3.1732 \(\int \frac {(a+b x)^2 (A+B x)}{(d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=124 \[ -\frac {2 b \sqrt {d+e x} (-2 a B e-A b e+3 b B d)}{e^4}-\frac {2 (b d-a e) (-a B e-2 A b e+3 b B d)}{e^4 \sqrt {d+e x}}+\frac {2 (b d-a e)^2 (B d-A e)}{3 e^4 (d+e x)^{3/2}}+\frac {2 b^2 B (d+e x)^{3/2}}{3 e^4} \]

[Out]

2/3*(-a*e+b*d)^2*(-A*e+B*d)/e^4/(e*x+d)^(3/2)+2/3*b^2*B*(e*x+d)^(3/2)/e^4-2*(-a*e+b*d)*(-2*A*b*e-B*a*e+3*B*b*d
)/e^4/(e*x+d)^(1/2)-2*b*(-A*b*e-2*B*a*e+3*B*b*d)*(e*x+d)^(1/2)/e^4

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Rubi [A]  time = 0.05, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {77} \[ -\frac {2 b \sqrt {d+e x} (-2 a B e-A b e+3 b B d)}{e^4}-\frac {2 (b d-a e) (-a B e-2 A b e+3 b B d)}{e^4 \sqrt {d+e x}}+\frac {2 (b d-a e)^2 (B d-A e)}{3 e^4 (d+e x)^{3/2}}+\frac {2 b^2 B (d+e x)^{3/2}}{3 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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)^2 (A+B x)}{(d+e x)^{5/2}} \, dx &=\int \left (\frac {(-b d+a e)^2 (-B d+A e)}{e^3 (d+e x)^{5/2}}+\frac {(-b d+a e) (-3 b B d+2 A b e+a B e)}{e^3 (d+e x)^{3/2}}+\frac {b (-3 b B d+A b e+2 a B e)}{e^3 \sqrt {d+e x}}+\frac {b^2 B \sqrt {d+e x}}{e^3}\right ) \, dx\\ &=\frac {2 (b d-a e)^2 (B d-A e)}{3 e^4 (d+e x)^{3/2}}-\frac {2 (b d-a e) (3 b B d-2 A b e-a B e)}{e^4 \sqrt {d+e x}}-\frac {2 b (3 b B d-A b e-2 a B e) \sqrt {d+e x}}{e^4}+\frac {2 b^2 B (d+e x)^{3/2}}{3 e^4}\\ \end {align*}

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Mathematica [A]  time = 0.14, size = 105, normalized size = 0.85 \[ \frac {2 \left (-3 b (d+e x)^2 (-2 a B e-A b e+3 b B d)-3 (d+e x) (b d-a e) (-a B e-2 A b e+3 b B d)+(b d-a e)^2 (B d-A e)+b^2 B (d+e x)^3\right )}{3 e^4 (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.77, size = 175, normalized size = 1.41 \[ \frac {2 \, {\left (B b^{2} e^{3} x^{3} - 16 \, B b^{2} d^{3} - A a^{2} e^{3} + 8 \, {\left (2 \, B a b + A b^{2}\right )} d^{2} e - 2 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{2} - 3 \, {\left (2 \, B b^{2} d e^{2} - {\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} - 3 \, {\left (8 \, B b^{2} d^{2} e - 4 \, {\left (2 \, B a b + A b^{2}\right )} d e^{2} + {\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x\right )} \sqrt {e x + d}}{3 \, {\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(B*x+A)/(e*x+d)^(5/2),x, algorithm="fricas")

[Out]

2/3*(B*b^2*e^3*x^3 - 16*B*b^2*d^3 - A*a^2*e^3 + 8*(2*B*a*b + A*b^2)*d^2*e - 2*(B*a^2 + 2*A*a*b)*d*e^2 - 3*(2*B
*b^2*d*e^2 - (2*B*a*b + A*b^2)*e^3)*x^2 - 3*(8*B*b^2*d^2*e - 4*(2*B*a*b + A*b^2)*d*e^2 + (B*a^2 + 2*A*a*b)*e^3
)*x)*sqrt(e*x + d)/(e^6*x^2 + 2*d*e^5*x + d^2*e^4)

