3.1400 \(\int (a+b x)^4 (c+d x)^{5/2} \, dx\)

Optimal. Leaf size=129 \[ -\frac{8 b^3 (c+d x)^{13/2} (b c-a d)}{13 d^5}+\frac{12 b^2 (c+d x)^{11/2} (b c-a d)^2}{11 d^5}-\frac{8 b (c+d x)^{9/2} (b c-a d)^3}{9 d^5}+\frac{2 (c+d x)^{7/2} (b c-a d)^4}{7 d^5}+\frac{2 b^4 (c+d x)^{15/2}}{15 d^5} \]

[Out]

(2*(b*c - a*d)^4*(c + d*x)^(7/2))/(7*d^5) - (8*b*(b*c - a*d)^3*(c + d*x)^(9/2))/
(9*d^5) + (12*b^2*(b*c - a*d)^2*(c + d*x)^(11/2))/(11*d^5) - (8*b^3*(b*c - a*d)*
(c + d*x)^(13/2))/(13*d^5) + (2*b^4*(c + d*x)^(15/2))/(15*d^5)

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Rubi [A]  time = 0.117842, antiderivative size = 129, 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 \[ -\frac{8 b^3 (c+d x)^{13/2} (b c-a d)}{13 d^5}+\frac{12 b^2 (c+d x)^{11/2} (b c-a d)^2}{11 d^5}-\frac{8 b (c+d x)^{9/2} (b c-a d)^3}{9 d^5}+\frac{2 (c+d x)^{7/2} (b c-a d)^4}{7 d^5}+\frac{2 b^4 (c+d x)^{15/2}}{15 d^5} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^4*(c + d*x)^(5/2),x]

[Out]

(2*(b*c - a*d)^4*(c + d*x)^(7/2))/(7*d^5) - (8*b*(b*c - a*d)^3*(c + d*x)^(9/2))/
(9*d^5) + (12*b^2*(b*c - a*d)^2*(c + d*x)^(11/2))/(11*d^5) - (8*b^3*(b*c - a*d)*
(c + d*x)^(13/2))/(13*d^5) + (2*b^4*(c + d*x)^(15/2))/(15*d^5)

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Rubi in Sympy [A]  time = 29.4065, size = 119, normalized size = 0.92 \[ \frac{2 b^{4} \left (c + d x\right )^{\frac{15}{2}}}{15 d^{5}} + \frac{8 b^{3} \left (c + d x\right )^{\frac{13}{2}} \left (a d - b c\right )}{13 d^{5}} + \frac{12 b^{2} \left (c + d x\right )^{\frac{11}{2}} \left (a d - b c\right )^{2}}{11 d^{5}} + \frac{8 b \left (c + d x\right )^{\frac{9}{2}} \left (a d - b c\right )^{3}}{9 d^{5}} + \frac{2 \left (c + d x\right )^{\frac{7}{2}} \left (a d - b c\right )^{4}}{7 d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**4*(d*x+c)**(5/2),x)

[Out]

2*b**4*(c + d*x)**(15/2)/(15*d**5) + 8*b**3*(c + d*x)**(13/2)*(a*d - b*c)/(13*d*
*5) + 12*b**2*(c + d*x)**(11/2)*(a*d - b*c)**2/(11*d**5) + 8*b*(c + d*x)**(9/2)*
(a*d - b*c)**3/(9*d**5) + 2*(c + d*x)**(7/2)*(a*d - b*c)**4/(7*d**5)

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Mathematica [A]  time = 0.172093, size = 154, normalized size = 1.19 \[ \frac{2 (c+d x)^{7/2} \left (6435 a^4 d^4+2860 a^3 b d^3 (7 d x-2 c)+390 a^2 b^2 d^2 \left (8 c^2-28 c d x+63 d^2 x^2\right )+60 a b^3 d \left (-16 c^3+56 c^2 d x-126 c d^2 x^2+231 d^3 x^3\right )+b^4 \left (128 c^4-448 c^3 d x+1008 c^2 d^2 x^2-1848 c d^3 x^3+3003 d^4 x^4\right )\right )}{45045 d^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^4*(c + d*x)^(5/2),x]

