3.2217 \(\int \frac{(a+b x)^{5/2} (A+B x)}{(d+e x)^{17/2}} \, dx\)

Optimal. Leaf size=255 \[ \frac{32 b^3 (a+b x)^{7/2} (-15 a B e+8 A b e+7 b B d)}{45045 e (d+e x)^{7/2} (b d-a e)^5}+\frac{16 b^2 (a+b x)^{7/2} (-15 a B e+8 A b e+7 b B d)}{6435 e (d+e x)^{9/2} (b d-a e)^4}+\frac{4 b (a+b x)^{7/2} (-15 a B e+8 A b e+7 b B d)}{715 e (d+e x)^{11/2} (b d-a e)^3}+\frac{2 (a+b x)^{7/2} (-15 a B e+8 A b e+7 b B d)}{195 e (d+e x)^{13/2} (b d-a e)^2}-\frac{2 (a+b x)^{7/2} (B d-A e)}{15 e (d+e x)^{15/2} (b d-a e)} \]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(7/2))/(15*e*(b*d - a*e)*(d + e*x)^(15/2)) + (2*(7*b*B
*d + 8*A*b*e - 15*a*B*e)*(a + b*x)^(7/2))/(195*e*(b*d - a*e)^2*(d + e*x)^(13/2))
+ (4*b*(7*b*B*d + 8*A*b*e - 15*a*B*e)*(a + b*x)^(7/2))/(715*e*(b*d - a*e)^3*(d
+ e*x)^(11/2)) + (16*b^2*(7*b*B*d + 8*A*b*e - 15*a*B*e)*(a + b*x)^(7/2))/(6435*e
*(b*d - a*e)^4*(d + e*x)^(9/2)) + (32*b^3*(7*b*B*d + 8*A*b*e - 15*a*B*e)*(a + b*
x)^(7/2))/(45045*e*(b*d - a*e)^5*(d + e*x)^(7/2))

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Rubi [A]  time = 0.469568, 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 \[ \frac{32 b^3 (a+b x)^{7/2} (-15 a B e+8 A b e+7 b B d)}{45045 e (d+e x)^{7/2} (b d-a e)^5}+\frac{16 b^2 (a+b x)^{7/2} (-15 a B e+8 A b e+7 b B d)}{6435 e (d+e x)^{9/2} (b d-a e)^4}+\frac{4 b (a+b x)^{7/2} (-15 a B e+8 A b e+7 b B d)}{715 e (d+e x)^{11/2} (b d-a e)^3}+\frac{2 (a+b x)^{7/2} (-15 a B e+8 A b e+7 b B d)}{195 e (d+e x)^{13/2} (b d-a e)^2}-\frac{2 (a+b x)^{7/2} (B d-A e)}{15 e (d+e x)^{15/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(17/2),x]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(7/2))/(15*e*(b*d - a*e)*(d + e*x)^(15/2)) + (2*(7*b*B
*d + 8*A*b*e - 15*a*B*e)*(a + b*x)^(7/2))/(195*e*(b*d - a*e)^2*(d + e*x)^(13/2))
+ (4*b*(7*b*B*d + 8*A*b*e - 15*a*B*e)*(a + b*x)^(7/2))/(715*e*(b*d - a*e)^3*(d
+ e*x)^(11/2)) + (16*b^2*(7*b*B*d + 8*A*b*e - 15*a*B*e)*(a + b*x)^(7/2))/(6435*e
*(b*d - a*e)^4*(d + e*x)^(9/2)) + (32*b^3*(7*b*B*d + 8*A*b*e - 15*a*B*e)*(a + b*
x)^(7/2))/(45045*e*(b*d - a*e)^5*(d + e*x)^(7/2))

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Rubi in Sympy [A]  time = 50.7058, size = 246, normalized size = 0.96 \[ - \frac{32 b^{3} \left (a + b x\right )^{\frac{7}{2}} \left (8 A b e - 15 B a e + 7 B b d\right )}{45045 e \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{5}} + \frac{16 b^{2} \left (a + b x\right )^{\frac{7}{2}} \left (8 A b e - 15 B a e + 7 B b d\right )}{6435 e \left (d + e x\right )^{\frac{9}{2}} \left (a e - b d\right )^{4}} - \frac{4 b \left (a + b x\right )^{\frac{7}{2}} \left (8 A b e - 15 B a e + 7 B b d\right )}{715 e \left (d + e x\right )^{\frac{11}{2}} \left (a e - b d\right )^{3}} + \frac{2 \left (a + b x\right )^{\frac{7}{2}} \left (8 A b e - 15 B a e + 7 B b d\right )}{195 e \left (d + e x\right )^{\frac{13}{2}} \left (a e - b d\right )^{2}} - \frac{2 \left (a + b x\right )^{\frac{7}{2}} \left (A e - B d\right )}{15 e \left (d + e x\right )^{\frac{15}{2}} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(5/2)*(B*x+A)/(e*x+d)**(17/2),x)

