3.653 \(\int x^4 (A+B x) \sqrt {a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=114 \[ \frac {x^6 \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{6 (a+b x)}+\frac {a A x^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac {b B x^7 \sqrt {a^2+2 a b x+b^2 x^2}}{7 (a+b x)} \]

[Out]

1/5*a*A*x^5*((b*x+a)^2)^(1/2)/(b*x+a)+1/6*(A*b+B*a)*x^6*((b*x+a)^2)^(1/2)/(b*x+a)+1/7*b*B*x^7*((b*x+a)^2)^(1/2
)/(b*x+a)

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Rubi [A]  time = 0.07, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {770, 76} \[ \frac {x^6 \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{6 (a+b x)}+\frac {a A x^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac {b B x^7 \sqrt {a^2+2 a b x+b^2 x^2}}{7 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[x^4*(A + B*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

(a*A*x^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*(a + b*x)) + ((A*b + a*B)*x^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*(a
+ b*x)) + (b*B*x^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*(a + b*x))

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rule 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rubi steps

\begin {align*} \int x^4 (A+B x) \sqrt {a^2+2 a b x+b^2 x^2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int x^4 \left (a b+b^2 x\right ) (A+B x) \, dx}{a b+b^2 x}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a A b x^4+b (A b+a B) x^5+b^2 B x^6\right ) \, dx}{a b+b^2 x}\\ &=\frac {a A x^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac {(A b+a B) x^6 \sqrt {a^2+2 a b x+b^2 x^2}}{6 (a+b x)}+\frac {b B x^7 \sqrt {a^2+2 a b x+b^2 x^2}}{7 (a+b x)}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 49, normalized size = 0.43 \[ \frac {x^5 \sqrt {(a+b x)^2} (7 a (6 A+5 B x)+5 b x (7 A+6 B x))}{210 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(A + B*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

(x^5*Sqrt[(a + b*x)^2]*(7*a*(6*A + 5*B*x) + 5*b*x*(7*A + 6*B*x)))/(210*(a + b*x))

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fricas [A]  time = 0.92, size = 27, normalized size = 0.24 \[ \frac {1}{7} \, B b x^{7} + \frac {1}{5} \, A a x^{5} + \frac {1}{6} \, {\left (B a + A b\right )} x^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x+A)*((b*x+a)^2)^(1/2),x, algorithm="fricas")

[Out]

1/7*B*b*x^7 + 1/5*A*a*x^5 + 1/6*(B*a + A*b)*x^6

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giac [A]  time = 0.20, size = 78, normalized size = 0.68 \[ \frac {1}{7} \, B b x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{6} \, B a x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{6} \, A b x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{5} \, A a x^{5} \mathrm {sgn}\left (b x + a\right ) - \frac {{\left (5 \, B a^{7} - 7 \, A a^{6} b\right )} \mathrm {sgn}\left (b x + a\right )}{210 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x+A)*((b*x+a)^2)^(1/2),x, algorithm="giac")

[Out]

1/7*B*b*x^7*sgn(b*x + a) + 1/6*B*a*x^6*sgn(b*x + a) + 1/6*A*b*x^6*sgn(b*x + a) + 1/5*A*a*x^5*sgn(b*x + a) - 1/
210*(5*B*a^7 - 7*A*a^6*b)*sgn(b*x + a)/b^6

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maple [A]  time = 0.06, size = 44, normalized size = 0.39 \[ \frac {\left (30 B b \,x^{2}+35 A b x +35 B a x +42 A a \right ) \sqrt {\left (b x +a \right )^{2}}\, x^{5}}{210 b x +210 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(B*x+A)*((b*x+a)^2)^(1/2),x)

[Out]

1/210*x^5*(30*B*b*x^2+35*A*b*x+35*B*a*x+42*A*a)*((b*x+a)^2)^(1/2)/(b*x+a)

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maxima [B]  time = 0.58, size = 361, normalized size = 3.17 \[ \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B x^{4}}{7 \, b^{2}} - \frac {11 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B a x^{3}}{42 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A x^{3}}{6 \, b^{2}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B a^{5} x}{2 \, b^{5}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A a^{4} x}{2 \, b^{4}} + \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B a^{2} x^{2}}{14 \, b^{4}} - \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A a x^{2}}{10 \, b^{3}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B a^{6}}{2 \, b^{6}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A a^{5}}{2 \, b^{5}} - \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B a^{3} x}{7 \, b^{5}} + \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A a^{2} x}{5 \, b^{4}} + \frac {10 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B a^{4}}{21 \, b^{6}} - \frac {7 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A a^{3}}{15 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x+A)*((b*x+a)^2)^(1/2),x, algorithm="maxima")

