3.1708 \(\int \frac{(a+b x) (A+B x)}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=81 \[ \frac{2 (-a B e-A b e+2 b B d)}{3 e^3 (d+e x)^{3/2}}-\frac{2 (b d-a e) (B d-A e)}{5 e^3 (d+e x)^{5/2}}-\frac{2 b B}{e^3 \sqrt{d+e x}} \]

[Out]

(-2*(b*d - a*e)*(B*d - A*e))/(5*e^3*(d + e*x)^(5/2)) + (2*(2*b*B*d - A*b*e - a*B
*e))/(3*e^3*(d + e*x)^(3/2)) - (2*b*B)/(e^3*Sqrt[d + e*x])

_______________________________________________________________________________________

Rubi [A]  time = 0.101409, antiderivative size = 81, 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 \[ \frac{2 (-a B e-A b e+2 b B d)}{3 e^3 (d+e x)^{3/2}}-\frac{2 (b d-a e) (B d-A e)}{5 e^3 (d+e x)^{5/2}}-\frac{2 b B}{e^3 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*(A + B*x))/(d + e*x)^(7/2),x]

[Out]

(-2*(b*d - a*e)*(B*d - A*e))/(5*e^3*(d + e*x)^(5/2)) + (2*(2*b*B*d - A*b*e - a*B
*e))/(3*e^3*(d + e*x)^(3/2)) - (2*b*B)/(e^3*Sqrt[d + e*x])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 16.6884, size = 80, normalized size = 0.99 \[ - \frac{2 B b}{e^{3} \sqrt{d + e x}} - \frac{2 \left (A b e + B a e - 2 B b d\right )}{3 e^{3} \left (d + e x\right )^{\frac{3}{2}}} - \frac{2 \left (A e - B d\right ) \left (a e - b d\right )}{5 e^{3} \left (d + e x\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(B*x+A)/(e*x+d)**(7/2),x)

[Out]

-2*B*b/(e**3*sqrt(d + e*x)) - 2*(A*b*e + B*a*e - 2*B*b*d)/(3*e**3*(d + e*x)**(3/
2)) - 2*(A*e - B*d)*(a*e - b*d)/(5*e**3*(d + e*x)**(5/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0873675, size = 68, normalized size = 0.84 \[ -\frac{2 \left (a e (3 A e+2 B d+5 B e x)+A b e (2 d+5 e x)+b B \left (8 d^2+20 d e x+15 e^2 x^2\right )\right )}{15 e^3 (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*(A + B*x))/(d + e*x)^(7/2),x]

[Out]

(-2*(A*b*e*(2*d + 5*e*x) + a*e*(2*B*d + 3*A*e + 5*B*e*x) + b*B*(8*d^2 + 20*d*e*x
+ 15*e^2*x^2)))/(15*e^3*(d + e*x)^(5/2))

_______________________________________________________________________________________

Maple [A]  time = 0.006, size = 73, normalized size = 0.9 \[ -{\frac{30\,bB{x}^{2}{e}^{2}+10\,Ab{e}^{2}x+10\,Ba{e}^{2}x+40\,Bbdex+6\,aA{e}^{2}+4\,Abde+4\,Bade+16\,bB{d}^{2}}{15\,{e}^{3}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(B*x+A)/(e*x+d)^(7/2),x)

[Out]

-2/15/(e*x+d)^(5/2)*(15*B*b*e^2*x^2+5*A*b*e^2*x+5*B*a*e^2*x+20*B*b*d*e*x+3*A*a*e
^2+2*A*b*d*e+2*B*a*d*e+8*B*b*d^2)/e^3

_______________________________________________________________________________________

Maxima [A]  time = 1.39192, size = 97, normalized size = 1.2 \[ -\frac{2 \,{\left (15 \,{\left (e x + d\right )}^{2} B b + 3 \, B b d^{2} + 3 \, A a e^{2} - 3 \,{\left (B a + A b\right )} d e - 5 \,{\left (2 \, B b d -{\left (B a + A b\right )} e\right )}{\left (e x + d\right )}\right )}}{15 \,{\left (e x + d\right )}^{\frac{5}{2}} e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)/(e*x + d)^(7/2),x, algorithm="maxima")