[Out]

-32*b**3*(a + b*x)**(7/2)*(8*A*b*e - 15*B*a*e + 7*B*b*d)/(45045*e*(d + e*x)**(7/
2)*(a*e - b*d)**5) + 16*b**2*(a + b*x)**(7/2)*(8*A*b*e - 15*B*a*e + 7*B*b*d)/(64
35*e*(d + e*x)**(9/2)*(a*e - b*d)**4) - 4*b*(a + b*x)**(7/2)*(8*A*b*e - 15*B*a*e
+ 7*B*b*d)/(715*e*(d + e*x)**(11/2)*(a*e - b*d)**3) + 2*(a + b*x)**(7/2)*(8*A*b
*e - 15*B*a*e + 7*B*b*d)/(195*e*(d + e*x)**(13/2)*(a*e - b*d)**2) - 2*(a + b*x)*
*(7/2)*(A*e - B*d)/(15*e*(d + e*x)**(15/2)*(a*e - b*d))

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Mathematica [A]  time = 0.732623, size = 292, normalized size = 1.15 \[ \frac{2 \sqrt{a+b x} \left (\frac{16 b^6 (d+e x)^7 (-15 a B e+8 A b e+7 b B d)}{(b d-a e)^5}+\frac{8 b^5 (d+e x)^6 (-15 a B e+8 A b e+7 b B d)}{(b d-a e)^4}+\frac{6 b^4 (d+e x)^5 (-15 a B e+8 A b e+7 b B d)}{(b d-a e)^3}+\frac{5 b^3 (d+e x)^4 (-15 a B e+8 A b e+7 b B d)}{(b d-a e)^2}-\frac{35 b^2 (d+e x)^3 (159 a B e+A b e-160 b B d)}{a e-b d}+63 b (d+e x)^2 (-135 a B e-71 A b e+206 b B d)-231 (d+e x) (a e-b d) (15 a B e+31 A b e-46 b B d)+3003 (b d-a e)^2 (B d-A e)\right )}{45045 e^4 (d+e x)^{15/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(17/2),x]

[Out]

(2*Sqrt[a + b*x]*(3003*(b*d - a*e)^2*(B*d - A*e) - 231*(-(b*d) + a*e)*(-46*b*B*d
+ 31*A*b*e + 15*a*B*e)*(d + e*x) + 63*b*(206*b*B*d - 71*A*b*e - 135*a*B*e)*(d +
e*x)^2 - (35*b^2*(-160*b*B*d + A*b*e + 159*a*B*e)*(d + e*x)^3)/(-(b*d) + a*e) +
(5*b^3*(7*b*B*d + 8*A*b*e - 15*a*B*e)*(d + e*x)^4)/(b*d - a*e)^2 + (6*b^4*(7*b*
B*d + 8*A*b*e - 15*a*B*e)*(d + e*x)^5)/(b*d - a*e)^3 + (8*b^5*(7*b*B*d + 8*A*b*e
- 15*a*B*e)*(d + e*x)^6)/(b*d - a*e)^4 + (16*b^6*(7*b*B*d + 8*A*b*e - 15*a*B*e)
*(d + e*x)^7)/(b*d - a*e)^5))/(45045*e^4*(d + e*x)^(15/2))

