3.2205 \(\int \frac{(a+b x)^{3/2} (A+B x)}{(d+e x)^{17/2}} \, dx\)

Optimal. Leaf size=304 \[ \frac{256 b^4 (a+b x)^{5/2} (-3 a B e+2 A b e+b B d)}{45045 e (d+e x)^{5/2} (b d-a e)^6}+\frac{128 b^3 (a+b x)^{5/2} (-3 a B e+2 A b e+b B d)}{9009 e (d+e x)^{7/2} (b d-a e)^5}+\frac{32 b^2 (a+b x)^{5/2} (-3 a B e+2 A b e+b B d)}{1287 e (d+e x)^{9/2} (b d-a e)^4}+\frac{16 b (a+b x)^{5/2} (-3 a B e+2 A b e+b B d)}{429 e (d+e x)^{11/2} (b d-a e)^3}+\frac{2 (a+b x)^{5/2} (-3 a B e+2 A b e+b B d)}{39 e (d+e x)^{13/2} (b d-a e)^2}-\frac{2 (a+b x)^{5/2} (B d-A e)}{15 e (d+e x)^{15/2} (b d-a e)} \]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(5/2))/(15*e*(b*d - a*e)*(d + e*x)^(15/2)) + (2*(b*B*d
+ 2*A*b*e - 3*a*B*e)*(a + b*x)^(5/2))/(39*e*(b*d - a*e)^2*(d + e*x)^(13/2)) + (
16*b*(b*B*d + 2*A*b*e - 3*a*B*e)*(a + b*x)^(5/2))/(429*e*(b*d - a*e)^3*(d + e*x)
^(11/2)) + (32*b^2*(b*B*d + 2*A*b*e - 3*a*B*e)*(a + b*x)^(5/2))/(1287*e*(b*d - a
*e)^4*(d + e*x)^(9/2)) + (128*b^3*(b*B*d + 2*A*b*e - 3*a*B*e)*(a + b*x)^(5/2))/(
9009*e*(b*d - a*e)^5*(d + e*x)^(7/2)) + (256*b^4*(b*B*d + 2*A*b*e - 3*a*B*e)*(a
+ b*x)^(5/2))/(45045*e*(b*d - a*e)^6*(d + e*x)^(5/2))

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Rubi [A]  time = 0.579331, antiderivative size = 304, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{256 b^4 (a+b x)^{5/2} (-3 a B e+2 A b e+b B d)}{45045 e (d+e x)^{5/2} (b d-a e)^6}+\frac{128 b^3 (a+b x)^{5/2} (-3 a B e+2 A b e+b B d)}{9009 e (d+e x)^{7/2} (b d-a e)^5}+\frac{32 b^2 (a+b x)^{5/2} (-3 a B e+2 A b e+b B d)}{1287 e (d+e x)^{9/2} (b d-a e)^4}+\frac{16 b (a+b x)^{5/2} (-3 a B e+2 A b e+b B d)}{429 e (d+e x)^{11/2} (b d-a e)^3}+\frac{2 (a+b x)^{5/2} (-3 a B e+2 A b e+b B d)}{39 e (d+e x)^{13/2} (b d-a e)^2}-\frac{2 (a+b x)^{5/2} (B d-A e)}{15 e (d+e x)^{15/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^(3/2)*(A + B*x))/(d + e*x)^(17/2),x]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(5/2))/(15*e*(b*d - a*e)*(d + e*x)^(15/2)) + (2*(b*B*d
+ 2*A*b*e - 3*a*B*e)*(a + b*x)^(5/2))/(39*e*(b*d - a*e)^2*(d + e*x)^(13/2)) + (
16*b*(b*B*d + 2*A*b*e - 3*a*B*e)*(a + b*x)^(5/2))/(429*e*(b*d - a*e)^3*(d + e*x)
^(11/2)) + (32*b^2*(b*B*d + 2*A*b*e - 3*a*B*e)*(a + b*x)^(5/2))/(1287*e*(b*d - a
*e)^4*(d + e*x)^(9/2)) + (128*b^3*(b*B*d + 2*A*b*e - 3*a*B*e)*(a + b*x)^(5/2))/(
9009*e*(b*d - a*e)^5*(d + e*x)^(7/2)) + (256*b^4*(b*B*d + 2*A*b*e - 3*a*B*e)*(a
+ b*x)^(5/2))/(45045*e*(b*d - a*e)^6*(d + e*x)^(5/2))

