3.7.11 \(\int x^5 (A+B x) (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\)

Optimal. Leaf size=306 \[ \frac {10 a^2 b^2 x^9 \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{9 (a+b x)}+\frac {b^4 x^{11} \sqrt {a^2+2 a b x+b^2 x^2} (5 a B+A b)}{11 (a+b x)}+\frac {a b^3 x^{10} \sqrt {a^2+2 a b x+b^2 x^2} (2 a B+A b)}{2 (a+b x)}+\frac {b^5 B x^{12} \sqrt {a^2+2 a b x+b^2 x^2}}{12 (a+b x)}+\frac {a^5 A x^6 \sqrt {a^2+2 a b x+b^2 x^2}}{6 (a+b x)}+\frac {a^4 x^7 \sqrt {a^2+2 a b x+b^2 x^2} (a B+5 A b)}{7 (a+b x)}+\frac {5 a^3 b x^8 \sqrt {a^2+2 a b x+b^2 x^2} (a B+2 A b)}{8 (a+b x)} \]

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Rubi [A]  time = 0.14, antiderivative size = 306, 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} \begin {gather*} \frac {b^4 x^{11} \sqrt {a^2+2 a b x+b^2 x^2} (5 a B+A b)}{11 (a+b x)}+\frac {a b^3 x^{10} \sqrt {a^2+2 a b x+b^2 x^2} (2 a B+A b)}{2 (a+b x)}+\frac {10 a^2 b^2 x^9 \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{9 (a+b x)}+\frac {5 a^3 b x^8 \sqrt {a^2+2 a b x+b^2 x^2} (a B+2 A b)}{8 (a+b x)}+\frac {a^4 x^7 \sqrt {a^2+2 a b x+b^2 x^2} (a B+5 A b)}{7 (a+b x)}+\frac {a^5 A x^6 \sqrt {a^2+2 a b x+b^2 x^2}}{6 (a+b x)}+\frac {b^5 B x^{12} \sqrt {a^2+2 a b x+b^2 x^2}}{12 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^5*A*x^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*(a + b*x)) + (a^4*(5*A*b + a*B)*x^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2]
)/(7*(a + b*x)) + (5*a^3*b*(2*A*b + a*B)*x^8*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*(a + b*x)) + (10*a^2*b^2*(A*b +
a*B)*x^9*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*(a + b*x)) + (a*b^3*(A*b + 2*a*B)*x^10*Sqrt[a^2 + 2*a*b*x + b^2*x^
2])/(2*(a + b*x)) + (b^4*(A*b + 5*a*B)*x^11*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*(a + b*x)) + (b^5*B*x^12*Sqrt[a
^2 + 2*a*b*x + b^2*x^2])/(12*(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^5 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int x^5 \left (a b+b^2 x\right )^5 (A+B x) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a^5 A b^5 x^5+a^4 b^5 (5 A b+a B) x^6+5 a^3 b^6 (2 A b+a B) x^7+10 a^2 b^7 (A b+a B) x^8+5 a b^8 (A b+2 a B) x^9+b^9 (A b+5 a B) x^{10}+b^{10} B x^{11}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {a^5 A x^6 \sqrt {a^2+2 a b x+b^2 x^2}}{6 (a+b x)}+\frac {a^4 (5 A b+a B) x^7 \sqrt {a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac {5 a^3 b (2 A b+a B) x^8 \sqrt {a^2+2 a b x+b^2 x^2}}{8 (a+b x)}+\frac {10 a^2 b^2 (A b+a B) x^9 \sqrt {a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac {a b^3 (A b+2 a B) x^{10} \sqrt {a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac {b^4 (A b+5 a B) x^{11} \sqrt {a^2+2 a b x+b^2 x^2}}{11 (a+b x)}+\frac {b^5 B x^{12} \sqrt {a^2+2 a b x+b^2 x^2}}{12 (a+b x)}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 125, normalized size = 0.41 \begin {gather*} \frac {x^6 \sqrt {(a+b x)^2} \left (132 a^5 (7 A+6 B x)+495 a^4 b x (8 A+7 B x)+770 a^3 b^2 x^2 (9 A+8 B x)+616 a^2 b^3 x^3 (10 A+9 B x)+252 a b^4 x^4 (11 A+10 B x)+42 b^5 x^5 (12 A+11 B x)\right )}{5544 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x^6*Sqrt[(a + b*x)^2]*(132*a^5*(7*A + 6*B*x) + 495*a^4*b*x*(8*A + 7*B*x) + 770*a^3*b^2*x^2*(9*A + 8*B*x) + 61
6*a^2*b^3*x^3*(10*A + 9*B*x) + 252*a*b^4*x^4*(11*A + 10*B*x) + 42*b^5*x^5*(12*A + 11*B*x)))/(5544*(a + b*x))

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IntegrateAlgebraic [F]  time = 1.36, size = 0, normalized size = 0.00 \begin {gather*} \int x^5 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.44, size = 119, normalized size = 0.39 \begin {gather*} \frac {1}{12} \, B b^{5} x^{12} + \frac {1}{6} \, A a^{5} x^{6} + \frac {1}{11} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{11} + \frac {1}{2} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{10} + \frac {10}{9} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + \frac {5}{8} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{8} + \frac {1}{7} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*B*b^5*x^12 + 1/6*A*a^5*x^6 + 1/11*(5*B*a*b^4 + A*b^5)*x^11 + 1/2*(2*B*a^2*b^3 + A*a*b^4)*x^10 + 10/9*(B*a
^3*b^2 + A*a^2*b^3)*x^9 + 5/8*(B*a^4*b + 2*A*a^3*b^2)*x^8 + 1/7*(B*a^5 + 5*A*a^4*b)*x^7

