3.2203 \(\int \frac{(a+b x)^{3/2} (A+B x)}{(d+e x)^{13/2}} \, dx\)

Optimal. Leaf size=201 \[ \frac{16 b^2 (a+b x)^{5/2} (-11 a B e+6 A b e+5 b B d)}{3465 e (d+e x)^{5/2} (b d-a e)^4}+\frac{8 b (a+b x)^{5/2} (-11 a B e+6 A b e+5 b B d)}{693 e (d+e x)^{7/2} (b d-a e)^3}+\frac{2 (a+b x)^{5/2} (-11 a B e+6 A b e+5 b B d)}{99 e (d+e x)^{9/2} (b d-a e)^2}-\frac{2 (a+b x)^{5/2} (B d-A e)}{11 e (d+e x)^{11/2} (b d-a e)} \]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(5/2))/(11*e*(b*d - a*e)*(d + e*x)^(11/2)) + (2*(5*b*B
*d + 6*A*b*e - 11*a*B*e)*(a + b*x)^(5/2))/(99*e*(b*d - a*e)^2*(d + e*x)^(9/2)) +
(8*b*(5*b*B*d + 6*A*b*e - 11*a*B*e)*(a + b*x)^(5/2))/(693*e*(b*d - a*e)^3*(d +
e*x)^(7/2)) + (16*b^2*(5*b*B*d + 6*A*b*e - 11*a*B*e)*(a + b*x)^(5/2))/(3465*e*(b
*d - a*e)^4*(d + e*x)^(5/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.366159, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{16 b^2 (a+b x)^{5/2} (-11 a B e+6 A b e+5 b B d)}{3465 e (d+e x)^{5/2} (b d-a e)^4}+\frac{8 b (a+b x)^{5/2} (-11 a B e+6 A b e+5 b B d)}{693 e (d+e x)^{7/2} (b d-a e)^3}+\frac{2 (a+b x)^{5/2} (-11 a B e+6 A b e+5 b B d)}{99 e (d+e x)^{9/2} (b d-a e)^2}-\frac{2 (a+b x)^{5/2} (B d-A e)}{11 e (d+e x)^{11/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^(3/2)*(A + B*x))/(d + e*x)^(13/2),x]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(5/2))/(11*e*(b*d - a*e)*(d + e*x)^(11/2)) + (2*(5*b*B
*d + 6*A*b*e - 11*a*B*e)*(a + b*x)^(5/2))/(99*e*(b*d - a*e)^2*(d + e*x)^(9/2)) +
(8*b*(5*b*B*d + 6*A*b*e - 11*a*B*e)*(a + b*x)^(5/2))/(693*e*(b*d - a*e)^3*(d +
e*x)^(7/2)) + (16*b^2*(5*b*B*d + 6*A*b*e - 11*a*B*e)*(a + b*x)^(5/2))/(3465*e*(b
*d - a*e)^4*(d + e*x)^(5/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 37.3252, size = 192, normalized size = 0.96 \[ \frac{16 b^{2} \left (a + b x\right )^{\frac{5}{2}} \left (6 A b e - 11 B a e + 5 B b d\right )}{3465 e \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{4}} - \frac{8 b \left (a + b x\right )^{\frac{5}{2}} \left (6 A b e - 11 B a e + 5 B b d\right )}{693 e \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{3}} + \frac{2 \left (a + b x\right )^{\frac{5}{2}} \left (6 A b e - 11 B a e + 5 B b d\right )}{99 e \left (d + e x\right )^{\frac{9}{2}} \left (a e - b d\right )^{2}} - \frac{2 \left (a + b x\right )^{\frac{5}{2}} \left (A e - B d\right )}{11 e \left (d + e x\right )^{\frac{11}{2}} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(3/2)*(B*x+A)/(e*x+d)**(13/2),x)

[Out]

16*b**2*(a + b*x)**(5/2)*(6*A*b*e - 11*B*a*e + 5*B*b*d)/(3465*e*(d + e*x)**(5/2)
*(a*e - b*d)**4) - 8*b*(a + b*x)**(5/2)*(6*A*b*e - 11*B*a*e + 5*B*b*d)/(693*e*(d
+ e*x)**(7/2)*(a*e - b*d)**3) + 2*(a + b*x)**(5/2)*(6*A*b*e - 11*B*a*e + 5*B*b*
d)/(99*e*(d + e*x)**(9/2)*(a*e - b*d)**2) - 2*(a + b*x)**(5/2)*(A*e - B*d)/(11*e
*(d + e*x)**(11/2)*(a*e - b*d))