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Rubi in Sympy [A]  time = 64.4818, size = 286, normalized size = 0.94 \[ - \frac{512 b^{4} \left (a + b x\right )^{\frac{5}{2}} \left (- A b e + \frac{B \left (3 a e - b d\right )}{2}\right )}{45045 e \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{6}} + \frac{256 b^{3} \left (a + b x\right )^{\frac{5}{2}} \left (- A b e + \frac{B \left (3 a e - b d\right )}{2}\right )}{9009 e \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{5}} - \frac{64 b^{2} \left (a + b x\right )^{\frac{5}{2}} \left (- A b e + \frac{B \left (3 a e - b d\right )}{2}\right )}{1287 e \left (d + e x\right )^{\frac{9}{2}} \left (a e - b d\right )^{4}} - \frac{16 b \left (a + b x\right )^{\frac{5}{2}} \left (2 A b e - 3 B a e + B b d\right )}{429 e \left (d + e x\right )^{\frac{11}{2}} \left (a e - b d\right )^{3}} - \frac{4 \left (a + b x\right )^{\frac{5}{2}} \left (- A b e + \frac{B \left (3 a e - b d\right )}{2}\right )}{39 e \left (d + e x\right )^{\frac{13}{2}} \left (a e - b d\right )^{2}} - \frac{2 \left (a + b x\right )^{\frac{5}{2}} \left (A e - B d\right )}{15 e \left (d + e x\right )^{\frac{15}{2}} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(3/2)*(B*x+A)/(e*x+d)**(17/2),x)

[Out]

-512*b**4*(a + b*x)**(5/2)*(-A*b*e + B*(3*a*e - b*d)/2)/(45045*e*(d + e*x)**(5/2
)*(a*e - b*d)**6) + 256*b**3*(a + b*x)**(5/2)*(-A*b*e + B*(3*a*e - b*d)/2)/(9009
*e*(d + e*x)**(7/2)*(a*e - b*d)**5) - 64*b**2*(a + b*x)**(5/2)*(-A*b*e + B*(3*a*
e - b*d)/2)/(1287*e*(d + e*x)**(9/2)*(a*e - b*d)**4) - 16*b*(a + b*x)**(5/2)*(2*
A*b*e - 3*B*a*e + B*b*d)/(429*e*(d + e*x)**(11/2)*(a*e - b*d)**3) - 4*(a + b*x)*
*(5/2)*(-A*b*e + B*(3*a*e - b*d)/2)/(39*e*(d + e*x)**(13/2)*(a*e - b*d)**2) - 2*
(a + b*x)**(5/2)*(A*e - B*d)/(15*e*(d + e*x)**(15/2)*(a*e - b*d))

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Mathematica [A]  time = 0.67981, size = 287, normalized size = 0.94 \[ \frac{2 \sqrt{a+b x} \left (\frac{128 b^6 (d+e x)^7 (-3 a B e+2 A b e+b B d)}{(b d-a e)^6}+\frac{64 b^5 (d+e x)^6 (-3 a B e+2 A b e+b B d)}{(b d-a e)^5}+\frac{48 b^4 (d+e x)^5 (-3 a B e+2 A b e+b B d)}{(b d-a e)^4}+\frac{40 b^3 (d+e x)^4 (-3 a B e+2 A b e+b B d)}{(b d-a e)^3}+\frac{35 b^2 (d+e x)^3 (-3 a B e+2 A b e+b B d)}{(b d-a e)^2}-\frac{63 b (d+e x)^2 (70 a B e+A b e-71 b B d)}{a e-b d}+231 (d+e x) (-15 a B e-16 A b e+31 b B d)-3003 (b d-a e) (B d-A e)\right )}{45045 e^3 (d+e x)^{15/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^(3/2)*(A + B*x))/(d + e*x)^(17/2),x]

[Out]

(2*Sqrt[a + b*x]*(-3003*(b*d - a*e)*(B*d - A*e) + 231*(31*b*B*d - 16*A*b*e - 15*
a*B*e)*(d + e*x) - (63*b*(-71*b*B*d + A*b*e + 70*a*B*e)*(d + e*x)^2)/(-(b*d) + a
*e) + (35*b^2*(b*B*d + 2*A*b*e - 3*a*B*e)*(d + e*x)^3)/(b*d - a*e)^2 + (40*b^3*(
b*B*d + 2*A*b*e - 3*a*B*e)*(d + e*x)^4)/(b*d - a*e)^3 + (48*b^4*(b*B*d + 2*A*b*e
- 3*a*B*e)*(d + e*x)^5)/(b*d - a*e)^4 + (64*b^5*(b*B*d + 2*A*b*e - 3*a*B*e)*(d
+ e*x)^6)/(b*d - a*e)^5 + (128*b^6*(b*B*d + 2*A*b*e - 3*a*B*e)*(d + e*x)^7)/(b*d
- a*e)^6))/(45045*e^3*(d + e*x)^(15/2))

