3.2204 \(\int \frac{(a+b x)^{3/2} (A+B x)}{(d+e x)^{15/2}} \, dx\)

Optimal. Leaf size=255 \[ \frac{32 b^3 (a+b x)^{5/2} (-13 a B e+8 A b e+5 b B d)}{15015 e (d+e x)^{5/2} (b d-a e)^5}+\frac{16 b^2 (a+b x)^{5/2} (-13 a B e+8 A b e+5 b B d)}{3003 e (d+e x)^{7/2} (b d-a e)^4}+\frac{4 b (a+b x)^{5/2} (-13 a B e+8 A b e+5 b B d)}{429 e (d+e x)^{9/2} (b d-a e)^3}+\frac{2 (a+b x)^{5/2} (-13 a B e+8 A b e+5 b B d)}{143 e (d+e x)^{11/2} (b d-a e)^2}-\frac{2 (a+b x)^{5/2} (B d-A e)}{13 e (d+e x)^{13/2} (b d-a e)} \]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(5/2))/(13*e*(b*d - a*e)*(d + e*x)^(13/2)) + (2*(5*b*B
*d + 8*A*b*e - 13*a*B*e)*(a + b*x)^(5/2))/(143*e*(b*d - a*e)^2*(d + e*x)^(11/2))
+ (4*b*(5*b*B*d + 8*A*b*e - 13*a*B*e)*(a + b*x)^(5/2))/(429*e*(b*d - a*e)^3*(d
+ e*x)^(9/2)) + (16*b^2*(5*b*B*d + 8*A*b*e - 13*a*B*e)*(a + b*x)^(5/2))/(3003*e*
(b*d - a*e)^4*(d + e*x)^(7/2)) + (32*b^3*(5*b*B*d + 8*A*b*e - 13*a*B*e)*(a + b*x
)^(5/2))/(15015*e*(b*d - a*e)^5*(d + e*x)^(5/2))

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Rubi [A]  time = 0.449307, antiderivative size = 255, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{32 b^3 (a+b x)^{5/2} (-13 a B e+8 A b e+5 b B d)}{15015 e (d+e x)^{5/2} (b d-a e)^5}+\frac{16 b^2 (a+b x)^{5/2} (-13 a B e+8 A b e+5 b B d)}{3003 e (d+e x)^{7/2} (b d-a e)^4}+\frac{4 b (a+b x)^{5/2} (-13 a B e+8 A b e+5 b B d)}{429 e (d+e x)^{9/2} (b d-a e)^3}+\frac{2 (a+b x)^{5/2} (-13 a B e+8 A b e+5 b B d)}{143 e (d+e x)^{11/2} (b d-a e)^2}-\frac{2 (a+b x)^{5/2} (B d-A e)}{13 e (d+e x)^{13/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^(3/2)*(A + B*x))/(d + e*x)^(15/2),x]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(5/2))/(13*e*(b*d - a*e)*(d + e*x)^(13/2)) + (2*(5*b*B
*d + 8*A*b*e - 13*a*B*e)*(a + b*x)^(5/2))/(143*e*(b*d - a*e)^2*(d + e*x)^(11/2))
+ (4*b*(5*b*B*d + 8*A*b*e - 13*a*B*e)*(a + b*x)^(5/2))/(429*e*(b*d - a*e)^3*(d
+ e*x)^(9/2)) + (16*b^2*(5*b*B*d + 8*A*b*e - 13*a*B*e)*(a + b*x)^(5/2))/(3003*e*
(b*d - a*e)^4*(d + e*x)^(7/2)) + (32*b^3*(5*b*B*d + 8*A*b*e - 13*a*B*e)*(a + b*x
)^(5/2))/(15015*e*(b*d - a*e)^5*(d + e*x)^(5/2))

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Rubi in Sympy [A]  time = 50.9976, size = 246, normalized size = 0.96 \[ - \frac{32 b^{3} \left (a + b x\right )^{\frac{5}{2}} \left (8 A b e - 13 B a e + 5 B b d\right )}{15015 e \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{5}} + \frac{16 b^{2} \left (a + b x\right )^{\frac{5}{2}} \left (8 A b e - 13 B a e + 5 B b d\right )}{3003 e \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{4}} - \frac{4 b \left (a + b x\right )^{\frac{5}{2}} \left (8 A b e - 13 B a e + 5 B b d\right )}{429 e \left (d + e x\right )^{\frac{9}{2}} \left (a e - b d\right )^{3}} + \frac{2 \left (a + b x\right )^{\frac{5}{2}} \left (8 A b e - 13 B a e + 5 B b d\right )}{143 e \left (d + e x\right )^{\frac{11}{2}} \left (a e - b d\right )^{2}} - \frac{2 \left (a + b x\right )^{\frac{5}{2}} \left (A e - B d\right )}{13 e \left (d + e x\right )^{\frac{13}{2}} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(3/2)*(B*x+A)/(e*x+d)**(15/2),x)

