{d}^{2}{x}^{2}+64350\,{a}^{4}b{d}^{5}x-57200\,{a}^{3}{b}^{2}c{d}^{4}x+31200\,{a}^{2}{b}^{3}{c}^{2}{d}^{3}x-9600\,a{b}^{4}{c}^{3}{d}^{2}x+1280\,{b}^{5}{c}^{4}dx+18018\,{a}^{5}{d}^{5}-25740\,{a}^{4}bc{d}^{4}+22880\,{a}^{3}{b}^{2}{c}^{2}{d}^{3}-12480\,{a}^{2}{b}^{3}{c}^{3}{d}^{2}+3840\,a{b}^{4}{c}^{4}d-512\,{b}^{5}{c}^{5}}{45045\,{d}^{6}} \left ( dx+c \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^5*(d*x+c)^(3/2),x)

[Out]

2/45045*(d*x+c)^(5/2)*(3003*b^5*d^5*x^5+17325*a*b^4*d^5*x^4-2310*b^5*c*d^4*x^4+4
0950*a^2*b^3*d^5*x^3-12600*a*b^4*c*d^4*x^3+1680*b^5*c^2*d^3*x^3+50050*a^3*b^2*d^
5*x^2-27300*a^2*b^3*c*d^4*x^2+8400*a*b^4*c^2*d^3*x^2-1120*b^5*c^3*d^2*x^2+32175*
a^4*b*d^5*x-28600*a^3*b^2*c*d^4*x+15600*a^2*b^3*c^2*d^3*x-4800*a*b^4*c^3*d^2*x+6
40*b^5*c^4*d*x+9009*a^5*d^5-12870*a^4*b*c*d^4+11440*a^3*b^2*c^2*d^3-6240*a^2*b^3
*c^3*d^2+1920*a*b^4*c^4*d-256*b^5*c^5)/d^6

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Maxima [A]  time = 1.39371, size = 350, normalized size = 2.22 \[ \frac{2 \,{\left (3003 \,{\left (d x + c\right )}^{\frac{15}{2}} b^{5} - 17325 \,{\left (b^{5} c - a b^{4} d\right )}{\left (d x + c\right )}^{\frac{13}{2}} + 40950 \,{\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )}{\left (d x + c\right )}^{\frac{11}{2}} - 50050 \,{\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )}{\left (d x + c\right )}^{\frac{9}{2}} + 32175 \,{\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )}{\left (d x + c\right )}^{\frac{7}{2}} - 9009 \,{\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )}{\left (d x + c\right )}^{\frac{5}{2}}\right )}}{45045 \, d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5*(d*x + c)^(3/2),x, algorithm="maxima")

[Out]

2/45045*(3003*(d*x + c)^(15/2)*b^5 - 17325*(b^5*c - a*b^4*d)*(d*x + c)^(13/2) +
40950*(b^5*c^2 - 2*a*b^4*c*d + a^2*b^3*d^2)*(d*x + c)^(11/2) - 50050*(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)^(9/2) + 32175*(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)^(7/
2) - 9009*(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)^(5/2))/d^6