[Out]

(2*(c + d*x)^(7/2)*(6435*a^4*d^4 + 2860*a^3*b*d^3*(-2*c + 7*d*x) + 390*a^2*b^2*d
^2*(8*c^2 - 28*c*d*x + 63*d^2*x^2) + 60*a*b^3*d*(-16*c^3 + 56*c^2*d*x - 126*c*d^
2*x^2 + 231*d^3*x^3) + b^4*(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)))/(45045*d^5)

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Maple [A]  time = 0.01, size = 186, normalized size = 1.4 \[{\frac{6006\,{x}^{4}{b}^{4}{d}^{4}+27720\,a{b}^{3}{d}^{4}{x}^{3}-3696\,{b}^{4}c{d}^{3}{x}^{3}+49140\,{a}^{2}{b}^{2}{d}^{4}{x}^{2}-15120\,a{b}^{3}c{d}^{3}{x}^{2}+2016\,{b}^{4}{c}^{2}{d}^{2}{x}^{2}+40040\,{a}^{3}b{d}^{4}x-21840\,{a}^{2}{b}^{2}c{d}^{3}x+6720\,a{b}^{3}{c}^{2}{d}^{2}x-896\,{b}^{4}{c}^{3}dx+12870\,{a}^{4}{d}^{4}-11440\,{a}^{3}bc{d}^{3}+6240\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}-1920\,a{b}^{3}{c}^{3}d+256\,{b}^{4}{c}^{4}}{45045\,{d}^{5}} \left ( dx+c \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^4*(d*x+c)^(5/2),x)

[Out]

2/45045*(d*x+c)^(7/2)*(3003*b^4*d^4*x^4+13860*a*b^3*d^4*x^3-1848*b^4*c*d^3*x^3+2
4570*a^2*b^2*d^4*x^2-7560*a*b^3*c*d^3*x^2+1008*b^4*c^2*d^2*x^2+20020*a^3*b*d^4*x
-10920*a^2*b^2*c*d^3*x+3360*a*b^3*c^2*d^2*x-448*b^4*c^3*d*x+6435*a^4*d^4-5720*a^
3*b*c*d^3+3120*a^2*b^2*c^2*d^2-960*a*b^3*c^3*d+128*b^4*c^4)/d^5

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Maxima [A]  time = 1.37173, size = 244, normalized size = 1.89 \[ \frac{2 \,{\left (3003 \,{\left (d x + c\right )}^{\frac{15}{2}} b^{4} - 13860 \,{\left (b^{4} c - a b^{3} d\right )}{\left (d x + c\right )}^{\frac{13}{2}} + 24570 \,{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )}{\left (d x + c\right )}^{\frac{11}{2}} - 20020 \,{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )}{\left (d x + c\right )}^{\frac{9}{2}} + 6435 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )}{\left (d x + c\right )}^{\frac{7}{2}}\right )}}{45045 \, d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^4*(d*x + c)^(5/2),x, algorithm="maxima")

[Out]

2/45045*(3003*(d*x + c)^(15/2)*b^4 - 13860*(b^4*c - a*b^3*d)*(d*x + c)^(13/2) +
24570*(b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*(d*x + c)^(11/2) - 20020*(b^4*c^3 -
3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*(d*x + c)^(9/2) + 6435*(b^4*c^4 - 4
*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*(d*x + c)^(7/2))/d^5

