3.7.82 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^3}{x^{9/2}} \, dx\)

Optimal. Leaf size=151 \[ -\frac {2 a^6 A}{7 x^{7/2}}-\frac {2 a^5 (a B+6 A b)}{5 x^{5/2}}-\frac {2 a^4 b (2 a B+5 A b)}{x^{3/2}}-\frac {10 a^3 b^2 (3 a B+4 A b)}{\sqrt {x}}+10 a^2 b^3 \sqrt {x} (4 a B+3 A b)+\frac {2}{5} b^5 x^{5/2} (6 a B+A b)+2 a b^4 x^{3/2} (5 a B+2 A b)+\frac {2}{7} b^6 B x^{7/2} \]

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Rubi [A]  time = 0.08, antiderivative size = 151, 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 = {27, 76} \begin {gather*} -\frac {10 a^3 b^2 (3 a B+4 A b)}{\sqrt {x}}+10 a^2 b^3 \sqrt {x} (4 a B+3 A b)-\frac {2 a^5 (a B+6 A b)}{5 x^{5/2}}-\frac {2 a^4 b (2 a B+5 A b)}{x^{3/2}}-\frac {2 a^6 A}{7 x^{7/2}}+2 a b^4 x^{3/2} (5 a B+2 A b)+\frac {2}{5} b^5 x^{5/2} (6 a B+A b)+\frac {2}{7} b^6 B x^{7/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^(9/2),x]

[Out]

(-2*a^6*A)/(7*x^(7/2)) - (2*a^5*(6*A*b + a*B))/(5*x^(5/2)) - (2*a^4*b*(5*A*b + 2*a*B))/x^(3/2) - (10*a^3*b^2*(
4*A*b + 3*a*B))/Sqrt[x] + 10*a^2*b^3*(3*A*b + 4*a*B)*Sqrt[x] + 2*a*b^4*(2*A*b + 5*a*B)*x^(3/2) + (2*b^5*(A*b +
6*a*B)*x^(5/2))/5 + (2*b^6*B*x^(7/2))/7

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

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])

Rubi steps

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

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Mathematica [A]  time = 0.04, size = 123, normalized size = 0.81 \begin {gather*} \frac {2 \left (-\left (a^6 (5 A+7 B x)\right )-14 a^5 b x (3 A+5 B x)-175 a^4 b^2 x^2 (A+3 B x)+700 a^3 b^3 x^3 (B x-A)+175 a^2 b^4 x^4 (3 A+B x)+14 a b^5 x^5 (5 A+3 B x)+b^6 x^6 (7 A+5 B x)\right )}{35 x^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^(9/2),x]

[Out]

(2*(700*a^3*b^3*x^3*(-A + B*x) + 175*a^2*b^4*x^4*(3*A + B*x) - 175*a^4*b^2*x^2*(A + 3*B*x) + 14*a*b^5*x^5*(5*A
+ 3*B*x) - 14*a^5*b*x*(3*A + 5*B*x) + b^6*x^6*(7*A + 5*B*x) - a^6*(5*A + 7*B*x)))/(35*x^(7/2))

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IntegrateAlgebraic [A]  time = 0.08, size = 151, normalized size = 1.00 \begin {gather*} \frac {2 \left (-5 a^6 A-7 a^6 B x-42 a^5 A b x-70 a^5 b B x^2-175 a^4 A b^2 x^2-525 a^4 b^2 B x^3-700 a^3 A b^3 x^3+700 a^3 b^3 B x^4+525 a^2 A b^4 x^4+175 a^2 b^4 B x^5+70 a A b^5 x^5+42 a b^5 B x^6+7 A b^6 x^6+5 b^6 B x^7\right )}{35 x^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^(9/2),x]

[Out]

(2*(-5*a^6*A - 42*a^5*A*b*x - 7*a^6*B*x - 175*a^4*A*b^2*x^2 - 70*a^5*b*B*x^2 - 700*a^3*A*b^3*x^3 - 525*a^4*b^2
*B*x^3 + 525*a^2*A*b^4*x^4 + 700*a^3*b^3*B*x^4 + 70*a*A*b^5*x^5 + 175*a^2*b^4*B*x^5 + 7*A*b^6*x^6 + 42*a*b^5*B
*x^6 + 5*b^6*B*x^7))/(35*x^(7/2))