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Fricas [A]  time = 0.213101, size = 564, normalized size = 3.57 \[ \frac{2 \,{\left (3003 \, b^{5} d^{7} x^{7} - 256 \, b^{5} c^{7} + 1920 \, a b^{4} c^{6} d - 6240 \, a^{2} b^{3} c^{5} d^{2} + 11440 \, a^{3} b^{2} c^{4} d^{3} - 12870 \, a^{4} b c^{3} d^{4} + 9009 \, a^{5} c^{2} d^{5} + 231 \,{\left (16 \, b^{5} c d^{6} + 75 \, a b^{4} d^{7}\right )} x^{6} + 63 \,{\left (b^{5} c^{2} d^{5} + 350 \, a b^{4} c d^{6} + 650 \, a^{2} b^{3} d^{7}\right )} x^{5} - 35 \,{\left (2 \, b^{5} c^{3} d^{4} - 15 \, a b^{4} c^{2} d^{5} - 1560 \, a^{2} b^{3} c d^{6} - 1430 \, a^{3} b^{2} d^{7}\right )} x^{4} + 5 \,{\left (16 \, b^{5} c^{4} d^{3} - 120 \, a b^{4} c^{3} d^{4} + 390 \, a^{2} b^{3} c^{2} d^{5} + 14300 \, a^{3} b^{2} c d^{6} + 6435 \, a^{4} b d^{7}\right )} x^{3} - 3 \,{\left (32 \, b^{5} c^{5} d^{2} - 240 \, a b^{4} c^{4} d^{3} + 780 \, a^{2} b^{3} c^{3} d^{4} - 1430 \, a^{3} b^{2} c^{2} d^{5} - 17160 \, a^{4} b c d^{6} - 3003 \, a^{5} d^{7}\right )} x^{2} +{\left (128 \, b^{5} c^{6} d - 960 \, a b^{4} c^{5} d^{2} + 3120 \, a^{2} b^{3} c^{4} d^{3} - 5720 \, a^{3} b^{2} c^{3} d^{4} + 6435 \, a^{4} b c^{2} d^{5} + 18018 \, a^{5} c d^{6}\right )} x\right )} \sqrt{d x + c}}{45045 \, d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5*(d*x + c)^(3/2),x, algorithm="fricas")

[Out]

2/45045*(3003*b^5*d^7*x^7 - 256*b^5*c^7 + 1920*a*b^4*c^6*d - 6240*a^2*b^3*c^5*d^
2 + 11440*a^3*b^2*c^4*d^3 - 12870*a^4*b*c^3*d^4 + 9009*a^5*c^2*d^5 + 231*(16*b^5
*c*d^6 + 75*a*b^4*d^7)*x^6 + 63*(b^5*c^2*d^5 + 350*a*b^4*c*d^6 + 650*a^2*b^3*d^7
)*x^5 - 35*(2*b^5*c^3*d^4 - 15*a*b^4*c^2*d^5 - 1560*a^2*b^3*c*d^6 - 1430*a^3*b^2
*d^7)*x^4 + 5*(16*b^5*c^4*d^3 - 120*a*b^4*c^3*d^4 + 390*a^2*b^3*c^2*d^5 + 14300*
a^3*b^2*c*d^6 + 6435*a^4*b*d^7)*x^3 - 3*(32*b^5*c^5*d^2 - 240*a*b^4*c^4*d^3 + 78
0*a^2*b^3*c^3*d^4 - 1430*a^3*b^2*c^2*d^5 - 17160*a^4*b*c*d^6 - 3003*a^5*d^7)*x^2
+ (128*b^5*c^6*d - 960*a*b^4*c^5*d^2 + 3120*a^2*b^3*c^4*d^3 - 5720*a^3*b^2*c^3*
d^4 + 6435*a^4*b*c^2*d^5 + 18018*a^5*c*d^6)*x)*sqrt(d*x + c)/d^6

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Sympy [A]  time = 4.77879, size = 763, normalized size = 4.83 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**5*(d*x+c)**(3/2),x)

[Out]

a**5*c*Piecewise((sqrt(c)*x, Eq(d, 0)), (2*(c + d*x)**(3/2)/(3*d), True)) + 2*a*
*5*(-c*(c + d*x)**(3/2)/3 + (c + d*x)**(5/2)/5)/d + 10*a**4*b*c*(-c*(c + d*x)**(
3/2)/3 + (c + d*x)**(5/2)/5)/d**2 + 10*a**4*b*(c**2*(c + d*x)**(3/2)/3 - 2*c*(c
+ d*x)**(5/2)/5 + (c + d*x)**(7/2)/7)/d**2 + 20*a**3*b**2*c*(c**2*(c + d*x)**(3/
2)/3 - 2*c*(c + d*x)**(5/2)/5 + (c + d*x)**(7/2)/7)/d**3 + 20*a**3*b**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**3 + 20*a**2*b**3*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**4 + 20*a**2*b**3*(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 + 10*a*b**4*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**5 + 10*a*b**4*(-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 + 2*b**5*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**6 + 2*b**5*
(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**6

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GIAC/XCAS [A]  time = 0.233464, 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)^(3/2),x, algorithm="giac")

[Out]

Done