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Maple [B]  time = 0.018, size = 722, normalized size = 2.4 \[ -{\frac{-512\,A{b}^{5}{e}^{5}{x}^{5}+768\,Ba{b}^{4}{e}^{5}{x}^{5}-256\,B{b}^{5}d{e}^{4}{x}^{5}+1280\,Aa{b}^{4}{e}^{5}{x}^{4}-3840\,A{b}^{5}d{e}^{4}{x}^{4}-1920\,B{a}^{2}{b}^{3}{e}^{5}{x}^{4}+6400\,Ba{b}^{4}d{e}^{4}{x}^{4}-1920\,B{b}^{5}{d}^{2}{e}^{3}{x}^{4}-2240\,A{a}^{2}{b}^{3}{e}^{5}{x}^{3}+9600\,Aa{b}^{4}d{e}^{4}{x}^{3}-12480\,A{b}^{5}{d}^{2}{e}^{3}{x}^{3}+3360\,B{a}^{3}{b}^{2}{e}^{5}{x}^{3}-15520\,B{a}^{2}{b}^{3}d{e}^{4}{x}^{3}+23520\,Ba{b}^{4}{d}^{2}{e}^{3}{x}^{3}-6240\,B{b}^{5}{d}^{3}{e}^{2}{x}^{3}+3360\,A{a}^{3}{b}^{2}{e}^{5}{x}^{2}-16800\,A{a}^{2}{b}^{3}d{e}^{4}{x}^{2}+31200\,Aa{b}^{4}{d}^{2}{e}^{3}{x}^{2}-22880\,A{b}^{5}{d}^{3}{e}^{2}{x}^{2}-5040\,B{a}^{4}b{e}^{5}{x}^{2}+26880\,B{a}^{3}{b}^{2}d{e}^{4}{x}^{2}-55200\,B{a}^{2}{b}^{3}{d}^{2}{e}^{3}{x}^{2}+49920\,Ba{b}^{4}{d}^{3}{e}^{2}{x}^{2}-11440\,B{b}^{5}{d}^{4}e{x}^{2}-4620\,A{a}^{4}b{e}^{5}x+25200\,A{a}^{3}{b}^{2}d{e}^{4}x-54600\,A{a}^{2}{b}^{3}{d}^{2}{e}^{3}x+57200\,Aa{b}^{4}{d}^{3}{e}^{2}x-25740\,A{b}^{5}{d}^{4}ex+6930\,B{a}^{5}{e}^{5}x-40110\,B{a}^{4}bd{e}^{4}x+94500\,B{a}^{3}{b}^{2}{d}^{2}{e}^{3}x-113100\,B{a}^{2}{b}^{3}{d}^{3}{e}^{2}x+67210\,Ba{b}^{4}{d}^{4}ex-12870\,B{b}^{5}{d}^{5}x+6006\,A{a}^{5}{e}^{5}-34650\,A{a}^{4}bd{e}^{4}+81900\,A{a}^{3}{b}^{2}{d}^{2}{e}^{3}-100100\,A{a}^{2}{b}^{3}{d}^{3}{e}^{2}+64350\,Aa{b}^{4}{d}^{4}e-18018\,A{b}^{5}{d}^{5}+924\,B{a}^{5}d{e}^{4}-5040\,B{a}^{4}b{d}^{2}{e}^{3}+10920\,B{a}^{3}{b}^{2}{d}^{3}{e}^{2}-11440\,B{a}^{2}{b}^{3}{d}^{4}e+5148\,Ba{b}^{4}{d}^{5}}{45045\,{a}^{6}{e}^{6}-270270\,{a}^{5}bd{e}^{5}+675675\,{a}^{4}{b}^{2}{d}^{2}{e}^{4}-900900\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}+675675\,{a}^{2}{b}^{4}{d}^{4}{e}^{2}-270270\,a{b}^{5}{d}^{5}e+45045\,{b}^{6}{d}^{6}} \left ( bx+a \right ) ^{{\frac{5}{2}}} \left ( ex+d \right ) ^{-{\frac{15}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(3/2)*(B*x+A)/(e*x+d)^(17/2),x)

[Out]