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fricas [A]  time = 0.42, size = 147, normalized size = 0.97 \begin {gather*} \frac {2 \, {\left (5 \, B b^{6} x^{7} - 5 \, A a^{6} + 7 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 35 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 175 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} - 175 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} - 35 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} - 7 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x\right )}}{35 \, x^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/x^(9/2),x, algorithm="fricas")

[Out]

2/35*(5*B*b^6*x^7 - 5*A*a^6 + 7*(6*B*a*b^5 + A*b^6)*x^6 + 35*(5*B*a^2*b^4 + 2*A*a*b^5)*x^5 + 175*(4*B*a^3*b^3
+ 3*A*a^2*b^4)*x^4 - 175*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 - 35*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 - 7*(B*a^6 + 6*A*a
^5*b)*x)/x^(7/2)

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giac [A]  time = 0.18, size = 148, normalized size = 0.98 \begin {gather*} \frac {2}{7} \, B b^{6} x^{\frac {7}{2}} + \frac {12}{5} \, B a b^{5} x^{\frac {5}{2}} + \frac {2}{5} \, A b^{6} x^{\frac {5}{2}} + 10 \, B a^{2} b^{4} x^{\frac {3}{2}} + 4 \, A a b^{5} x^{\frac {3}{2}} + 40 \, B a^{3} b^{3} \sqrt {x} + 30 \, A a^{2} b^{4} \sqrt {x} - \frac {2 \, {\left (525 \, B a^{4} b^{2} x^{3} + 700 \, A a^{3} b^{3} x^{3} + 70 \, B a^{5} b x^{2} + 175 \, A a^{4} b^{2} x^{2} + 7 \, B a^{6} x + 42 \, A a^{5} b x + 5 \, A a^{6}\right )}}{35 \, x^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/x^(9/2),x, algorithm="giac")

[Out]

2/7*B*b^6*x^(7/2) + 12/5*B*a*b^5*x^(5/2) + 2/5*A*b^6*x^(5/2) + 10*B*a^2*b^4*x^(3/2) + 4*A*a*b^5*x^(3/2) + 40*B
*a^3*b^3*sqrt(x) + 30*A*a^2*b^4*sqrt(x) - 2/35*(525*B*a^4*b^2*x^3 + 700*A*a^3*b^3*x^3 + 70*B*a^5*b*x^2 + 175*A
*a^4*b^2*x^2 + 7*B*a^6*x + 42*A*a^5*b*x + 5*A*a^6)/x^(7/2)

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maple [A]  time = 0.06, size = 148, normalized size = 0.98 \begin {gather*} -\frac {2 \left (-5 B \,b^{6} x^{7}-7 A \,b^{6} x^{6}-42 x^{6} B a \,b^{5}-70 A a \,b^{5} x^{5}-175 x^{5} B \,a^{2} b^{4}-525 A \,a^{2} b^{4} x^{4}-700 x^{4} B \,a^{3} b^{3}+700 A \,a^{3} b^{3} x^{3}+525 B \,a^{4} b^{2} x^{3}+175 A \,a^{4} b^{2} x^{2}+70 x^{2} B \,a^{5} b +42 A \,a^{5} b x +7 x B \,a^{6}+5 A \,a^{6}\right )}{35 x^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/x^(9/2),x)

[Out]

-2/35*(-5*B*b^6*x^7-7*A*b^6*x^6-42*B*a*b^5*x^6-70*A*a*b^5*x^5-175*B*a^2*b^4*x^5-525*A*a^2*b^4*x^4-700*B*a^3*b^
3*x^4+700*A*a^3*b^3*x^3+525*B*a^4*b^2*x^3+175*A*a^4*b^2*x^2+70*B*a^5*b*x^2+42*A*a^5*b*x+7*B*a^6*x+5*A*a^6)/x^(
7/2)

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maxima [A]  time = 0.56, size = 148, normalized size = 0.98 \begin {gather*} \frac {2}{7} \, B b^{6} x^{\frac {7}{2}} + \frac {2}{5} \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{\frac {5}{2}} + 2 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{\frac {3}{2}} + 10 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} \sqrt {x} - \frac {2 \, {\left (5 \, A a^{6} + 175 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 35 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 7 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x\right )}}{35 \, x^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/x^(9/2),x, algorithm="maxima")

