3.1706 \(\int \frac{(a+b x) (A+B x)}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=79 \[ -\frac{2 \sqrt{d+e x} (-a B e-A b e+2 b B d)}{e^3}-\frac{2 (b d-a e) (B d-A e)}{e^3 \sqrt{d+e x}}+\frac{2 b B (d+e x)^{3/2}}{3 e^3} \]

[Out]

(-2*(b*d - a*e)*(B*d - A*e))/(e^3*Sqrt[d + e*x]) - (2*(2*b*B*d - A*b*e - a*B*e)*
Sqrt[d + e*x])/e^3 + (2*b*B*(d + e*x)^(3/2))/(3*e^3)

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Rubi [A]  time = 0.101986, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ -\frac{2 \sqrt{d+e x} (-a B e-A b e+2 b B d)}{e^3}-\frac{2 (b d-a e) (B d-A e)}{e^3 \sqrt{d+e x}}+\frac{2 b B (d+e x)^{3/2}}{3 e^3} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(b*d - a*e)*(B*d - A*e))/(e^3*Sqrt[d + e*x]) - (2*(2*b*B*d - A*b*e - a*B*e)*
Sqrt[d + e*x])/e^3 + (2*b*B*(d + e*x)^(3/2))/(3*e^3)

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Rubi in Sympy [A]  time = 16.3844, size = 75, normalized size = 0.95 \[ \frac{2 B b \left (d + e x\right )^{\frac{3}{2}}}{3 e^{3}} + \frac{2 \sqrt{d + e x} \left (A b e + B a e - 2 B b d\right )}{e^{3}} - \frac{2 \left (A e - B d\right ) \left (a e - b d\right )}{e^{3} \sqrt{d + e x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*b*(d + e*x)**(3/2)/(3*e**3) + 2*sqrt(d + e*x)*(A*b*e + B*a*e - 2*B*b*d)/e**3
- 2*(A*e - B*d)*(a*e - b*d)/(e**3*sqrt(d + e*x))

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Mathematica [A]  time = 0.0883281, size = 68, normalized size = 0.86 \[ \frac{6 a e (-A e+2 B d+B e x)+6 A b e (2 d+e x)+2 b B \left (-8 d^2-4 d e x+e^2 x^2\right )}{3 e^3 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(6*A*b*e*(2*d + e*x) + 6*a*e*(2*B*d - A*e + B*e*x) + 2*b*B*(-8*d^2 - 4*d*e*x + e
^2*x^2))/(3*e^3*Sqrt[d + e*x])

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Maple [A]  time = 0.006, size = 73, normalized size = 0.9 \[ -{\frac{-2\,bB{x}^{2}{e}^{2}-6\,Ab{e}^{2}x-6\,Ba{e}^{2}x+8\,Bbdex+6\,aA{e}^{2}-12\,Abde-12\,Bade+16\,bB{d}^{2}}{3\,{e}^{3}}{\frac{1}{\sqrt{ex+d}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3/(e*x+d)^(1/2)*(-B*b*e^2*x^2-3*A*b*e^2*x-3*B*a*e^2*x+4*B*b*d*e*x+3*A*a*e^2-6
*A*b*d*e-6*B*a*d*e+8*B*b*d^2)/e^3

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Maxima [A]  time = 1.35156, size = 111, normalized size = 1.41 \[ \frac{2 \,{\left (\frac{{\left (e x + d\right )}^{\frac{3}{2}} B b - 3 \,{\left (2 \, B b d -{\left (B a + A b\right )} e\right )} \sqrt{e x + d}}{e^{2}} - \frac{3 \,{\left (B b d^{2} + A a e^{2} -{\left (B a + A b\right )} d e\right )}}{\sqrt{e x + d} e^{2}}\right )}}{3 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(((e*x + d)^(3/2)*B*b - 3*(2*B*b*d - (B*a + A*b)*e)*sqrt(e*x + d))/e^2 - 3*(
B*b*d^2 + A*a*e^2 - (B*a + A*b)*d*e)/(sqrt(e*x + d)*e^2))/e

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Fricas [A]  time = 0.231779, size = 93, normalized size = 1.18 \[ \frac{2 \,{\left (B b e^{2} x^{2} - 8 \, B b d^{2} - 3 \, A a e^{2} + 6 \,{\left (B a + A b\right )} d e -{\left (4 \, B b d e - 3 \,{\left (B a + A b\right )} e^{2}\right )} x\right )}}{3 \, \sqrt{e x + d} e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(B*b*e^2*x^2 - 8*B*b*d^2 - 3*A*a*e^2 + 6*(B*a + A*b)*d*e - (4*B*b*d*e - 3*(B
*a + A*b)*e^2)*x)/(sqrt(e*x + d)*e^3)

