3.1725 \(\int \frac{(a+b x) (A+B x)}{(d+e x)^{7/2}} \, dx\)

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])

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Rubi [A]  time = 0.0376425, 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, Rules used = {77} \[ \frac{2 (-a B e-A b e+2 b B d)}{3 e^3 (d+e x)^{3/2}}-\frac{2 (b d-a e) (B d-A e)}{5 e^3 (d+e x)^{5/2}}-\frac{2 b B}{e^3 \sqrt{d+e x}} \]

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])

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{(a+b x) (A+B x)}{(d+e x)^{7/2}} \, dx &=\int \left (\frac{(-b d+a e) (-B d+A e)}{e^2 (d+e x)^{7/2}}+\frac{-2 b B d+A b e+a B e}{e^2 (d+e x)^{5/2}}+\frac{b B}{e^2 (d+e x)^{3/2}}\right ) \, dx\\ &=-\frac{2 (b d-a e) (B d-A e)}{5 e^3 (d+e x)^{5/2}}+\frac{2 (2 b B d-A b e-a B e)}{3 e^3 (d+e x)^{3/2}}-\frac{2 b B}{e^3 \sqrt{d+e x}}\\ \end{align*}

Mathematica [A]  time = 0.0482524, 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))

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Maple [A]  time = 0.003, size = 73, normalized size = 0.9 \begin{align*} -{\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}}}} \end{align*}

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

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Maxima [A]  time = 2.34526, size = 97, normalized size = 1.2 \begin{align*} -\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}} \end{align*}

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)

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Fricas [A]  time = 1.57907, size = 224, normalized size = 2.77 \begin{align*} -\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 )} \sqrt{e x + d}}{15 \,{\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \end{align*}

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)*sqrt(e*
x + d)/(e^6*x^3 + 3*d*e^5*x^2 + 3*d^2*e^4*x + d^3*e^3)

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Sympy [A]  time = 2.99948, size = 520, normalized size = 6.42 \begin{align*} \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} \end{align*}

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*sqrt(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*sqr
t(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*sqrt(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))

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Giac [A]  time = 2.34809, size = 117, normalized size = 1.44 \begin{align*} -\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}}} \end{align*}

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)