[Out]

2/7*B*b^6*x^(7/2) + 2/5*(6*B*a*b^5 + A*b^6)*x^(5/2) + 2*(5*B*a^2*b^4 + 2*A*a*b^5)*x^(3/2) + 10*(4*B*a^3*b^3 +
3*A*a^2*b^4)*sqrt(x) - 2/35*(5*A*a^6 + 175*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 + 35*(2*B*a^5*b + 5*A*a^4*b^2)*x^2
+ 7*(B*a^6 + 6*A*a^5*b)*x)/x^(7/2)

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mupad [B]  time = 1.15, size = 138, normalized size = 0.91 \begin {gather*} x^{5/2}\,\left (\frac {2\,A\,b^6}{5}+\frac {12\,B\,a\,b^5}{5}\right )-\frac {x\,\left (\frac {2\,B\,a^6}{5}+\frac {12\,A\,b\,a^5}{5}\right )+\frac {2\,A\,a^6}{7}+x^2\,\left (4\,B\,a^5\,b+10\,A\,a^4\,b^2\right )+x^3\,\left (30\,B\,a^4\,b^2+40\,A\,a^3\,b^3\right )}{x^{7/2}}+\frac {2\,B\,b^6\,x^{7/2}}{7}+10\,a^2\,b^3\,\sqrt {x}\,\left (3\,A\,b+4\,B\,a\right )+2\,a\,b^4\,x^{3/2}\,\left (2\,A\,b+5\,B\,a\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^3)/x^(9/2),x)

[Out]

x^(5/2)*((2*A*b^6)/5 + (12*B*a*b^5)/5) - (x*((2*B*a^6)/5 + (12*A*a^5*b)/5) + (2*A*a^6)/7 + x^2*(10*A*a^4*b^2 +
4*B*a^5*b) + x^3*(40*A*a^3*b^3 + 30*B*a^4*b^2))/x^(7/2) + (2*B*b^6*x^(7/2))/7 + 10*a^2*b^3*x^(1/2)*(3*A*b + 4
*B*a) + 2*a*b^4*x^(3/2)*(2*A*b + 5*B*a)

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sympy [A]  time = 9.61, size = 202, normalized size = 1.34 \begin {gather*} - \frac {2 A a^{6}}{7 x^{\frac {7}{2}}} - \frac {12 A a^{5} b}{5 x^{\frac {5}{2}}} - \frac {10 A a^{4} b^{2}}{x^{\frac {3}{2}}} - \frac {40 A a^{3} b^{3}}{\sqrt {x}} + 30 A a^{2} b^{4} \sqrt {x} + 4 A a b^{5} x^{\frac {3}{2}} + \frac {2 A b^{6} x^{\frac {5}{2}}}{5} - \frac {2 B a^{6}}{5 x^{\frac {5}{2}}} - \frac {4 B a^{5} b}{x^{\frac {3}{2}}} - \frac {30 B a^{4} b^{2}}{\sqrt {x}} + 40 B a^{3} b^{3} \sqrt {x} + 10 B a^{2} b^{4} x^{\frac {3}{2}} + \frac {12 B a b^{5} x^{\frac {5}{2}}}{5} + \frac {2 B b^{6} x^{\frac {7}{2}}}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**3/x**(9/2),x)

[Out]

-2*A*a**6/(7*x**(7/2)) - 12*A*a**5*b/(5*x**(5/2)) - 10*A*a**4*b**2/x**(3/2) - 40*A*a**3*b**3/sqrt(x) + 30*A*a*
*2*b**4*sqrt(x) + 4*A*a*b**5*x**(3/2) + 2*A*b**6*x**(5/2)/5 - 2*B*a**6/(5*x**(5/2)) - 4*B*a**5*b/x**(3/2) - 30
*B*a**4*b**2/sqrt(x) + 40*B*a**3*b**3*sqrt(x) + 10*B*a**2*b**4*x**(3/2) + 12*B*a*b**5*x**(5/2)/5 + 2*B*b**6*x*
*(7/2)/7

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