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giac [A]  time = 0.17, size = 220, normalized size = 0.72 \begin {gather*} \frac {1}{12} \, B b^{5} x^{12} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{11} \, B a b^{4} x^{11} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{11} \, A b^{5} x^{11} \mathrm {sgn}\left (b x + a\right ) + B a^{2} b^{3} x^{10} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, A a b^{4} x^{10} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{9} \, B a^{3} b^{2} x^{9} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{9} \, A a^{2} b^{3} x^{9} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{8} \, B a^{4} b x^{8} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{4} \, A a^{3} b^{2} x^{8} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{7} \, B a^{5} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{7} \, A a^{4} b x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{6} \, A a^{5} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {{\left (B a^{12} - 2 \, A a^{11} b\right )} \mathrm {sgn}\left (b x + a\right )}{5544 \, b^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*B*b^5*x^12*sgn(b*x + a) + 5/11*B*a*b^4*x^11*sgn(b*x + a) + 1/11*A*b^5*x^11*sgn(b*x + a) + B*a^2*b^3*x^10*
sgn(b*x + a) + 1/2*A*a*b^4*x^10*sgn(b*x + a) + 10/9*B*a^3*b^2*x^9*sgn(b*x + a) + 10/9*A*a^2*b^3*x^9*sgn(b*x +
a) + 5/8*B*a^4*b*x^8*sgn(b*x + a) + 5/4*A*a^3*b^2*x^8*sgn(b*x + a) + 1/7*B*a^5*x^7*sgn(b*x + a) + 5/7*A*a^4*b*
x^7*sgn(b*x + a) + 1/6*A*a^5*x^6*sgn(b*x + a) + 1/5544*(B*a^12 - 2*A*a^11*b)*sgn(b*x + a)/b^7

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maple [A]  time = 0.07, size = 140, normalized size = 0.46 \begin {gather*} \frac {\left (462 B \,b^{5} x^{6}+504 x^{5} A \,b^{5}+2520 x^{5} B a \,b^{4}+2772 x^{4} A a \,b^{4}+5544 x^{4} B \,a^{2} b^{3}+6160 A \,a^{2} b^{3} x^{3}+6160 B \,a^{3} b^{2} x^{3}+6930 x^{2} A \,a^{3} b^{2}+3465 x^{2} B \,a^{4} b +3960 x A \,a^{4} b +792 x B \,a^{5}+924 A \,a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} x^{6}}{5544 \left (b x +a \right )^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5544*x^6*(462*B*b^5*x^6+504*A*b^5*x^5+2520*B*a*b^4*x^5+2772*A*a*b^4*x^4+5544*B*a^2*b^3*x^4+6160*A*a^2*b^3*x^
3+6160*B*a^3*b^2*x^3+6930*A*a^3*b^2*x^2+3465*B*a^4*b*x^2+3960*A*a^4*b*x+792*B*a^5*x+924*A*a^5)*((b*x+a)^2)^(5/
2)/(b*x+a)^5

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maxima [A]  time = 0.52, size = 421, normalized size = 1.38 \begin {gather*} \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B x^{5}}{12 \, b^{2}} - \frac {17 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B a x^{4}}{132 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A x^{4}}{11 \, b^{2}} + \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B a^{2} x^{3}}{33 \, b^{4}} - \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A a x^{3}}{22 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B a^{6} x}{6 \, b^{6}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A a^{5} x}{6 \, b^{5}} - \frac {16 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B a^{3} x^{2}}{99 \, b^{5}} + \frac {31 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A a^{2} x^{2}}{198 \, b^{4}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B a^{7}}{6 \, b^{7}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A a^{6}}{6 \, b^{6}} + \frac {131 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B a^{4} x}{792 \, b^{6}} - \frac {65 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A a^{3} x}{396 \, b^{5}} - \frac {923 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B a^{5}}{5544 \, b^{7}} + \frac {461 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A a^{4}}{2772 \, b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*x^5/b^2 - 17/132*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*a*x^4/b^3 + 1/11*(b^
2*x^2 + 2*a*b*x + a^2)^(7/2)*A*x^4/b^2 + 5/33*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*a^2*x^3/b^4 - 3/22*(b^2*x^2 +
2*a*b*x + a^2)^(7/2)*A*a*x^3/b^3 + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*a^6*x/b^6 - 1/6*(b^2*x^2 + 2*a*b*x +
a^2)^(5/2)*A*a^5*x/b^5 - 16/99*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*a^3*x^2/b^5 + 31/198*(b^2*x^2 + 2*a*b*x + a^2
)^(7/2)*A*a^2*x^2/b^4 + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*a^7/b^7 - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*
a^6/b^6 + 131/792*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*a^4*x/b^6 - 65/396*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*a^3*x
/b^5 - 923/5544*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*a^5/b^7 + 461/2772*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*a^4/b^6

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^5\,\left (A+B\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{5} \left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**5*(A + B*x)*((a + b*x)**2)**(5/2), x)

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