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

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

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Rubi [A]  time = 0.03, antiderivative size = 79, 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} \begin {gather*} \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} \end {gather*}

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

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

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Mathematica [A]  time = 0.09, size = 68, normalized size = 0.86 \begin {gather*} -\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}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.06, size = 82, normalized size = 1.04 \begin {gather*} \frac {2 \left (-a A e^2-3 a B e (d+e x)+a B d e-3 A b e (d+e x)+A b d e-b B d^2+6 b B d (d+e x)+3 b B (d+e x)^2\right )}{3 e^3 (d+e x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.44, size = 91, normalized size = 1.15 \begin {gather*} \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 )} \sqrt {e x + d}}{3 \, {\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )}} \end {gather*}

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

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giac [A]  time = 1.24, size = 88, normalized size = 1.11 \begin {gather*} 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}}} \end {gather*}

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)

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maple [A]  time = 0.00, size = 72, normalized size = 0.91 \begin {gather*} -\frac {2 \left (-3 B b \,x^{2} e^{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}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{3}} \end {gather*}

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 = 0.47, size = 79, normalized size = 1.00 \begin {gather*} \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} \end {gather*}

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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mupad [B]  time = 0.07, size = 72, normalized size = 0.91 \begin {gather*} -\frac {2\,A\,a\,e^2-16\,B\,b\,d^2+6\,A\,b\,e^2\,x+6\,B\,a\,e^2\,x-6\,B\,b\,e^2\,x^2+4\,A\,b\,d\,e+4\,B\,a\,d\,e-24\,B\,b\,d\,e\,x}{3\,e^3\,{\left (d+e\,x\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*A*a*e^2 - 16*B*b*d^2 + 6*A*b*e^2*x + 6*B*a*e^2*x - 6*B*b*e^2*x^2 + 4*A*b*d*e + 4*B*a*d*e - 24*B*b*d*e*x)/(
3*e^3*(d + e*x)^(3/2))

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sympy [A]  time = 1.36, size = 355, normalized size = 4.49 \begin {gather*} \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} \end {gather*}

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)) + 1
6*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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