3.16.92 \(\int \frac {(a+b x)^2 (A+B x)}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=124 \[ -\frac {2 b (d+e x)^{3/2} (-2 a B e-A b e+3 b B d)}{3 e^4}+\frac {2 \sqrt {d+e x} (b d-a e) (-a B e-2 A b e+3 b B d)}{e^4}+\frac {2 (b d-a e)^2 (B d-A e)}{e^4 \sqrt {d+e x}}+\frac {2 b^2 B (d+e x)^{5/2}}{5 e^4} \]

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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} \begin {gather*} -\frac {2 b (d+e x)^{3/2} (-2 a B e-A b e+3 b B d)}{3 e^4}+\frac {2 \sqrt {d+e x} (b d-a e) (-a B e-2 A b e+3 b B d)}{e^4}+\frac {2 (b d-a e)^2 (B d-A e)}{e^4 \sqrt {d+e x}}+\frac {2 b^2 B (d+e x)^{5/2}}{5 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^2*(A + B*x))/(d + e*x)^(3/2),x]

[Out]

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

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Mathematica [A]  time = 0.13, size = 107, normalized size = 0.86 \begin {gather*} \frac {2 \left (-5 b (d+e x)^2 (-2 a B e-A b e+3 b B d)+15 (d+e x) (b d-a e) (-a B e-2 A b e+3 b B d)+15 (b d-a e)^2 (B d-A e)+3 b^2 B (d+e x)^3\right )}{15 e^4 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^2*(A + B*x))/(d + e*x)^(3/2),x]

[Out]

(2*(15*(b*d - a*e)^2*(B*d - A*e) + 15*(b*d - a*e)*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x) - 5*b*(3*b*B*d - A*b*e
- 2*a*B*e)*(d + e*x)^2 + 3*b^2*B*(d + e*x)^3))/(15*e^4*Sqrt[d + e*x])

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IntegrateAlgebraic [A]  time = 0.10, size = 193, normalized size = 1.56 \begin {gather*} \frac {2 \left (-15 a^2 A e^3+15 a^2 B e^2 (d+e x)+15 a^2 B d e^2+30 a A b e^2 (d+e x)+30 a A b d e^2-30 a b B d^2 e-60 a b B d e (d+e x)+10 a b B e (d+e x)^2-15 A b^2 d^2 e-30 A b^2 d e (d+e x)+5 A b^2 e (d+e x)^2+15 b^2 B d^3+45 b^2 B d^2 (d+e x)-15 b^2 B d (d+e x)^2+3 b^2 B (d+e x)^3\right )}{15 e^4 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((a + b*x)^2*(A + B*x))/(d + e*x)^(3/2),x]

[Out]

(2*(15*b^2*B*d^3 - 15*A*b^2*d^2*e - 30*a*b*B*d^2*e + 30*a*A*b*d*e^2 + 15*a^2*B*d*e^2 - 15*a^2*A*e^3 + 45*b^2*B
*d^2*(d + e*x) - 30*A*b^2*d*e*(d + e*x) - 60*a*b*B*d*e*(d + e*x) + 30*a*A*b*e^2*(d + e*x) + 15*a^2*B*e^2*(d +
e*x) - 15*b^2*B*d*(d + e*x)^2 + 5*A*b^2*e*(d + e*x)^2 + 10*a*b*B*e*(d + e*x)^2 + 3*b^2*B*(d + e*x)^3))/(15*e^4
*Sqrt[d + e*x])

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fricas [A]  time = 1.26, size = 165, normalized size = 1.33 \begin {gather*} \frac {2 \, {\left (3 \, B b^{2} e^{3} x^{3} + 48 \, B b^{2} d^{3} - 15 \, A a^{2} e^{3} - 40 \, {\left (2 \, B a b + A b^{2}\right )} d^{2} e + 30 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{2} - {\left (6 \, B b^{2} d e^{2} - 5 \, {\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} + {\left (24 \, B b^{2} d^{2} e - 20 \, {\left (2 \, B a b + A b^{2}\right )} d e^{2} + 15 \, {\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x\right )} \sqrt {e x + d}}{15 \, {\left (e^{5} x + d e^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(B*x+A)/(e*x+d)^(3/2),x, algorithm="fricas")