[Out]

-32*b**3*(a + b*x)**(5/2)*(8*A*b*e - 13*B*a*e + 5*B*b*d)/(15015*e*(d + e*x)**(5/
2)*(a*e - b*d)**5) + 16*b**2*(a + b*x)**(5/2)*(8*A*b*e - 13*B*a*e + 5*B*b*d)/(30
03*e*(d + e*x)**(7/2)*(a*e - b*d)**4) - 4*b*(a + b*x)**(5/2)*(8*A*b*e - 13*B*a*e
+ 5*B*b*d)/(429*e*(d + e*x)**(9/2)*(a*e - b*d)**3) + 2*(a + b*x)**(5/2)*(8*A*b*
e - 13*B*a*e + 5*B*b*d)/(143*e*(d + e*x)**(11/2)*(a*e - b*d)**2) - 2*(a + b*x)**
(5/2)*(A*e - B*d)/(13*e*(d + e*x)**(13/2)*(a*e - b*d))

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Mathematica [A]  time = 0.54001, size = 254, normalized size = 1. \[ \frac{2 \sqrt{a+b x} \left (\frac{16 b^5 (d+e x)^6 (-13 a B e+8 A b e+5 b B d)}{(b d-a e)^5}+\frac{8 b^4 (d+e x)^5 (-13 a B e+8 A b e+5 b B d)}{(b d-a e)^4}+\frac{6 b^3 (d+e x)^4 (-13 a B e+8 A b e+5 b B d)}{(b d-a e)^3}+\frac{5 b^2 (d+e x)^3 (-13 a B e+8 A b e+5 b B d)}{(b d-a e)^2}-\frac{35 b (d+e x)^2 (52 a B e+A b e-53 b B d)}{a e-b d}+105 (d+e x) (-13 a B e-14 A b e+27 b B d)-1155 (b d-a e) (B d-A e)\right )}{15015 e^3 (d+e x)^{13/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^(3/2)*(A + B*x))/(d + e*x)^(15/2),x]

[Out]

(2*Sqrt[a + b*x]*(-1155*(b*d - a*e)*(B*d - A*e) + 105*(27*b*B*d - 14*A*b*e - 13*
a*B*e)*(d + e*x) - (35*b*(-53*b*B*d + A*b*e + 52*a*B*e)*(d + e*x)^2)/(-(b*d) + a
*e) + (5*b^2*(5*b*B*d + 8*A*b*e - 13*a*B*e)*(d + e*x)^3)/(b*d - a*e)^2 + (6*b^3*
(5*b*B*d + 8*A*b*e - 13*a*B*e)*(d + e*x)^4)/(b*d - a*e)^3 + (8*b^4*(5*b*B*d + 8*
A*b*e - 13*a*B*e)*(d + e*x)^5)/(b*d - a*e)^4 + (16*b^5*(5*b*B*d + 8*A*b*e - 13*a
*B*e)*(d + e*x)^6)/(b*d - a*e)^5))/(15015*e^3*(d + e*x)^(13/2))