[Out]

1/7*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*B*x^4/b^2 - 11/42*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*B*a*x^3/b^3 + 1/6*(b^2*x
^2 + 2*a*b*x + a^2)^(3/2)*A*x^3/b^2 - 1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*B*a^5*x/b^5 + 1/2*sqrt(b^2*x^2 + 2*a*b
*x + a^2)*A*a^4*x/b^4 + 5/14*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*B*a^2*x^2/b^4 - 3/10*(b^2*x^2 + 2*a*b*x + a^2)^(3
/2)*A*a*x^2/b^3 - 1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*B*a^6/b^6 + 1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*A*a^5/b^5 -
3/7*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*B*a^3*x/b^5 + 2/5*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*A*a^2*x/b^4 + 10/21*(b^2
*x^2 + 2*a*b*x + a^2)^(3/2)*B*a^4/b^6 - 7/15*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*A*a^3/b^5

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mupad [B]  time = 1.87, size = 431, normalized size = 3.78 \[ \frac {A\,x^3\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{6\,b^2}+\frac {B\,x^4\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{7\,b^2}-\frac {11\,B\,a\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (a^5+5\,b^3\,x^3\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )-14\,a^3\,b^2\,x^2-13\,a^4\,b\,x-9\,a\,b^2\,x^2\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )+12\,a^2\,b\,x\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )\right )}{210\,b^6}-\frac {B\,a^2\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (4\,b^2\,x^2\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )-a^4+9\,a^2\,b^2\,x^2+8\,a^3\,b\,x-7\,a\,b\,x\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )\right )}{35\,b^6}-\frac {A\,a^2\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (a^3-5\,a\,b^2\,x^2+3\,b\,x\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )-4\,a^2\,b\,x\right )}{24\,b^5}-\frac {3\,A\,a\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (4\,b^2\,x^2\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )-a^4+9\,a^2\,b^2\,x^2+8\,a^3\,b\,x-7\,a\,b\,x\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )\right )}{40\,b^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*((a + b*x)^2)^(1/2)*(A + B*x),x)

[Out]

(A*x^3*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/(6*b^2) + (B*x^4*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/(7*b^2) - (11*B*a*(a
^2 + b^2*x^2 + 2*a*b*x)^(1/2)*(a^5 + 5*b^3*x^3*(a^2 + b^2*x^2 + 2*a*b*x) - 14*a^3*b^2*x^2 - 13*a^4*b*x - 9*a*b
^2*x^2*(a^2 + b^2*x^2 + 2*a*b*x) + 12*a^2*b*x*(a^2 + b^2*x^2 + 2*a*b*x)))/(210*b^6) - (B*a^2*(a^2 + b^2*x^2 +
2*a*b*x)^(1/2)*(4*b^2*x^2*(a^2 + b^2*x^2 + 2*a*b*x) - a^4 + 9*a^2*b^2*x^2 + 8*a^3*b*x - 7*a*b*x*(a^2 + b^2*x^2
+ 2*a*b*x)))/(35*b^6) - (A*a^2*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2)*(a^3 - 5*a*b^2*x^2 + 3*b*x*(a^2 + b^2*x^2 + 2*
a*b*x) - 4*a^2*b*x))/(24*b^5) - (3*A*a*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2)*(4*b^2*x^2*(a^2 + b^2*x^2 + 2*a*b*x) -
a^4 + 9*a^2*b^2*x^2 + 8*a^3*b*x - 7*a*b*x*(a^2 + b^2*x^2 + 2*a*b*x)))/(40*b^5)

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sympy [A]  time = 0.10, size = 29, normalized size = 0.25 \[ \frac {A a x^{5}}{5} + \frac {B b x^{7}}{7} + x^{6} \left (\frac {A b}{6} + \frac {B a}{6}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(B*x+A)*((b*x+a)**2)**(1/2),x)

[Out]

A*a*x**5/5 + B*b*x**7/7 + x**6*(A*b/6 + B*a/6)

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