_______________________________________________________________________________________

Mathematica [A]  time = 0.431784, size = 217, normalized size = 1.08 \[ \frac{2 \sqrt{a+b x} \left (\frac{8 b^4 (d+e x)^5 (-11 a B e+6 A b e+5 b B d)}{(b d-a e)^4}+\frac{4 b^3 (d+e x)^4 (-11 a B e+6 A b e+5 b B d)}{(b d-a e)^3}+\frac{3 b^2 (d+e x)^3 (-11 a B e+6 A b e+5 b B d)}{(b d-a e)^2}-\frac{5 b (d+e x)^2 (110 a B e+3 A b e-113 b B d)}{a e-b d}+35 (d+e x) (-11 a B e-12 A b e+23 b B d)-315 (b d-a e) (B d-A e)\right )}{3465 e^3 (d+e x)^{11/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^(3/2)*(A + B*x))/(d + e*x)^(13/2),x]

[Out]

(2*Sqrt[a + b*x]*(-315*(b*d - a*e)*(B*d - A*e) + 35*(23*b*B*d - 12*A*b*e - 11*a*
B*e)*(d + e*x) - (5*b*(-113*b*B*d + 3*A*b*e + 110*a*B*e)*(d + e*x)^2)/(-(b*d) +
a*e) + (3*b^2*(5*b*B*d + 6*A*b*e - 11*a*B*e)*(d + e*x)^3)/(b*d - a*e)^2 + (4*b^3
*(5*b*B*d + 6*A*b*e - 11*a*B*e)*(d + e*x)^4)/(b*d - a*e)^3 + (8*b^4*(5*b*B*d + 6
*A*b*e - 11*a*B*e)*(d + e*x)^5)/(b*d - a*e)^4))/(3465*e^3*(d + e*x)^(11/2))

_______________________________________________________________________________________

Maple [A]  time = 0.014, size = 322, normalized size = 1.6 \[ -{\frac{-96\,A{b}^{3}{e}^{3}{x}^{3}+176\,Ba{b}^{2}{e}^{3}{x}^{3}-80\,B{b}^{3}d{e}^{2}{x}^{3}+240\,Aa{b}^{2}{e}^{3}{x}^{2}-528\,A{b}^{3}d{e}^{2}{x}^{2}-440\,B{a}^{2}b{e}^{3}{x}^{2}+1168\,Ba{b}^{2}d{e}^{2}{x}^{2}-440\,B{b}^{3}{d}^{2}e{x}^{2}-420\,A{a}^{2}b{e}^{3}x+1320\,Aa{b}^{2}d{e}^{2}x-1188\,A{b}^{3}{d}^{2}ex+770\,B{a}^{3}{e}^{3}x-2770\,B{a}^{2}bd{e}^{2}x+3278\,Ba{b}^{2}{d}^{2}ex-990\,B{b}^{3}{d}^{3}x+630\,A{a}^{3}{e}^{3}-2310\,A{a}^{2}bd{e}^{2}+2970\,Aa{b}^{2}{d}^{2}e-1386\,A{b}^{3}{d}^{3}+140\,B{a}^{3}d{e}^{2}-440\,B{a}^{2}b{d}^{2}e+396\,Ba{b}^{2}{d}^{3}}{3465\,{e}^{4}{a}^{4}-13860\,b{e}^{3}d{a}^{3}+20790\,{b}^{2}{e}^{2}{d}^{2}{a}^{2}-13860\,a{b}^{3}{d}^{3}e+3465\,{b}^{4}{d}^{4}} \left ( bx+a \right ) ^{{\frac{5}{2}}} \left ( ex+d \right ) ^{-{\frac{11}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(3/2)*(B*x+A)/(e*x+d)^(13/2),x)

[Out]