[Out]

2/15*(3*B*b^2*e^3*x^3 + 48*B*b^2*d^3 - 15*A*a^2*e^3 - 40*(2*B*a*b + A*b^2)*d^2*e + 30*(B*a^2 + 2*A*a*b)*d*e^2
- (6*B*b^2*d*e^2 - 5*(2*B*a*b + A*b^2)*e^3)*x^2 + (24*B*b^2*d^2*e - 20*(2*B*a*b + A*b^2)*d*e^2 + 15*(B*a^2 + 2
*A*a*b)*e^3)*x)*sqrt(e*x + d)/(e^5*x + d*e^4)

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giac [A]  time = 1.34, size = 219, normalized size = 1.77 \begin {gather*} \frac {2}{15} \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{2} e^{16} - 15 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{2} d e^{16} + 45 \, \sqrt {x e + d} B b^{2} d^{2} e^{16} + 10 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b e^{17} + 5 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{2} e^{17} - 60 \, \sqrt {x e + d} B a b d e^{17} - 30 \, \sqrt {x e + d} A b^{2} d e^{17} + 15 \, \sqrt {x e + d} B a^{2} e^{18} + 30 \, \sqrt {x e + d} A a b e^{18}\right )} e^{\left (-20\right )} + \frac {2 \, {\left (B b^{2} d^{3} - 2 \, B a b d^{2} e - A b^{2} d^{2} e + B a^{2} d e^{2} + 2 \, A a b d e^{2} - A a^{2} e^{3}\right )} e^{\left (-4\right )}}{\sqrt {x e + d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(B*x+A)/(e*x+d)^(3/2),x, algorithm="giac")

[Out]

2/15*(3*(x*e + d)^(5/2)*B*b^2*e^16 - 15*(x*e + d)^(3/2)*B*b^2*d*e^16 + 45*sqrt(x*e + d)*B*b^2*d^2*e^16 + 10*(x
*e + d)^(3/2)*B*a*b*e^17 + 5*(x*e + d)^(3/2)*A*b^2*e^17 - 60*sqrt(x*e + d)*B*a*b*d*e^17 - 30*sqrt(x*e + d)*A*b
^2*d*e^17 + 15*sqrt(x*e + d)*B*a^2*e^18 + 30*sqrt(x*e + d)*A*a*b*e^18)*e^(-20) + 2*(B*b^2*d^3 - 2*B*a*b*d^2*e
- A*b^2*d^2*e + B*a^2*d*e^2 + 2*A*a*b*d*e^2 - A*a^2*e^3)*e^(-4)/sqrt(x*e + d)

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maple [A]  time = 0.01, size = 169, normalized size = 1.36 \begin {gather*} -\frac {2 \left (-3 b^{2} B \,x^{3} e^{3}-5 A \,b^{2} e^{3} x^{2}-10 B a b \,e^{3} x^{2}+6 B \,b^{2} d \,e^{2} x^{2}-30 A a b \,e^{3} x +20 A \,b^{2} d \,e^{2} x -15 B \,a^{2} e^{3} x +40 B a b d \,e^{2} x -24 B \,b^{2} d^{2} e x +15 a^{2} A \,e^{3}-60 A a b d \,e^{2}+40 A \,b^{2} d^{2} e -30 B \,a^{2} d \,e^{2}+80 B a b \,d^{2} e -48 B \,b^{2} d^{3}\right )}{15 \sqrt {e x +d}\, e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^2*(B*x+A)/(e*x+d)^(3/2),x)

[Out]