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giac [A]  time = 1.26, size = 206, normalized size = 1.66 \[ \frac {2}{3} \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} B b^{2} e^{8} - 9 \, \sqrt {x e + d} B b^{2} d e^{8} + 6 \, \sqrt {x e + d} B a b e^{9} + 3 \, \sqrt {x e + d} A b^{2} e^{9}\right )} e^{\left (-12\right )} - \frac {2 \, {\left (9 \, {\left (x e + d\right )} B b^{2} d^{2} - B b^{2} d^{3} - 12 \, {\left (x e + d\right )} B a b d e - 6 \, {\left (x e + d\right )} A b^{2} d e + 2 \, B a b d^{2} e + A b^{2} d^{2} e + 3 \, {\left (x e + d\right )} B a^{2} e^{2} + 6 \, {\left (x e + d\right )} A a b e^{2} - B a^{2} d e^{2} - 2 \, A a b d e^{2} + A a^{2} e^{3}\right )} e^{\left (-4\right )}}{3 \, {\left (x e + d\right )}^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(B*x+A)/(e*x+d)^(5/2),x, algorithm="giac")

[Out]

2/3*((x*e + d)^(3/2)*B*b^2*e^8 - 9*sqrt(x*e + d)*B*b^2*d*e^8 + 6*sqrt(x*e + d)*B*a*b*e^9 + 3*sqrt(x*e + d)*A*b
^2*e^9)*e^(-12) - 2/3*(9*(x*e + d)*B*b^2*d^2 - B*b^2*d^3 - 12*(x*e + d)*B*a*b*d*e - 6*(x*e + d)*A*b^2*d*e + 2*
B*a*b*d^2*e + A*b^2*d^2*e + 3*(x*e + d)*B*a^2*e^2 + 6*(x*e + d)*A*a*b*e^2 - B*a^2*d*e^2 - 2*A*a*b*d*e^2 + A*a^
2*e^3)*e^(-4)/(x*e + d)^(3/2)

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maple [A]  time = 0.01, size = 168, normalized size = 1.35 \[ -\frac {2 \left (-b^{2} B \,x^{3} e^{3}-3 A \,b^{2} e^{3} x^{2}-6 B a b \,e^{3} x^{2}+6 B \,b^{2} d \,e^{2} x^{2}+6 A a b \,e^{3} x -12 A \,b^{2} d \,e^{2} x +3 B \,a^{2} e^{3} x -24 B a b d \,e^{2} x +24 B \,b^{2} d^{2} e x +a^{2} A \,e^{3}+4 A a b d \,e^{2}-8 A \,b^{2} d^{2} e +2 B \,a^{2} d \,e^{2}-16 B a b \,d^{2} e +16 B \,b^{2} d^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3/(e*x+d)^(3/2)*(-B*b^2*e^3*x^3-3*A*b^2*e^3*x^2-6*B*a*b*e^3*x^2+6*B*b^2*d*e^2*x^2+6*A*a*b*e^3*x-12*A*b^2*d*
e^2*x+3*B*a^2*e^3*x-24*B*a*b*d*e^2*x+24*B*b^2*d^2*e*x+A*a^2*e^3+4*A*a*b*d*e^2-8*A*b^2*d^2*e+2*B*a^2*d*e^2-16*B
*a*b*d^2*e+16*B*b^2*d^3)/e^4

