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]

-2/5*(-a*e+b*d)*(-A*e+B*d)/e^3/(e*x+d)^(5/2)+2/3*(-A*b*e-B*a*e+2*B*b*d)/e^3/(e*x+d)^(3/2)-2*b*B/e^3/(e*x+d)^(1
/2)

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Rubi [A]  time = 0.04, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, 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*}

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Mathematica [A]  time = 0.07, 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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fricas [A]  time = 0.75, size = 101, normalized size = 1.25 \[ -\frac {2 \, {\left (15 \, B b e^{2} x^{2} + 8 \, B b d^{2} + 3 \, A a e^{2} + 2 \, {\left (B a + A b\right )} d e + 5 \, {\left (4 \, B b d e + {\left (B a + A b\right )} e^{2}\right )} x\right )} \sqrt {e x + d}}{15 \, {\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \]

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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giac [A]  time = 1.30, size = 87, normalized size = 1.07 \[ -\frac {2 \, {\left (15 \, {\left (x e + d\right )}^{2} B b - 10 \, {\left (x e + d\right )} B b d + 3 \, B b d^{2} + 5 \, {\left (x e + d\right )} B a e + 5 \, {\left (x e + d\right )} A b e - 3 \, B a d e - 3 \, A b d e + 3 \, A a e^{2}\right )} e^{\left (-3\right )}}{15 \, {\left (x e + d\right )}^{\frac {5}{2}}} \]

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)

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maple [A]  time = 0.00, size = 73, normalized size = 0.90 \[ -\frac {2 \left (15 B b \,x^{2} e^{2}+5 A b \,e^{2} x +5 B a \,e^{2} x +20 B b d e x +3 A a \,e^{2}+2 A b d e +2 B a d e +8 B b \,d^{2}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} e^{3}} \]

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 = 0.51, size = 72, normalized size = 0.89 \[ -\frac {2 \, {\left (15 \, {\left (e x + d\right )}^{2} B b + 3 \, B b d^{2} + 3 \, A a e^{2} - 3 \, {\left (B a + A b\right )} d e - 5 \, {\left (2 \, B b d - {\left (B a + A b\right )} e\right )} {\left (e x + d\right )}\right )}}{15 \, {\left (e x + d\right )}^{\frac {5}{2}} e^{3}} \]

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)

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mupad [B]  time = 0.08, size = 72, normalized size = 0.89 \[ -\frac {\left (d+e\,x\right )\,\left (\frac {2\,A\,b\,e}{3}+\frac {2\,B\,a\,e}{3}-\frac {4\,B\,b\,d}{3}\right )+2\,B\,b\,{\left (d+e\,x\right )}^2+\frac {2\,A\,a\,e^2}{5}+\frac {2\,B\,b\,d^2}{5}-\frac {2\,A\,b\,d\,e}{5}-\frac {2\,B\,a\,d\,e}{5}}{e^3\,{\left (d+e\,x\right )}^{5/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 3.09, 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*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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