3.1707 \(\int \frac{(a+b x) (A+B x)}{(d+e x)^{5/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.100903, 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 \[ \frac{2 (-a B e-A b e+2 b B d)}{e^3 \sqrt{d+e x}}-\frac{2 (b d-a e) (B d-A e)}{3 e^3 (d+e x)^{3/2}}+\frac{2 b B \sqrt{d+e x}}{e^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.5992, size = 76, normalized size = 0.96 \[ \frac{2 B b \sqrt{d + e x}}{e^{3}} - \frac{2 \left (A b e + B a e - 2 B b d\right )}{e^{3} \sqrt{d + e x}} - \frac{2 \left (A e - B d\right ) \left (a e - b d\right )}{3 e^{3} \left (d + e x\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0930066, size = 68, normalized size = 0.86 \[ -\frac{2 \left (a e (A e+2 B d+3 B e x)+A b e (2 d+3 e x)-b B \left (8 d^2+12 d e x+3 e^2 x^2\right )\right )}{3 e^3 (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(A*b*e*(2*d + 3*e*x) + a*e*(2*B*d + A*e + 3*B*e*x) - b*B*(8*d^2 + 12*d*e*x +
3*e^2*x^2)))/(3*e^3*(d + e*x)^(3/2))

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Maple [A]  time = 0.007, size = 72, normalized size = 0.9 \[ -{\frac{-6\,bB{x}^{2}{e}^{2}+6\,Ab{e}^{2}x+6\,Ba{e}^{2}x-24\,Bbdex+2\,aA{e}^{2}+4\,Abde+4\,Bade-16\,bB{d}^{2}}{3\,{e}^{3}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3/(e*x+d)^(3/2)*(-3*B*b*e^2*x^2+3*A*b*e^2*x+3*B*a*e^2*x-12*B*b*d*e*x+A*a*e^2+
2*A*b*d*e+2*B*a*d*e-8*B*b*d^2)/e^3

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Maxima [A]  time = 1.34278, size = 107, normalized size = 1.35 \[ \frac{2 \,{\left (\frac{3 \, \sqrt{e x + d} B b}{e^{2}} - \frac{B b d^{2} + A a e^{2} -{\left (B a + A b\right )} d e - 3 \,{\left (2 \, B b d -{\left (B a + A b\right )} e\right )}{\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac{3}{2}} e^{2}}\right )}}{3 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.224134, size = 108, normalized size = 1.37 \[ \frac{2 \,{\left (3 \, B b e^{2} x^{2} + 8 \, B b d^{2} - A a e^{2} - 2 \,{\left (B a + A b\right )} d e + 3 \,{\left (4 \, B b d e -{\left (B a + A b\right )} e^{2}\right )} x\right )}}{3 \,{\left (e^{4} x + d e^{3}\right )} \sqrt{e x + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.34213, size = 355, normalized size = 4.49 \[ \begin{cases} - \frac{2 A a e^{2}}{3 d e^{3} \sqrt{d + e x} + 3 e^{4} x \sqrt{d + e x}} - \frac{4 A b d e}{3 d e^{3} \sqrt{d + e x} + 3 e^{4} x \sqrt{d + e x}} - \frac{6 A b e^{2} x}{3 d e^{3} \sqrt{d + e x} + 3 e^{4} x \sqrt{d + e x}} - \frac{4 B a d e}{3 d e^{3} \sqrt{d + e x} + 3 e^{4} x \sqrt{d + e x}} - \frac{6 B a e^{2} x}{3 d e^{3} \sqrt{d + e x} + 3 e^{4} x \sqrt{d + e x}} + \frac{16 B b d^{2}}{3 d e^{3} \sqrt{d + e x} + 3 e^{4} x \sqrt{d + e x}} + \frac{24 B b d e x}{3 d e^{3} \sqrt{d + e x} + 3 e^{4} x \sqrt{d + e x}} + \frac{6 B b e^{2} x^{2}}{3 d e^{3} \sqrt{d + e x} + 3 e^{4} x \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{A a x + \frac{A b x^{2}}{2} + \frac{B a x^{2}}{2} + \frac{B b x^{3}}{3}}{d^{\frac{5}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-2*A*a*e**2/(3*d*e**3*sqrt(d + e*x) + 3*e**4*x*sqrt(d + e*x)) - 4*A*b
*d*e/(3*d*e**3*sqrt(d + e*x) + 3*e**4*x*sqrt(d + e*x)) - 6*A*b*e**2*x/(3*d*e**3*
sqrt(d + e*x) + 3*e**4*x*sqrt(d + e*x)) - 4*B*a*d*e/(3*d*e**3*sqrt(d + e*x) + 3*
e**4*x*sqrt(d + e*x)) - 6*B*a*e**2*x/(3*d*e**3*sqrt(d + e*x) + 3*e**4*x*sqrt(d +
e*x)) + 16*B*b*d**2/(3*d*e**3*sqrt(d + e*x) + 3*e**4*x*sqrt(d + e*x)) + 24*B*b*
d*e*x/(3*d*e**3*sqrt(d + e*x) + 3*e**4*x*sqrt(d + e*x)) + 6*B*b*e**2*x**2/(3*d*e
**3*sqrt(d + e*x) + 3*e**4*x*sqrt(d + e*x)), Ne(e, 0)), ((A*a*x + A*b*x**2/2 + B
*a*x**2/2 + B*b*x**3/3)/d**(5/2), True))

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GIAC/XCAS [A]  time = 0.210119, size = 119, normalized size = 1.51 \[ 2 \, \sqrt{x e + d} B b e^{\left (-3\right )} + \frac{2 \,{\left (6 \,{\left (x e + d\right )} B b d - B b d^{2} - 3 \,{\left (x e + d\right )} B a e - 3 \,{\left (x e + d\right )} A b e + B a d e + A b d e - A a e^{2}\right )} e^{\left (-3\right )}}{3 \,{\left (x e + d\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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