3.2192 \(\int \frac{\sqrt{a+b x} (A+B x)}{(d+e x)^{13/2}} \, dx\)

Optimal. Leaf size=255 \[ \frac{32 b^3 (a+b x)^{3/2} (-11 a B e+8 A b e+3 b B d)}{3465 e (d+e x)^{3/2} (b d-a e)^5}+\frac{16 b^2 (a+b x)^{3/2} (-11 a B e+8 A b e+3 b B d)}{1155 e (d+e x)^{5/2} (b d-a e)^4}+\frac{4 b (a+b x)^{3/2} (-11 a B e+8 A b e+3 b B d)}{231 e (d+e x)^{7/2} (b d-a e)^3}+\frac{2 (a+b x)^{3/2} (-11 a B e+8 A b e+3 b B d)}{99 e (d+e x)^{9/2} (b d-a e)^2}-\frac{2 (a+b x)^{3/2} (B d-A e)}{11 e (d+e x)^{11/2} (b d-a e)} \]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(3/2))/(11*e*(b*d - a*e)*(d + e*x)^(11/2)) + (2*(3*b*B
*d + 8*A*b*e - 11*a*B*e)*(a + b*x)^(3/2))/(99*e*(b*d - a*e)^2*(d + e*x)^(9/2)) +
(4*b*(3*b*B*d + 8*A*b*e - 11*a*B*e)*(a + b*x)^(3/2))/(231*e*(b*d - a*e)^3*(d +
e*x)^(7/2)) + (16*b^2*(3*b*B*d + 8*A*b*e - 11*a*B*e)*(a + b*x)^(3/2))/(1155*e*(b
*d - a*e)^4*(d + e*x)^(5/2)) + (32*b^3*(3*b*B*d + 8*A*b*e - 11*a*B*e)*(a + b*x)^
(3/2))/(3465*e*(b*d - a*e)^5*(d + e*x)^(3/2))

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Rubi [A]  time = 0.466685, 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 \[ \frac{32 b^3 (a+b x)^{3/2} (-11 a B e+8 A b e+3 b B d)}{3465 e (d+e x)^{3/2} (b d-a e)^5}+\frac{16 b^2 (a+b x)^{3/2} (-11 a B e+8 A b e+3 b B d)}{1155 e (d+e x)^{5/2} (b d-a e)^4}+\frac{4 b (a+b x)^{3/2} (-11 a B e+8 A b e+3 b B d)}{231 e (d+e x)^{7/2} (b d-a e)^3}+\frac{2 (a+b x)^{3/2} (-11 a B e+8 A b e+3 b B d)}{99 e (d+e x)^{9/2} (b d-a e)^2}-\frac{2 (a+b x)^{3/2} (B d-A e)}{11 e (d+e x)^{11/2} (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(B*d - A*e)*(a + b*x)^(3/2))/(11*e*(b*d - a*e)*(d + e*x)^(11/2)) + (2*(3*b*B
*d + 8*A*b*e - 11*a*B*e)*(a + b*x)^(3/2))/(99*e*(b*d - a*e)^2*(d + e*x)^(9/2)) +
(4*b*(3*b*B*d + 8*A*b*e - 11*a*B*e)*(a + b*x)^(3/2))/(231*e*(b*d - a*e)^3*(d +
e*x)^(7/2)) + (16*b^2*(3*b*B*d + 8*A*b*e - 11*a*B*e)*(a + b*x)^(3/2))/(1155*e*(b
*d - a*e)^4*(d + e*x)^(5/2)) + (32*b^3*(3*b*B*d + 8*A*b*e - 11*a*B*e)*(a + b*x)^
(3/2))/(3465*e*(b*d - a*e)^5*(d + e*x)^(3/2))