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Maple [B]  time = 0.017, size = 505, normalized size = 2. \[ -{\frac{256\,A{b}^{4}{e}^{4}{x}^{4}-416\,Ba{b}^{3}{e}^{4}{x}^{4}+160\,B{b}^{4}d{e}^{3}{x}^{4}-640\,Aa{b}^{3}{e}^{4}{x}^{3}+1664\,A{b}^{4}d{e}^{3}{x}^{3}+1040\,B{a}^{2}{b}^{2}{e}^{4}{x}^{3}-3104\,Ba{b}^{3}d{e}^{3}{x}^{3}+1040\,B{b}^{4}{d}^{2}{e}^{2}{x}^{3}+1120\,A{a}^{2}{b}^{2}{e}^{4}{x}^{2}-4160\,Aa{b}^{3}d{e}^{3}{x}^{2}+4576\,A{b}^{4}{d}^{2}{e}^{2}{x}^{2}-1820\,B{a}^{3}b{e}^{4}{x}^{2}+7460\,B{a}^{2}{b}^{2}d{e}^{3}{x}^{2}-10036\,Ba{b}^{3}{d}^{2}{e}^{2}{x}^{2}+2860\,B{b}^{4}{d}^{3}e{x}^{2}-1680\,A{a}^{3}b{e}^{4}x+7280\,A{a}^{2}{b}^{2}d{e}^{3}x-11440\,Aa{b}^{3}{d}^{2}{e}^{2}x+6864\,A{b}^{4}{d}^{3}ex+2730\,B{a}^{4}{e}^{4}x-12880\,B{a}^{3}bd{e}^{3}x+23140\,B{a}^{2}{b}^{2}{d}^{2}{e}^{2}x-18304\,Ba{b}^{3}{d}^{3}ex+4290\,B{b}^{4}{d}^{4}x+2310\,A{a}^{4}{e}^{4}-10920\,A{a}^{3}bd{e}^{3}+20020\,A{a}^{2}{b}^{2}{d}^{2}{e}^{2}-17160\,Aa{b}^{3}{d}^{3}e+6006\,A{b}^{4}{d}^{4}+420\,B{a}^{4}d{e}^{3}-1820\,B{a}^{3}b{d}^{2}{e}^{2}+2860\,B{a}^{2}{b}^{2}{d}^{3}e-1716\,Ba{b}^{3}{d}^{4}}{15015\,{a}^{5}{e}^{5}-75075\,{a}^{4}bd{e}^{4}+150150\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}-150150\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+75075\,a{b}^{4}{d}^{4}e-15015\,{b}^{5}{d}^{5}} \left ( bx+a \right ) ^{{\frac{5}{2}}} \left ( ex+d \right ) ^{-{\frac{13}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(3/2)*(B*x+A)/(e*x+d)^(15/2),x)

[Out]

-2/15015*(b*x+a)^(5/2)*(128*A*b^4*e^4*x^4-208*B*a*b^3*e^4*x^4+80*B*b^4*d*e^3*x^4
-320*A*a*b^3*e^4*x^3+832*A*b^4*d*e^3*x^3+520*B*a^2*b^2*e^4*x^3-1552*B*a*b^3*d*e^
3*x^3+520*B*b^4*d^2*e^2*x^3+560*A*a^2*b^2*e^4*x^2-2080*A*a*b^3*d*e^3*x^2+2288*A*
b^4*d^2*e^2*x^2-910*B*a^3*b*e^4*x^2+3730*B*a^2*b^2*d*e^3*x^2-5018*B*a*b^3*d^2*e^
2*x^2+1430*B*b^4*d^3*e*x^2-840*A*a^3*b*e^4*x+3640*A*a^2*b^2*d*e^3*x-5720*A*a*b^3
*d^2*e^2*x+3432*A*b^4*d^3*e*x+1365*B*a^4*e^4*x-6440*B*a^3*b*d*e^3*x+11570*B*a^2*
b^2*d^2*e^2*x-9152*B*a*b^3*d^3*e*x+2145*B*b^4*d^4*x+1155*A*a^4*e^4-5460*A*a^3*b*
d*e^3+10010*A*a^2*b^2*d^2*e^2-8580*A*a*b^3*d^3*e+3003*A*b^4*d^4+210*B*a^4*d*e^3-
910*B*a^3*b*d^2*e^2+1430*B*a^2*b^2*d^3*e-858*B*a*b^3*d^4)/(e*x+d)^(13/2)/(a^5*e^
5-5*a^4*b*d*e^4+10*a^3*b^2*d^2*e^3-10*a^2*b^3*d^3*e^2+5*a*b^4*d^4*e-b^5*d^5)

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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)^(15/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 14.3208, size = 1690, normalized size = 6.63 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(3/2)/(e*x + d)^(15/2),x, algorithm="fricas")

[Out]