-2/3465*(b*x+a)^(5/2)*(-48*A*b^3*e^3*x^3+88*B*a*b^2*e^3*x^3-40*B*b^3*d*e^2*x^3+1
20*A*a*b^2*e^3*x^2-264*A*b^3*d*e^2*x^2-220*B*a^2*b*e^3*x^2+584*B*a*b^2*d*e^2*x^2
-220*B*b^3*d^2*e*x^2-210*A*a^2*b*e^3*x+660*A*a*b^2*d*e^2*x-594*A*b^3*d^2*e*x+385
*B*a^3*e^3*x-1385*B*a^2*b*d*e^2*x+1639*B*a*b^2*d^2*e*x-495*B*b^3*d^3*x+315*A*a^3
*e^3-1155*A*a^2*b*d*e^2+1485*A*a*b^2*d^2*e-693*A*b^3*d^3+70*B*a^3*d*e^2-220*B*a^
2*b*d^2*e+198*B*a*b^2*d^3)/(e*x+d)^(11/2)/(a^4*e^4-4*a^3*b*d*e^3+6*a^2*b^2*d^2*e
^2-4*a*b^3*d^3*e+b^4*d^4)

_______________________________________________________________________________________

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)^(13/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 6.09255, size = 1188, normalized size = 5.91 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(3/2)/(e*x + d)^(13/2),x, algorithm="fricas")

[Out]

-2/3465*(315*A*a^5*e^3 - 8*(5*B*b^5*d*e^2 - (11*B*a*b^4 - 6*A*b^5)*e^3)*x^5 - 4*
(55*B*b^5*d^2*e - 6*(21*B*a*b^4 - 11*A*b^5)*d*e^2 + (11*B*a^2*b^3 - 6*A*a*b^4)*e
^3)*x^4 + 99*(2*B*a^3*b^2 - 7*A*a^2*b^3)*d^3 - 55*(4*B*a^4*b - 27*A*a^3*b^2)*d^2
*e + 35*(2*B*a^5 - 33*A*a^4*b)*d*e^2 - (495*B*b^5*d^3 - 11*(109*B*a*b^4 - 54*A*b
^5)*d^2*e + (257*B*a^2*b^3 - 132*A*a*b^4)*d*e^2 - 3*(11*B*a^3*b^2 - 6*A*a^2*b^3)
*e^3)*x^3 - (99*(8*B*a*b^4 + 7*A*b^5)*d^3 - 33*(86*B*a^2*b^3 + 9*A*a*b^4)*d^2*e
+ (2116*B*a^3*b^2 + 99*A*a^2*b^3)*d*e^2 - 5*(110*B*a^4*b + 3*A*a^3*b^2)*e^3)*x^2
- (99*(B*a^2*b^3 + 14*A*a*b^4)*d^3 - 11*(109*B*a^3*b^2 + 216*A*a^2*b^3)*d^2*e +
15*(83*B*a^4*b + 110*A*a^3*b^2)*d*e^2 - 35*(11*B*a^5 + 12*A*a^4*b)*e^3)*x)*sqrt
(b*x + a)*sqrt(e*x + d)/(b^4*d^10 - 4*a*b^3*d^9*e + 6*a^2*b^2*d^8*e^2 - 4*a^3*b*
d^7*e^3 + a^4*d^6*e^4 + (b^4*d^4*e^6 - 4*a*b^3*d^3*e^7 + 6*a^2*b^2*d^2*e^8 - 4*a
^3*b*d*e^9 + a^4*e^10)*x^6 + 6*(b^4*d^5*e^5 - 4*a*b^3*d^4*e^6 + 6*a^2*b^2*d^3*e^
7 - 4*a^3*b*d^2*e^8 + a^4*d*e^9)*x^5 + 15*(b^4*d^6*e^4 - 4*a*b^3*d^5*e^5 + 6*a^2
*b^2*d^4*e^6 - 4*a^3*b*d^3*e^7 + a^4*d^2*e^8)*x^4 + 20*(b^4*d^7*e^3 - 4*a*b^3*d^
6*e^4 + 6*a^2*b^2*d^5*e^5 - 4*a^3*b*d^4*e^6 + a^4*d^3*e^7)*x^3 + 15*(b^4*d^8*e^2
- 4*a*b^3*d^7*e^3 + 6*a^2*b^2*d^6*e^4 - 4*a^3*b*d^5*e^5 + a^4*d^4*e^6)*x^2 + 6*
(b^4*d^9*e - 4*a*b^3*d^8*e^2 + 6*a^2*b^2*d^7*e^3 - 4*a^3*b*d^6*e^4 + a^4*d^5*e^5
)*x)

_______________________________________________________________________________________

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)**(13/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.410804, 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)^(13/2),x, algorithm="giac")

[Out]

Done