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Rubi in Sympy [A]  time = 50.8173, size = 246, normalized size = 0.96 \[ - \frac{32 b^{3} \left (a + b x\right )^{\frac{3}{2}} \left (8 A b e - 11 B a e + 3 B b d\right )}{3465 e \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{5}} + \frac{16 b^{2} \left (a + b x\right )^{\frac{3}{2}} \left (8 A b e - 11 B a e + 3 B b d\right )}{1155 e \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{4}} - \frac{4 b \left (a + b x\right )^{\frac{3}{2}} \left (8 A b e - 11 B a e + 3 B b d\right )}{231 e \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{3}} + \frac{2 \left (a + b x\right )^{\frac{3}{2}} \left (8 A b e - 11 B a e + 3 B b d\right )}{99 e \left (d + e x\right )^{\frac{9}{2}} \left (a e - b d\right )^{2}} - \frac{2 \left (a + b x\right )^{\frac{3}{2}} \left (A e - B d\right )}{11 e \left (d + e x\right )^{\frac{11}{2}} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(b*x+a)**(1/2)/(e*x+d)**(13/2),x)

[Out]

-32*b**3*(a + b*x)**(3/2)*(8*A*b*e - 11*B*a*e + 3*B*b*d)/(3465*e*(d + e*x)**(3/2
)*(a*e - b*d)**5) + 16*b**2*(a + b*x)**(3/2)*(8*A*b*e - 11*B*a*e + 3*B*b*d)/(115
5*e*(d + e*x)**(5/2)*(a*e - b*d)**4) - 4*b*(a + b*x)**(3/2)*(8*A*b*e - 11*B*a*e
+ 3*B*b*d)/(231*e*(d + e*x)**(7/2)*(a*e - b*d)**3) + 2*(a + b*x)**(3/2)*(8*A*b*e
- 11*B*a*e + 3*B*b*d)/(99*e*(d + e*x)**(9/2)*(a*e - b*d)**2) - 2*(a + b*x)**(3/
2)*(A*e - B*d)/(11*e*(d + e*x)**(11/2)*(a*e - b*d))

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Mathematica [A]  time = 0.409761, size = 218, normalized size = 0.85 \[ \frac{2 \sqrt{a+b x} \left (\frac{16 b^4 (d+e x)^5 (-11 a B e+8 A b e+3 b B d)}{(b d-a e)^5}+\frac{8 b^3 (d+e x)^4 (-11 a B e+8 A b e+3 b B d)}{(b d-a e)^4}+\frac{6 b^2 (d+e x)^3 (-11 a B e+8 A b e+3 b B d)}{(b d-a e)^3}+\frac{5 b (d+e x)^2 (-11 a B e+8 A b e+3 b B d)}{(b d-a e)^2}-\frac{35 (d+e x) (11 a B e+A b e-12 b B d)}{a e-b d}+315 (B d-A e)\right )}{3465 e^2 (d+e x)^{11/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[a + b*x]*(315*(B*d - A*e) - (35*(-12*b*B*d + A*b*e + 11*a*B*e)*(d + e*x)
)/(-(b*d) + a*e) + (5*b*(3*b*B*d + 8*A*b*e - 11*a*B*e)*(d + e*x)^2)/(b*d - a*e)^
2 + (6*b^2*(3*b*B*d + 8*A*b*e - 11*a*B*e)*(d + e*x)^3)/(b*d - a*e)^3 + (8*b^3*(3
*b*B*d + 8*A*b*e - 11*a*B*e)*(d + e*x)^4)/(b*d - a*e)^4 + (16*b^4*(3*b*B*d + 8*A
*b*e - 11*a*B*e)*(d + e*x)^5)/(b*d - a*e)^5))/(3465*e^2*(d + e*x)^(11/2))