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Fricas [A]  time = 0.219374, size = 509, normalized size = 3.95 \[ \frac{2 \,{\left (3003 \, b^{4} d^{7} x^{7} + 128 \, b^{4} c^{7} - 960 \, a b^{3} c^{6} d + 3120 \, a^{2} b^{2} c^{5} d^{2} - 5720 \, a^{3} b c^{4} d^{3} + 6435 \, a^{4} c^{3} d^{4} + 231 \,{\left (31 \, b^{4} c d^{6} + 60 \, a b^{3} d^{7}\right )} x^{6} + 63 \,{\left (71 \, b^{4} c^{2} d^{5} + 540 \, a b^{3} c d^{6} + 390 \, a^{2} b^{2} d^{7}\right )} x^{5} + 35 \,{\left (b^{4} c^{3} d^{4} + 636 \, a b^{3} c^{2} d^{5} + 1794 \, a^{2} b^{2} c d^{6} + 572 \, a^{3} b d^{7}\right )} x^{4} - 5 \,{\left (8 \, b^{4} c^{4} d^{3} - 60 \, a b^{3} c^{3} d^{4} - 8814 \, a^{2} b^{2} c^{2} d^{5} - 10868 \, a^{3} b c d^{6} - 1287 \, a^{4} d^{7}\right )} x^{3} + 3 \,{\left (16 \, b^{4} c^{5} d^{2} - 120 \, a b^{3} c^{4} d^{3} + 390 \, a^{2} b^{2} c^{3} d^{4} + 14300 \, a^{3} b c^{2} d^{5} + 6435 \, a^{4} c d^{6}\right )} x^{2} -{\left (64 \, b^{4} c^{6} d - 480 \, a b^{3} c^{5} d^{2} + 1560 \, a^{2} b^{2} c^{4} d^{3} - 2860 \, a^{3} b c^{3} d^{4} - 19305 \, a^{4} c^{2} d^{5}\right )} x\right )} \sqrt{d x + c}}{45045 \, d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^4*(d*x + c)^(5/2),x, algorithm="fricas")

[Out]

2/45045*(3003*b^4*d^7*x^7 + 128*b^4*c^7 - 960*a*b^3*c^6*d + 3120*a^2*b^2*c^5*d^2
- 5720*a^3*b*c^4*d^3 + 6435*a^4*c^3*d^4 + 231*(31*b^4*c*d^6 + 60*a*b^3*d^7)*x^6
+ 63*(71*b^4*c^2*d^5 + 540*a*b^3*c*d^6 + 390*a^2*b^2*d^7)*x^5 + 35*(b^4*c^3*d^4
+ 636*a*b^3*c^2*d^5 + 1794*a^2*b^2*c*d^6 + 572*a^3*b*d^7)*x^4 - 5*(8*b^4*c^4*d^
3 - 60*a*b^3*c^3*d^4 - 8814*a^2*b^2*c^2*d^5 - 10868*a^3*b*c*d^6 - 1287*a^4*d^7)*
x^3 + 3*(16*b^4*c^5*d^2 - 120*a*b^3*c^4*d^3 + 390*a^2*b^2*c^3*d^4 + 14300*a^3*b*
c^2*d^5 + 6435*a^4*c*d^6)*x^2 - (64*b^4*c^6*d - 480*a*b^3*c^5*d^2 + 1560*a^2*b^2
*c^4*d^3 - 2860*a^3*b*c^3*d^4 - 19305*a^4*c^2*d^5)*x)*sqrt(d*x + c)/d^5

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Sympy [A]  time = 6.27257, size = 960, normalized size = 7.44 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**4*(d*x+c)**(5/2),x)

[Out]

a**4*c**2*Piecewise((sqrt(c)*x, Eq(d, 0)), (2*(c + d*x)**(3/2)/(3*d), True)) + 4
*a**4*c*(-c*(c + d*x)**(3/2)/3 + (c + d*x)**(5/2)/5)/d + 2*a**4*(c**2*(c + d*x)*
*(3/2)/3 - 2*c*(c + d*x)**(5/2)/5 + (c + d*x)**(7/2)/7)/d + 8*a**3*b*c**2*(-c*(c
+ d*x)**(3/2)/3 + (c + d*x)**(5/2)/5)/d**2 + 16*a**3*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 + 8*a**3*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 + 12*a**2*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 + 24*a**2*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 + 12*a**
2*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 + 8*a*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 + 16*a*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 + 8*a*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 + 2*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 + 4*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 + 2*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

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GIAC/XCAS [A]  time = 0.234996, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^4*(d*x + c)^(5/2),x, algorithm="giac")

[Out]

Done