2/15015*(1155*A*a^6*e^4 + 16*(5*B*b^6*d*e^3 - (13*B*a*b^5 - 8*A*b^6)*e^4)*x^6 +
8*(65*B*b^6*d^2*e^2 - 2*(87*B*a*b^5 - 52*A*b^6)*d*e^3 + (13*B*a^2*b^4 - 8*A*a*b^
5)*e^4)*x^5 - 429*(2*B*a^3*b^3 - 7*A*a^2*b^4)*d^4 + 1430*(B*a^4*b^2 - 6*A*a^3*b^
3)*d^3*e - 910*(B*a^5*b - 11*A*a^4*b^2)*d^2*e^2 + 210*(B*a^6 - 26*A*a^5*b)*d*e^3
+ 2*(715*B*b^6*d^3*e - 13*(153*B*a*b^5 - 88*A*b^6)*d^2*e^2 + (353*B*a^2*b^4 - 2
08*A*a*b^5)*d*e^3 - 3*(13*B*a^3*b^3 - 8*A*a^2*b^4)*e^4)*x^4 + (2145*B*b^6*d^4 -
572*(11*B*a*b^5 - 6*A*b^6)*d^3*e + 26*(79*B*a^2*b^4 - 44*A*a*b^5)*d^2*e^2 - 4*(1
33*B*a^3*b^3 - 78*A*a^2*b^4)*d*e^3 + 5*(13*B*a^4*b^2 - 8*A*a^3*b^3)*e^4)*x^3 + (
429*(8*B*a*b^5 + 7*A*b^6)*d^4 - 1716*(9*B*a^2*b^4 + A*a*b^5)*d^3*e + 26*(662*B*a
^3*b^3 + 33*A*a^2*b^4)*d^2*e^2 - 20*(447*B*a^4*b^2 + 13*A*a^3*b^3)*d*e^3 + 35*(5
2*B*a^5*b + A*a^4*b^2)*e^4)*x^2 + (429*(B*a^2*b^4 + 14*A*a*b^5)*d^4 - 572*(11*B*
a^3*b^3 + 24*A*a^2*b^4)*d^3*e + 650*(15*B*a^4*b^2 + 22*A*a^3*b^3)*d^2*e^2 - 140*
(43*B*a^5*b + 52*A*a^4*b^2)*d*e^3 + 105*(13*B*a^6 + 14*A*a^5*b)*e^4)*x)*sqrt(b*x
+ a)*sqrt(e*x + d)/(b^5*d^12 - 5*a*b^4*d^11*e + 10*a^2*b^3*d^10*e^2 - 10*a^3*b^
2*d^9*e^3 + 5*a^4*b*d^8*e^4 - a^5*d^7*e^5 + (b^5*d^5*e^7 - 5*a*b^4*d^4*e^8 + 10*
a^2*b^3*d^3*e^9 - 10*a^3*b^2*d^2*e^10 + 5*a^4*b*d*e^11 - a^5*e^12)*x^7 + 7*(b^5*
d^6*e^6 - 5*a*b^4*d^5*e^7 + 10*a^2*b^3*d^4*e^8 - 10*a^3*b^2*d^3*e^9 + 5*a^4*b*d^
2*e^10 - a^5*d*e^11)*x^6 + 21*(b^5*d^7*e^5 - 5*a*b^4*d^6*e^6 + 10*a^2*b^3*d^5*e^
7 - 10*a^3*b^2*d^4*e^8 + 5*a^4*b*d^3*e^9 - a^5*d^2*e^10)*x^5 + 35*(b^5*d^8*e^4 -
5*a*b^4*d^7*e^5 + 10*a^2*b^3*d^6*e^6 - 10*a^3*b^2*d^5*e^7 + 5*a^4*b*d^4*e^8 - a
^5*d^3*e^9)*x^4 + 35*(b^5*d^9*e^3 - 5*a*b^4*d^8*e^4 + 10*a^2*b^3*d^7*e^5 - 10*a^
3*b^2*d^6*e^6 + 5*a^4*b*d^5*e^7 - a^5*d^4*e^8)*x^3 + 21*(b^5*d^10*e^2 - 5*a*b^4*
d^9*e^3 + 10*a^2*b^3*d^8*e^4 - 10*a^3*b^2*d^7*e^5 + 5*a^4*b*d^6*e^6 - a^5*d^5*e^
7)*x^2 + 7*(b^5*d^11*e - 5*a*b^4*d^10*e^2 + 10*a^2*b^3*d^9*e^3 - 10*a^3*b^2*d^8*
e^4 + 5*a^4*b*d^7*e^5 - a^5*d^6*e^6)*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)**(15/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.535647, 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)^(15/2),x, algorithm="giac")

[Out]

Done