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Maple [B]  time = 0.017, size = 505, normalized size = 2. \[ -{\frac{256\,A{b}^{4}{e}^{4}{x}^{4}-352\,Ba{b}^{3}{e}^{4}{x}^{4}+96\,B{b}^{4}d{e}^{3}{x}^{4}-384\,Aa{b}^{3}{e}^{4}{x}^{3}+1408\,A{b}^{4}d{e}^{3}{x}^{3}+528\,B{a}^{2}{b}^{2}{e}^{4}{x}^{3}-2080\,Ba{b}^{3}d{e}^{3}{x}^{3}+528\,B{b}^{4}{d}^{2}{e}^{2}{x}^{3}+480\,A{a}^{2}{b}^{2}{e}^{4}{x}^{2}-2112\,Aa{b}^{3}d{e}^{3}{x}^{2}+3168\,A{b}^{4}{d}^{2}{e}^{2}{x}^{2}-660\,B{a}^{3}b{e}^{4}{x}^{2}+3084\,B{a}^{2}{b}^{2}d{e}^{3}{x}^{2}-5148\,Ba{b}^{3}{d}^{2}{e}^{2}{x}^{2}+1188\,B{b}^{4}{d}^{3}e{x}^{2}-560\,A{a}^{3}b{e}^{4}x+2640\,A{a}^{2}{b}^{2}d{e}^{3}x-4752\,Aa{b}^{3}{d}^{2}{e}^{2}x+3696\,A{b}^{4}{d}^{3}ex+770\,B{a}^{4}{e}^{4}x-3840\,B{a}^{3}bd{e}^{3}x+7524\,B{a}^{2}{b}^{2}{d}^{2}{e}^{2}x-6864\,Ba{b}^{3}{d}^{3}ex+1386\,B{b}^{4}{d}^{4}x+630\,A{a}^{4}{e}^{4}-3080\,A{a}^{3}bd{e}^{3}+5940\,A{a}^{2}{b}^{2}{d}^{2}{e}^{2}-5544\,Aa{b}^{3}{d}^{3}e+2310\,A{b}^{4}{d}^{4}+140\,B{a}^{4}d{e}^{3}-660\,B{a}^{3}b{d}^{2}{e}^{2}+1188\,B{a}^{2}{b}^{2}{d}^{3}e-924\,Ba{b}^{3}{d}^{4}}{3465\,{a}^{5}{e}^{5}-17325\,{a}^{4}bd{e}^{4}+34650\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}-34650\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+17325\,a{b}^{4}{d}^{4}e-3465\,{b}^{5}{d}^{5}} \left ( bx+a \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{11}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(b*x+a)^(1/2)/(e*x+d)^(13/2),x)

[Out]

