3.18.24 \(\int \frac {(a+b x) (A+B x)}{(d+e x)^{5/2}} \, dx\) [1724]

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

[Out]

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

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Rubi [A]
time = 0.02, 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 = {78} \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 78

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 = 0.07, size = 68, normalized size = 0.86 \begin {gather*} -\frac {2 \left (A b e (2 d+3 e x)+a e (2 B d+A e+3 B 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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Maple [A]
time = 0.12, size = 74, normalized size = 0.94

method	result	size
gosper	\(-\frac {2 \left (-3 b B \,x^{2} e^{2}+3 A b \,e^{2} x +3 B a \,e^{2} x -12 B b d e x +A a \,e^{2}+2 A b d e +2 B a d e -8 B b \,d^{2}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{3}}\)	\(72\)
trager	\(-\frac {2 \left (-3 b B \,x^{2} e^{2}+3 A b \,e^{2} x +3 B a \,e^{2} x -12 B b d e x +A a \,e^{2}+2 A b d e +2 B a d e -8 B b \,d^{2}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{3}}\)	\(72\)
derivativedivides	\(\frac {2 B b \sqrt {e x +d}-\frac {2 \left (A b e +B a e -2 B b d \right )}{\sqrt {e x +d}}-\frac {2 \left (A a \,e^{2}-A b d e -B a d e +B b \,d^{2}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{3}}\)	\(74\)
default	\(\frac {2 B b \sqrt {e x +d}-\frac {2 \left (A b e +B a e -2 B b d \right )}{\sqrt {e x +d}}-\frac {2 \left (A a \,e^{2}-A b d e -B a d e +B b \,d^{2}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{3}}\)	\(74\)
risch	\(\frac {2 b B \sqrt {e x +d}}{e^{3}}-\frac {2 \left (3 A b \,e^{2} x +3 B a \,e^{2} x -6 B b d e x +A a \,e^{2}+2 A b d e +2 B a d e -5 B b \,d^{2}\right )}{3 e^{3} \left (e x +d \right )^{\frac {3}{2}}}\)	\(77\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.31, size = 83, normalized size = 1.05 \begin {gather*} \frac {2}{3} \, {\left (3 \, \sqrt {x e + d} B b e^{\left (-2\right )} - \frac {{\left (B b d^{2} + A a e^{2} - 3 \, {\left (2 \, B b d - B a e - A b e\right )} {\left (x e + d\right )} - {\left (B a e + A b e\right )} d\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{\frac {3}{2}}}\right )} e^{\left (-1\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="maxima")

[Out]

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

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Fricas [A]
time = 1.42, size = 85, normalized size = 1.08 \begin {gather*} \frac {2 \, {\left (8 \, B b d^{2} + {\left (3 \, B b x^{2} - A a - 3 \, {\left (B a + A b\right )} x\right )} e^{2} + 2 \, {\left (6 \, B b d x - {\left (B a + A b\right )} d\right )} e\right )} \sqrt {x e + d}}{3 \, {\left (x^{2} e^{5} + 2 \, d x e^{4} + 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*(8*B*b*d^2 + (3*B*b*x^2 - A*a - 3*(B*a + A*b)*x)*e^2 + 2*(6*B*b*d*x - (B*a + A*b)*d)*e)*sqrt(x*e + d)/(x^2
*e^5 + 2*d*x*e^4 + d^2*e^3)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 355 vs. \(2 (80) = 160\).
time = 0.43, 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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Giac [A]
time = 0.59, 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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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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