-2/15/(e*x+d)^(1/2)*(-3*B*b^2*e^3*x^3-5*A*b^2*e^3*x^2-10*B*a*b*e^3*x^2+6*B*b^2*d*e^2*x^2-30*A*a*b*e^3*x+20*A*b
^2*d*e^2*x-15*B*a^2*e^3*x+40*B*a*b*d*e^2*x-24*B*b^2*d^2*e*x+15*A*a^2*e^3-60*A*a*b*d*e^2+40*A*b^2*d^2*e-30*B*a^
2*d*e^2+80*B*a*b*d^2*e-48*B*b^2*d^3)/e^4

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maxima [A]  time = 0.53, size = 167, normalized size = 1.35 \begin {gather*} \frac {2 \, {\left (\frac {3 \, {\left (e x + d\right )}^{\frac {5}{2}} B b^{2} - 5 \, {\left (3 \, B b^{2} d - {\left (2 \, B a b + A b^{2}\right )} e\right )} {\left (e x + d\right )}^{\frac {3}{2}} + 15 \, {\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 )} \sqrt {e x + d}}{e^{3}} + \frac {15 \, {\left (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}\right )}}{\sqrt {e x + d} e^{3}}\right )}}{15 \, e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(B*x+A)/(e*x+d)^(3/2),x, algorithm="maxima")

[Out]

2/15*((3*(e*x + d)^(5/2)*B*b^2 - 5*(3*B*b^2*d - (2*B*a*b + A*b^2)*e)*(e*x + d)^(3/2) + 15*(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)*sqrt(e*x + d))/e^3 + 15*(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)/(sqrt(e*x + d)*e^3))/e

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mupad [B]  time = 0.08, size = 154, normalized size = 1.24 \begin {gather*} \frac {{\left (d+e\,x\right )}^{3/2}\,\left (2\,A\,b^2\,e-6\,B\,b^2\,d+4\,B\,a\,b\,e\right )}{3\,e^4}-\frac {-2\,B\,a^2\,d\,e^2+2\,A\,a^2\,e^3+4\,B\,a\,b\,d^2\,e-4\,A\,a\,b\,d\,e^2-2\,B\,b^2\,d^3+2\,A\,b^2\,d^2\,e}{e^4\,\sqrt {d+e\,x}}+\frac {2\,B\,b^2\,{\left (d+e\,x\right )}^{5/2}}{5\,e^4}+\frac {2\,\left (a\,e-b\,d\right )\,\sqrt {d+e\,x}\,\left (2\,A\,b\,e+B\,a\,e-3\,B\,b\,d\right )}{e^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^2)/(d + e*x)^(3/2),x)

[Out]

((d + e*x)^(3/2)*(2*A*b^2*e - 6*B*b^2*d + 4*B*a*b*e))/(3*e^4) - (2*A*a^2*e^3 - 2*B*b^2*d^3 + 2*A*b^2*d^2*e - 2
*B*a^2*d*e^2 - 4*A*a*b*d*e^2 + 4*B*a*b*d^2*e)/(e^4*(d + e*x)^(1/2)) + (2*B*b^2*(d + e*x)^(5/2))/(5*e^4) + (2*(
a*e - b*d)*(d + e*x)^(1/2)*(2*A*b*e + B*a*e - 3*B*b*d))/e^4

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sympy [A]  time = 34.81, size = 150, normalized size = 1.21 \begin {gather*} \frac {2 B b^{2} \left (d + e x\right )^{\frac {5}{2}}}{5 e^{4}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (2 A b^{2} e + 4 B a b e - 6 B b^{2} d\right )}{3 e^{4}} + \frac {\sqrt {d + e x} \left (4 A a b e^{2} - 4 A b^{2} d e + 2 B a^{2} e^{2} - 8 B a b d e + 6 B b^{2} d^{2}\right )}{e^{4}} + \frac {2 \left (- A e + B d\right ) \left (a e - b d\right )^{2}}{e^{4} \sqrt {d + e x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**2*(B*x+A)/(e*x+d)**(3/2),x)

[Out]

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

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