-2/45045*(b*x+a)^(5/2)*(-256*A*b^5*e^5*x^5+384*B*a*b^4*e^5*x^5-128*B*b^5*d*e^4*x
^5+640*A*a*b^4*e^5*x^4-1920*A*b^5*d*e^4*x^4-960*B*a^2*b^3*e^5*x^4+3200*B*a*b^4*d
*e^4*x^4-960*B*b^5*d^2*e^3*x^4-1120*A*a^2*b^3*e^5*x^3+4800*A*a*b^4*d*e^4*x^3-624
0*A*b^5*d^2*e^3*x^3+1680*B*a^3*b^2*e^5*x^3-7760*B*a^2*b^3*d*e^4*x^3+11760*B*a*b^
4*d^2*e^3*x^3-3120*B*b^5*d^3*e^2*x^3+1680*A*a^3*b^2*e^5*x^2-8400*A*a^2*b^3*d*e^4
*x^2+15600*A*a*b^4*d^2*e^3*x^2-11440*A*b^5*d^3*e^2*x^2-2520*B*a^4*b*e^5*x^2+1344
0*B*a^3*b^2*d*e^4*x^2-27600*B*a^2*b^3*d^2*e^3*x^2+24960*B*a*b^4*d^3*e^2*x^2-5720
*B*b^5*d^4*e*x^2-2310*A*a^4*b*e^5*x+12600*A*a^3*b^2*d*e^4*x-27300*A*a^2*b^3*d^2*
e^3*x+28600*A*a*b^4*d^3*e^2*x-12870*A*b^5*d^4*e*x+3465*B*a^5*e^5*x-20055*B*a^4*b
*d*e^4*x+47250*B*a^3*b^2*d^2*e^3*x-56550*B*a^2*b^3*d^3*e^2*x+33605*B*a*b^4*d^4*e
*x-6435*B*b^5*d^5*x+3003*A*a^5*e^5-17325*A*a^4*b*d*e^4+40950*A*a^3*b^2*d^2*e^3-5
0050*A*a^2*b^3*d^3*e^2+32175*A*a*b^4*d^4*e-9009*A*b^5*d^5+462*B*a^5*d*e^4-2520*B
*a^4*b*d^2*e^3+5460*B*a^3*b^2*d^3*e^2-5720*B*a^2*b^3*d^4*e+2574*B*a*b^4*d^5)/(e*
x+d)^(15/2)/(a^6*e^6-6*a^5*b*d*e^5+15*a^4*b^2*d^2*e^4-20*a^3*b^3*d^3*e^3+15*a^2*
b^4*d^4*e^2-6*a*b^5*d^5*e+b^6*d^6)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(3/2)/(e*x + d)^(17/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 30.843, size = 2260, normalized size = 7.43 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(3/2)/(e*x + d)^(17/2),x, algorithm="fricas")

[Out]