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maxima [A]  time = 0.61, size = 163, normalized size = 1.31 \[ \frac {2 \, {\left (\frac {{\left (e x + d\right )}^{\frac {3}{2}} B b^{2} - 3 \, {\left (3 \, B b^{2} d - {\left (2 \, B a b + A b^{2}\right )} e\right )} \sqrt {e x + d}}{e^{3}} + \frac {B b^{2} d^{3} - A a^{2} e^{3} - {\left (2 \, B a b + A b^{2}\right )} d^{2} e + {\left (B a^{2} + 2 \, A a b\right )} d e^{2} - 3 \, {\left (3 \, B b^{2} d^{2} - 2 \, {\left (2 \, B a b + A b^{2}\right )} d e + {\left (B a^{2} + 2 \, A a b\right )} e^{2}\right )} {\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac {3}{2}} e^{3}}\right )}}{3 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(B*x+A)/(e*x+d)^(5/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.26, size = 189, normalized size = 1.52 \[ \frac {2\,B\,b^2\,d^3-2\,A\,a^2\,e^3+2\,B\,b^2\,{\left (d+e\,x\right )}^3+6\,A\,b^2\,e\,{\left (d+e\,x\right )}^2-6\,B\,a^2\,e^2\,\left (d+e\,x\right )-18\,B\,b^2\,d\,{\left (d+e\,x\right )}^2-18\,B\,b^2\,d^2\,\left (d+e\,x\right )-2\,A\,b^2\,d^2\,e+2\,B\,a^2\,d\,e^2-12\,A\,a\,b\,e^2\,\left (d+e\,x\right )+12\,B\,a\,b\,e\,{\left (d+e\,x\right )}^2+12\,A\,b^2\,d\,e\,\left (d+e\,x\right )+4\,A\,a\,b\,d\,e^2-4\,B\,a\,b\,d^2\,e+24\,B\,a\,b\,d\,e\,\left (d+e\,x\right )}{3\,e^4\,{\left (d+e\,x\right )}^{3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*B*b^2*d^3 - 2*A*a^2*e^3 + 2*B*b^2*(d + e*x)^3 + 6*A*b^2*e*(d + e*x)^2 - 6*B*a^2*e^2*(d + e*x) - 18*B*b^2*d*
(d + e*x)^2 - 18*B*b^2*d^2*(d + e*x) - 2*A*b^2*d^2*e + 2*B*a^2*d*e^2 - 12*A*a*b*e^2*(d + e*x) + 12*B*a*b*e*(d
+ e*x)^2 + 12*A*b^2*d*e*(d + e*x) + 4*A*a*b*d*e^2 - 4*B*a*b*d^2*e + 24*B*a*b*d*e*(d + e*x))/(3*e^4*(d + e*x)^(
3/2))

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sympy [A]  time = 1.67, size = 709, normalized size = 5.72 \[ \begin {cases} - \frac {2 A a^{2} e^{3}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {8 A a b d e^{2}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {12 A a b e^{3} x}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {16 A b^{2} d^{2} e}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {24 A b^{2} d e^{2} x}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {6 A b^{2} e^{3} x^{2}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {4 B a^{2} d e^{2}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {6 B a^{2} e^{3} x}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {32 B a b d^{2} e}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {48 B a b d e^{2} x}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {12 B a b e^{3} x^{2}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {32 B b^{2} d^{3}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {48 B b^{2} d^{2} e x}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} - \frac {12 B b^{2} d e^{2} x^{2}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} + \frac {2 B b^{2} e^{3} x^{3}}{3 d e^{4} \sqrt {d + e x} + 3 e^{5} x \sqrt {d + e x}} & \text {for}\: e \neq 0 \\\frac {A a^{2} x + A a b x^{2} + \frac {A b^{2} x^{3}}{3} + \frac {B a^{2} x^{2}}{2} + \frac {2 B a b x^{3}}{3} + \frac {B b^{2} x^{4}}{4}}{d^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-2*A*a**2*e**3/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) - 8*A*a*b*d*e**2/(3*d*e**4*sqrt(d
+ e*x) + 3*e**5*x*sqrt(d + e*x)) - 12*A*a*b*e**3*x/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) + 16*A*b*
*2*d**2*e/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) + 24*A*b**2*d*e**2*x/(3*d*e**4*sqrt(d + e*x) + 3*e
**5*x*sqrt(d + e*x)) + 6*A*b**2*e**3*x**2/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) - 4*B*a**2*d*e**2/
(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) - 6*B*a**2*e**3*x/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d
+ e*x)) + 32*B*a*b*d**2*e/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) + 48*B*a*b*d*e**2*x/(3*d*e**4*sqrt
(d + e*x) + 3*e**5*x*sqrt(d + e*x)) + 12*B*a*b*e**3*x**2/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) - 3
2*B*b**2*d**3/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) - 48*B*b**2*d**2*e*x/(3*d*e**4*sqrt(d + e*x) +
3*e**5*x*sqrt(d + e*x)) - 12*B*b**2*d*e**2*x**2/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) + 2*B*b**2*
e**3*x**3/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)), Ne(e, 0)), ((A*a**2*x + A*a*b*x**2 + A*b**2*x**3/
3 + B*a**2*x**2/2 + 2*B*a*b*x**3/3 + B*b**2*x**4/4)/d**(5/2), True))

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