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Sympy [A]  time = 14.0894, size = 638, normalized size = 8.08 \[ - \frac{2 A a}{e \sqrt{d + e x}} + A b \left (\begin{cases} \frac{4 d}{e^{2} \sqrt{d + e x}} + \frac{2 x}{e \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{x^{2}}{2 d^{\frac{3}{2}}} & \text{otherwise} \end{cases}\right ) + B a \left (\begin{cases} \frac{4 d}{e^{2} \sqrt{d + e x}} + \frac{2 x}{e \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{x^{2}}{2 d^{\frac{3}{2}}} & \text{otherwise} \end{cases}\right ) + B b \left (- \frac{16 d^{\frac{19}{2}} \sqrt{1 + \frac{e x}{d}}}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}} + \frac{16 d^{\frac{19}{2}}}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}} - \frac{40 d^{\frac{17}{2}} e x \sqrt{1 + \frac{e x}{d}}}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}} + \frac{48 d^{\frac{17}{2}} e x}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}} - \frac{30 d^{\frac{15}{2}} e^{2} x^{2} \sqrt{1 + \frac{e x}{d}}}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}} + \frac{48 d^{\frac{15}{2}} e^{2} x^{2}}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}} - \frac{4 d^{\frac{13}{2}} e^{3} x^{3} \sqrt{1 + \frac{e x}{d}}}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}} + \frac{16 d^{\frac{13}{2}} e^{3} x^{3}}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}} + \frac{2 d^{\frac{11}{2}} e^{4} x^{4} \sqrt{1 + \frac{e x}{d}}}{3 d^{8} e^{3} + 9 d^{7} e^{4} x + 9 d^{6} e^{5} x^{2} + 3 d^{5} e^{6} x^{3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A*a/(e*sqrt(d + e*x)) + A*b*Piecewise((4*d/(e**2*sqrt(d + e*x)) + 2*x/(e*sqrt
(d + e*x)), Ne(e, 0)), (x**2/(2*d**(3/2)), True)) + B*a*Piecewise((4*d/(e**2*sqr
t(d + e*x)) + 2*x/(e*sqrt(d + e*x)), Ne(e, 0)), (x**2/(2*d**(3/2)), True)) + B*b
*(-16*d**(19/2)*sqrt(1 + e*x/d)/(3*d**8*e**3 + 9*d**7*e**4*x + 9*d**6*e**5*x**2
+ 3*d**5*e**6*x**3) + 16*d**(19/2)/(3*d**8*e**3 + 9*d**7*e**4*x + 9*d**6*e**5*x*
*2 + 3*d**5*e**6*x**3) - 40*d**(17/2)*e*x*sqrt(1 + e*x/d)/(3*d**8*e**3 + 9*d**7*
e**4*x + 9*d**6*e**5*x**2 + 3*d**5*e**6*x**3) + 48*d**(17/2)*e*x/(3*d**8*e**3 +
9*d**7*e**4*x + 9*d**6*e**5*x**2 + 3*d**5*e**6*x**3) - 30*d**(15/2)*e**2*x**2*sq
rt(1 + e*x/d)/(3*d**8*e**3 + 9*d**7*e**4*x + 9*d**6*e**5*x**2 + 3*d**5*e**6*x**3
) + 48*d**(15/2)*e**2*x**2/(3*d**8*e**3 + 9*d**7*e**4*x + 9*d**6*e**5*x**2 + 3*d
**5*e**6*x**3) - 4*d**(13/2)*e**3*x**3*sqrt(1 + e*x/d)/(3*d**8*e**3 + 9*d**7*e**
4*x + 9*d**6*e**5*x**2 + 3*d**5*e**6*x**3) + 16*d**(13/2)*e**3*x**3/(3*d**8*e**3
+ 9*d**7*e**4*x + 9*d**6*e**5*x**2 + 3*d**5*e**6*x**3) + 2*d**(11/2)*e**4*x**4*
sqrt(1 + e*x/d)/(3*d**8*e**3 + 9*d**7*e**4*x + 9*d**6*e**5*x**2 + 3*d**5*e**6*x*
*3))

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GIAC/XCAS [A]  time = 0.210336, size = 135, normalized size = 1.71 \[ \frac{2}{3} \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} B b e^{6} - 6 \, \sqrt{x e + d} B b d e^{6} + 3 \, \sqrt{x e + d} B a e^{7} + 3 \, \sqrt{x e + d} A b e^{7}\right )} e^{\left (-9\right )} - \frac{2 \,{\left (B b d^{2} - B a d e - A b d e + A a e^{2}\right )} e^{\left (-3\right )}}{\sqrt{x e + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*((x*e + d)^(3/2)*B*b*e^6 - 6*sqrt(x*e + d)*B*b*d*e^6 + 3*sqrt(x*e + d)*B*a*e
^7 + 3*sqrt(x*e + d)*A*b*e^7)*e^(-9) - 2*(B*b*d^2 - B*a*d*e - A*b*d*e + A*a*e^2)
*e^(-3)/sqrt(x*e + d)