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Maple [B]  time = 0.015, size = 505, normalized size = 2. \[ -{\frac{256\,A{b}^{4}{e}^{4}{x}^{4}-480\,Ba{b}^{3}{e}^{4}{x}^{4}+224\,B{b}^{4}d{e}^{3}{x}^{4}-896\,Aa{b}^{3}{e}^{4}{x}^{3}+1920\,A{b}^{4}d{e}^{3}{x}^{3}+1680\,B{a}^{2}{b}^{2}{e}^{4}{x}^{3}-4384\,Ba{b}^{3}d{e}^{3}{x}^{3}+1680\,B{b}^{4}{d}^{2}{e}^{2}{x}^{3}+2016\,A{a}^{2}{b}^{2}{e}^{4}{x}^{2}-6720\,Aa{b}^{3}d{e}^{3}{x}^{2}+6240\,A{b}^{4}{d}^{2}{e}^{2}{x}^{2}-3780\,B{a}^{3}b{e}^{4}{x}^{2}+14364\,B{a}^{2}{b}^{2}d{e}^{3}{x}^{2}-17580\,Ba{b}^{3}{d}^{2}{e}^{2}{x}^{2}+5460\,B{b}^{4}{d}^{3}e{x}^{2}-3696\,A{a}^{3}b{e}^{4}x+15120\,A{a}^{2}{b}^{2}d{e}^{3}x-21840\,Aa{b}^{3}{d}^{2}{e}^{2}x+11440\,A{b}^{4}{d}^{3}ex+6930\,B{a}^{4}{e}^{4}x-31584\,B{a}^{3}bd{e}^{3}x+54180\,B{a}^{2}{b}^{2}{d}^{2}{e}^{2}x-40560\,Ba{b}^{3}{d}^{3}ex+10010\,B{b}^{4}{d}^{4}x+6006\,A{a}^{4}{e}^{4}-27720\,A{a}^{3}bd{e}^{3}+49140\,A{a}^{2}{b}^{2}{d}^{2}{e}^{2}-40040\,Aa{b}^{3}{d}^{3}e+12870\,A{b}^{4}{d}^{4}+924\,B{a}^{4}d{e}^{3}-3780\,B{a}^{3}b{d}^{2}{e}^{2}+5460\,B{a}^{2}{b}^{2}{d}^{3}e-2860\,Ba{b}^{3}{d}^{4}}{45045\,{a}^{5}{e}^{5}-225225\,{a}^{4}bd{e}^{4}+450450\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}-450450\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+225225\,a{b}^{4}{d}^{4}e-45045\,{b}^{5}{d}^{5}} \left ( bx+a \right ) ^{{\frac{7}{2}}} \left ( ex+d \right ) ^{-{\frac{15}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(17/2),x)

[Out]

-2/45045*(b*x+a)^(7/2)*(128*A*b^4*e^4*x^4-240*B*a*b^3*e^4*x^4+112*B*b^4*d*e^3*x^
4-448*A*a*b^3*e^4*x^3+960*A*b^4*d*e^3*x^3+840*B*a^2*b^2*e^4*x^3-2192*B*a*b^3*d*e
^3*x^3+840*B*b^4*d^2*e^2*x^3+1008*A*a^2*b^2*e^4*x^2-3360*A*a*b^3*d*e^3*x^2+3120*
A*b^4*d^2*e^2*x^2-1890*B*a^3*b*e^4*x^2+7182*B*a^2*b^2*d*e^3*x^2-8790*B*a*b^3*d^2
*e^2*x^2+2730*B*b^4*d^3*e*x^2-1848*A*a^3*b*e^4*x+7560*A*a^2*b^2*d*e^3*x-10920*A*
a*b^3*d^2*e^2*x+5720*A*b^4*d^3*e*x+3465*B*a^4*e^4*x-15792*B*a^3*b*d*e^3*x+27090*
B*a^2*b^2*d^2*e^2*x-20280*B*a*b^3*d^3*e*x+5005*B*b^4*d^4*x+3003*A*a^4*e^4-13860*
A*a^3*b*d*e^3+24570*A*a^2*b^2*d^2*e^2-20020*A*a*b^3*d^3*e+6435*A*b^4*d^4+462*B*a
^4*d*e^3-1890*B*a^3*b*d^2*e^2+2730*B*a^2*b^2*d^3*e-1430*B*a*b^3*d^4)/(e*x+d)^(15
/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)^(5/2)/(e*x + d)^(17/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 32.7565, size = 1978, normalized size = 7.76 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(5/2)/(e*x + d)^(17/2),x, algorithm="fricas")

[Out]