-2/45045*(3003*A*a^7*e^5 - 128*(B*b^7*d*e^4 - (3*B*a*b^6 - 2*A*b^7)*e^5)*x^7 - 6
4*(15*B*b^7*d^2*e^3 - 2*(23*B*a*b^6 - 15*A*b^7)*d*e^4 + (3*B*a^2*b^5 - 2*A*a*b^6
)*e^5)*x^6 + 1287*(2*B*a^3*b^4 - 7*A*a^2*b^5)*d^5 - 715*(8*B*a^4*b^3 - 45*A*a^3*
b^4)*d^4*e + 910*(6*B*a^5*b^2 - 55*A*a^4*b^3)*d^3*e^2 - 630*(4*B*a^6*b - 65*A*a^
5*b^2)*d^2*e^3 + 231*(2*B*a^7 - 75*A*a^6*b)*d*e^4 - 48*(65*B*b^7*d^3*e^2 - 5*(41
*B*a*b^6 - 26*A*b^7)*d^2*e^3 + (31*B*a^2*b^5 - 20*A*a*b^6)*d*e^4 - (3*B*a^3*b^4
- 2*A*a^2*b^5)*e^5)*x^5 - 40*(143*B*b^7*d^4*e - 26*(18*B*a*b^6 - 11*A*b^7)*d^3*e
^2 + 6*(21*B*a^2*b^5 - 13*A*a*b^6)*d^2*e^3 - 2*(14*B*a^3*b^4 - 9*A*a^2*b^5)*d*e^
4 + (3*B*a^4*b^3 - 2*A*a^3*b^4)*e^5)*x^4 - 5*(1287*B*b^7*d^5 - 143*(31*B*a*b^6 -
18*A*b^7)*d^4*e + 26*(75*B*a^2*b^5 - 44*A*a*b^6)*d^3*e^2 - 6*(127*B*a^3*b^4 - 7
8*A*a^2*b^5)*d^2*e^3 + (187*B*a^4*b^3 - 120*A*a^3*b^4)*d*e^4 - 7*(3*B*a^5*b^2 -
2*A*a^4*b^3)*e^5)*x^3 - 3*(429*(8*B*a*b^6 + 7*A*b^7)*d^5 - 715*(26*B*a^2*b^5 + 3
*A*a*b^6)*d^4*e + 130*(212*B*a^3*b^4 + 11*A*a^2*b^5)*d^3*e^2 - 10*(2146*B*a^4*b^
3 + 65*A*a^3*b^4)*d^2*e^3 + 7*(1248*B*a^5*b^2 + 25*A*a^4*b^3)*d*e^4 - 21*(70*B*a
^6*b + A*a^5*b^2)*e^5)*x^2 - (1287*(B*a^2*b^5 + 14*A*a*b^6)*d^5 - 715*(31*B*a^3*
b^4 + 72*A*a^2*b^5)*d^4*e + 130*(351*B*a^4*b^3 + 550*A*a^3*b^4)*d^3*e^2 - 210*(2
01*B*a^5*b^2 + 260*A*a^4*b^3)*d^2*e^3 + 21*(911*B*a^6*b + 1050*A*a^5*b^2)*d*e^4
- 231*(15*B*a^7 + 16*A*a^6*b)*e^5)*x)*sqrt(b*x + a)*sqrt(e*x + d)/(b^6*d^14 - 6*
a*b^5*d^13*e + 15*a^2*b^4*d^12*e^2 - 20*a^3*b^3*d^11*e^3 + 15*a^4*b^2*d^10*e^4 -
6*a^5*b*d^9*e^5 + a^6*d^8*e^6 + (b^6*d^6*e^8 - 6*a*b^5*d^5*e^9 + 15*a^2*b^4*d^4
*e^10 - 20*a^3*b^3*d^3*e^11 + 15*a^4*b^2*d^2*e^12 - 6*a^5*b*d*e^13 + a^6*e^14)*x
^8 + 8*(b^6*d^7*e^7 - 6*a*b^5*d^6*e^8 + 15*a^2*b^4*d^5*e^9 - 20*a^3*b^3*d^4*e^10
+ 15*a^4*b^2*d^3*e^11 - 6*a^5*b*d^2*e^12 + a^6*d*e^13)*x^7 + 28*(b^6*d^8*e^6 -
6*a*b^5*d^7*e^7 + 15*a^2*b^4*d^6*e^8 - 20*a^3*b^3*d^5*e^9 + 15*a^4*b^2*d^4*e^10
- 6*a^5*b*d^3*e^11 + a^6*d^2*e^12)*x^6 + 56*(b^6*d^9*e^5 - 6*a*b^5*d^8*e^6 + 15*
a^2*b^4*d^7*e^7 - 20*a^3*b^3*d^6*e^8 + 15*a^4*b^2*d^5*e^9 - 6*a^5*b*d^4*e^10 + a
^6*d^3*e^11)*x^5 + 70*(b^6*d^10*e^4 - 6*a*b^5*d^9*e^5 + 15*a^2*b^4*d^8*e^6 - 20*
a^3*b^3*d^7*e^7 + 15*a^4*b^2*d^6*e^8 - 6*a^5*b*d^5*e^9 + a^6*d^4*e^10)*x^4 + 56*
(b^6*d^11*e^3 - 6*a*b^5*d^10*e^4 + 15*a^2*b^4*d^9*e^5 - 20*a^3*b^3*d^8*e^6 + 15*
a^4*b^2*d^7*e^7 - 6*a^5*b*d^6*e^8 + a^6*d^5*e^9)*x^3 + 28*(b^6*d^12*e^2 - 6*a*b^
5*d^11*e^3 + 15*a^2*b^4*d^10*e^4 - 20*a^3*b^3*d^9*e^5 + 15*a^4*b^2*d^8*e^6 - 6*a
^5*b*d^7*e^7 + a^6*d^6*e^8)*x^2 + 8*(b^6*d^13*e - 6*a*b^5*d^12*e^2 + 15*a^2*b^4*
d^11*e^3 - 20*a^3*b^3*d^10*e^4 + 15*a^4*b^2*d^9*e^5 - 6*a^5*b*d^8*e^6 + a^6*d^7*
e^7)*x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(3/2)*(B*x+A)/(e*x+d)**(17/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.698724, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(3/2)/(e*x + d)^(17/2),x, algorithm="giac")

[Out]

Done