[Out]

-2/15*(15*(e*x + d)^2*B*b + 3*B*b*d^2 + 3*A*a*e^2 - 3*(B*a + A*b)*d*e - 5*(2*B*b
*d - (B*a + A*b)*e)*(e*x + d))/((e*x + d)^(5/2)*e^3)

_______________________________________________________________________________________

Fricas [A]  time = 0.227252, size = 122, normalized size = 1.51 \[ -\frac{2 \,{\left (15 \, B b e^{2} x^{2} + 8 \, B b d^{2} + 3 \, A a e^{2} + 2 \,{\left (B a + A b\right )} d e + 5 \,{\left (4 \, B b d e +{\left (B a + A b\right )} e^{2}\right )} x\right )}}{15 \,{\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )} \sqrt{e x + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)/(e*x + d)^(7/2),x, algorithm="fricas")

[Out]

-2/15*(15*B*b*e^2*x^2 + 8*B*b*d^2 + 3*A*a*e^2 + 2*(B*a + A*b)*d*e + 5*(4*B*b*d*e
+ (B*a + A*b)*e^2)*x)/((e^5*x^2 + 2*d*e^4*x + d^2*e^3)*sqrt(e*x + d))

_______________________________________________________________________________________

Sympy [A]  time = 9.28145, size = 520, normalized size = 6.42 \[ \begin{cases} - \frac{6 A a e^{2}}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} - \frac{4 A b d e}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} - \frac{10 A b e^{2} x}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} - \frac{4 B a d e}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} - \frac{10 B a e^{2} x}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} - \frac{16 B b d^{2}}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} - \frac{40 B b d e x}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} - \frac{30 B b e^{2} x^{2}}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{A a x + \frac{A b x^{2}}{2} + \frac{B a x^{2}}{2} + \frac{B b x^{3}}{3}}{d^{\frac{7}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*(B*x+A)/(e*x+d)**(7/2),x)

[Out]

Piecewise((-6*A*a*e**2/(15*d**2*e**3*sqrt(d + e*x) + 30*d*e**4*x*sqrt(d + e*x) +
15*e**5*x**2*sqrt(d + e*x)) - 4*A*b*d*e/(15*d**2*e**3*sqrt(d + e*x) + 30*d*e**4
*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x)) - 10*A*b*e**2*x/(15*d**2*e**3*sqr
t(d + e*x) + 30*d*e**4*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x)) - 4*B*a*d*e
/(15*d**2*e**3*sqrt(d + e*x) + 30*d*e**4*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d +
e*x)) - 10*B*a*e**2*x/(15*d**2*e**3*sqrt(d + e*x) + 30*d*e**4*x*sqrt(d + e*x) +
15*e**5*x**2*sqrt(d + e*x)) - 16*B*b*d**2/(15*d**2*e**3*sqrt(d + e*x) + 30*d*e*
*4*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x)) - 40*B*b*d*e*x/(15*d**2*e**3*sq
rt(d + e*x) + 30*d*e**4*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x)) - 30*B*b*e
**2*x**2/(15*d**2*e**3*sqrt(d + e*x) + 30*d*e**4*x*sqrt(d + e*x) + 15*e**5*x**2*
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**(7
/2), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.215903, size = 117, normalized size = 1.44 \[ -\frac{2 \,{\left (15 \,{\left (x e + d\right )}^{2} B b - 10 \,{\left (x e + d\right )} B b d + 3 \, B b d^{2} + 5 \,{\left (x e + d\right )} B a e + 5 \,{\left (x e + d\right )} A b e - 3 \, B a d e - 3 \, A b d e + 3 \, A a e^{2}\right )} e^{\left (-3\right )}}{15 \,{\left (x e + d\right )}^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)/(e*x + d)^(7/2),x, algorithm="giac")

[Out]

-2/15*(15*(x*e + d)^2*B*b - 10*(x*e + d)*B*b*d + 3*B*b*d^2 + 5*(x*e + d)*B*a*e +
5*(x*e + d)*A*b*e - 3*B*a*d*e - 3*A*b*d*e + 3*A*a*e^2)*e^(-3)/(x*e + d)^(5/2)