2/45045*(3003*A*a^7*e^4 + 16*(7*B*b^7*d*e^3 - (15*B*a*b^6 - 8*A*b^7)*e^4)*x^7 +
8*(105*B*b^7*d^2*e^2 - 8*(29*B*a*b^6 - 15*A*b^7)*d*e^3 + (15*B*a^2*b^5 - 8*A*a*b
^6)*e^4)*x^6 + 6*(455*B*b^7*d^3*e - 5*(209*B*a*b^6 - 104*A*b^7)*d^2*e^2 + (157*B
*a^2*b^5 - 80*A*a*b^6)*d*e^3 - (15*B*a^3*b^4 - 8*A*a^2*b^5)*e^4)*x^5 - 715*(2*B*
a^4*b^3 - 9*A*a^3*b^4)*d^4 + 910*(3*B*a^5*b^2 - 22*A*a^4*b^3)*d^3*e - 1890*(B*a^
6*b - 13*A*a^5*b^2)*d^2*e^2 + 462*(B*a^7 - 30*A*a^6*b)*d*e^3 + 5*(1001*B*b^7*d^4
- 26*(93*B*a*b^6 - 44*A*b^7)*d^3*e + 24*(27*B*a^2*b^5 - 13*A*a*b^6)*d^2*e^2 - 2
*(71*B*a^3*b^4 - 36*A*a^2*b^5)*d*e^3 + (15*B*a^4*b^3 - 8*A*a^3*b^4)*e^4)*x^4 + 5
*(143*(19*B*a*b^6 + 9*A*b^7)*d^4 - 52*(192*B*a^2*b^5 + 11*A*a*b^6)*d^3*e + 6*(17
95*B*a^3*b^4 + 39*A*a^2*b^5)*d^2*e^2 - 4*(1378*B*a^4*b^3 + 15*A*a^3*b^4)*d*e^3 +
7*(159*B*a^5*b^2 + A*a^4*b^3)*e^4)*x^3 + 3*(715*(5*B*a^2*b^5 + 9*A*a*b^6)*d^4 -
260*(64*B*a^3*b^4 + 55*A*a^2*b^5)*d^3*e + 10*(2227*B*a^4*b^3 + 1469*A*a^3*b^4)*
d^2*e^2 - 28*(462*B*a^5*b^2 + 265*A*a^4*b^3)*d*e^3 + 21*(135*B*a^6*b + 71*A*a^5*
b^2)*e^4)*x^2 + (715*(B*a^3*b^4 + 27*A*a^2*b^5)*d^4 - 130*(93*B*a^4*b^3 + 418*A*
a^3*b^4)*d^3*e + 210*(102*B*a^5*b^2 + 299*A*a^4*b^3)*d^2*e^2 - 42*(343*B*a^6*b +
810*A*a^5*b^2)*d*e^3 + 231*(15*B*a^7 + 31*A*a^6*b)*e^4)*x)*sqrt(b*x + a)*sqrt(e
*x + d)/(b^5*d^13 - 5*a*b^4*d^12*e + 10*a^2*b^3*d^11*e^2 - 10*a^3*b^2*d^10*e^3 +
5*a^4*b*d^9*e^4 - a^5*d^8*e^5 + (b^5*d^5*e^8 - 5*a*b^4*d^4*e^9 + 10*a^2*b^3*d^3
*e^10 - 10*a^3*b^2*d^2*e^11 + 5*a^4*b*d*e^12 - a^5*e^13)*x^8 + 8*(b^5*d^6*e^7 -
5*a*b^4*d^5*e^8 + 10*a^2*b^3*d^4*e^9 - 10*a^3*b^2*d^3*e^10 + 5*a^4*b*d^2*e^11 -
a^5*d*e^12)*x^7 + 28*(b^5*d^7*e^6 - 5*a*b^4*d^6*e^7 + 10*a^2*b^3*d^5*e^8 - 10*a^
3*b^2*d^4*e^9 + 5*a^4*b*d^3*e^10 - a^5*d^2*e^11)*x^6 + 56*(b^5*d^8*e^5 - 5*a*b^4
*d^7*e^6 + 10*a^2*b^3*d^6*e^7 - 10*a^3*b^2*d^5*e^8 + 5*a^4*b*d^4*e^9 - a^5*d^3*e
^10)*x^5 + 70*(b^5*d^9*e^4 - 5*a*b^4*d^8*e^5 + 10*a^2*b^3*d^7*e^6 - 10*a^3*b^2*d
^6*e^7 + 5*a^4*b*d^5*e^8 - a^5*d^4*e^9)*x^4 + 56*(b^5*d^10*e^3 - 5*a*b^4*d^9*e^4
+ 10*a^2*b^3*d^8*e^5 - 10*a^3*b^2*d^7*e^6 + 5*a^4*b*d^6*e^7 - a^5*d^5*e^8)*x^3
+ 28*(b^5*d^11*e^2 - 5*a*b^4*d^10*e^3 + 10*a^2*b^3*d^9*e^4 - 10*a^3*b^2*d^8*e^5
+ 5*a^4*b*d^7*e^6 - a^5*d^6*e^7)*x^2 + 8*(b^5*d^12*e - 5*a*b^4*d^11*e^2 + 10*a^2
*b^3*d^10*e^3 - 10*a^3*b^2*d^9*e^4 + 5*a^4*b*d^8*e^5 - a^5*d^7*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)**(5/2)*(B*x+A)/(e*x+d)**(17/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.750125, 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)^(5/2)/(e*x + d)^(17/2),x, algorithm="giac")

[Out]

Done