-2/3465*(b*x+a)^(3/2)*(128*A*b^4*e^4*x^4-176*B*a*b^3*e^4*x^4+48*B*b^4*d*e^3*x^4-
192*A*a*b^3*e^4*x^3+704*A*b^4*d*e^3*x^3+264*B*a^2*b^2*e^4*x^3-1040*B*a*b^3*d*e^3
*x^3+264*B*b^4*d^2*e^2*x^3+240*A*a^2*b^2*e^4*x^2-1056*A*a*b^3*d*e^3*x^2+1584*A*b
^4*d^2*e^2*x^2-330*B*a^3*b*e^4*x^2+1542*B*a^2*b^2*d*e^3*x^2-2574*B*a*b^3*d^2*e^2
*x^2+594*B*b^4*d^3*e*x^2-280*A*a^3*b*e^4*x+1320*A*a^2*b^2*d*e^3*x-2376*A*a*b^3*d
^2*e^2*x+1848*A*b^4*d^3*e*x+385*B*a^4*e^4*x-1920*B*a^3*b*d*e^3*x+3762*B*a^2*b^2*
d^2*e^2*x-3432*B*a*b^3*d^3*e*x+693*B*b^4*d^4*x+315*A*a^4*e^4-1540*A*a^3*b*d*e^3+
2970*A*a^2*b^2*d^2*e^2-2772*A*a*b^3*d^3*e+1155*A*b^4*d^4+70*B*a^4*d*e^3-330*B*a^
3*b*d^2*e^2+594*B*a^2*b^2*d^3*e-462*B*a*b^3*d^4)/(e*x+d)^(11/2)/(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)*sqrt(b*x + a)/(e*x + d)^(13/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 6.02967, size = 1411, normalized size = 5.53 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3465*(315*A*a^5*e^4 + 16*(3*B*b^5*d*e^3 - (11*B*a*b^4 - 8*A*b^5)*e^4)*x^5 - 23
1*(2*B*a^2*b^3 - 5*A*a*b^4)*d^4 + 198*(3*B*a^3*b^2 - 14*A*a^2*b^3)*d^3*e - 330*(
B*a^4*b - 9*A*a^3*b^2)*d^2*e^2 + 70*(B*a^5 - 22*A*a^4*b)*d*e^3 + 8*(33*B*b^5*d^2
*e^2 - 4*(31*B*a*b^4 - 22*A*b^5)*d*e^3 + (11*B*a^2*b^3 - 8*A*a*b^4)*e^4)*x^4 + 2
*(297*B*b^5*d^3*e - 33*(35*B*a*b^4 - 24*A*b^5)*d^2*e^2 + (251*B*a^2*b^3 - 176*A*
a*b^4)*d*e^3 - 3*(11*B*a^3*b^2 - 8*A*a^2*b^3)*e^4)*x^3 + (693*B*b^5*d^4 - 66*(43
*B*a*b^4 - 28*A*b^5)*d^3*e + 396*(3*B*a^2*b^3 - 2*A*a*b^4)*d^2*e^2 - 6*(63*B*a^3
*b^2 - 44*A*a^2*b^3)*d*e^3 + 5*(11*B*a^4*b - 8*A*a^3*b^2)*e^4)*x^2 + (231*(B*a*b
^4 + 5*A*b^5)*d^4 - 66*(43*B*a^2*b^3 + 14*A*a*b^4)*d^3*e + 66*(52*B*a^3*b^2 + 9*
A*a^2*b^3)*d^2*e^2 - 10*(185*B*a^4*b + 22*A*a^3*b^2)*d*e^3 + 35*(11*B*a^5 + A*a^
4*b)*e^4)*x)*sqrt(b*x + a)*sqrt(e*x + d)/(b^5*d^11 - 5*a*b^4*d^10*e + 10*a^2*b^3
*d^9*e^2 - 10*a^3*b^2*d^8*e^3 + 5*a^4*b*d^7*e^4 - a^5*d^6*e^5 + (b^5*d^5*e^6 - 5
*a*b^4*d^4*e^7 + 10*a^2*b^3*d^3*e^8 - 10*a^3*b^2*d^2*e^9 + 5*a^4*b*d*e^10 - a^5*
e^11)*x^6 + 6*(b^5*d^6*e^5 - 5*a*b^4*d^5*e^6 + 10*a^2*b^3*d^4*e^7 - 10*a^3*b^2*d
^3*e^8 + 5*a^4*b*d^2*e^9 - a^5*d*e^10)*x^5 + 15*(b^5*d^7*e^4 - 5*a*b^4*d^6*e^5 +
10*a^2*b^3*d^5*e^6 - 10*a^3*b^2*d^4*e^7 + 5*a^4*b*d^3*e^8 - a^5*d^2*e^9)*x^4 +
20*(b^5*d^8*e^3 - 5*a*b^4*d^7*e^4 + 10*a^2*b^3*d^6*e^5 - 10*a^3*b^2*d^5*e^6 + 5*
a^4*b*d^4*e^7 - a^5*d^3*e^8)*x^3 + 15*(b^5*d^9*e^2 - 5*a*b^4*d^8*e^3 + 10*a^2*b^
3*d^7*e^4 - 10*a^3*b^2*d^6*e^5 + 5*a^4*b*d^5*e^6 - a^5*d^4*e^7)*x^2 + 6*(b^5*d^1
0*e - 5*a*b^4*d^9*e^2 + 10*a^2*b^3*d^8*e^3 - 10*a^3*b^2*d^7*e^4 + 5*a^4*b*d^6*e^
5 - a^5*d^5*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)*(b*x+a)**(1/2)/(e*x+d)**(13/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.401902, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done