3.7.80 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^3}{x^{5/2}} \, dx\)

Optimal. Leaf size=153 \[ -\frac {2 a^6 A}{3 x^{3/2}}-\frac {2 a^5 (a B+6 A b)}{\sqrt {x}}+6 a^4 b \sqrt {x} (2 a B+5 A b)+\frac {10}{3} a^3 b^2 x^{3/2} (3 a B+4 A b)+2 a^2 b^3 x^{5/2} (4 a B+3 A b)+\frac {2}{9} b^5 x^{9/2} (6 a B+A b)+\frac {6}{7} a b^4 x^{7/2} (5 a B+2 A b)+\frac {2}{11} b^6 B x^{11/2} \]

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Rubi [A]  time = 0.08, antiderivative size = 153, 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}{3} a^3 b^2 x^{3/2} (3 a B+4 A b)+2 a^2 b^3 x^{5/2} (4 a B+3 A b)-\frac {2 a^5 (a B+6 A b)}{\sqrt {x}}+6 a^4 b \sqrt {x} (2 a B+5 A b)-\frac {2 a^6 A}{3 x^{3/2}}+\frac {6}{7} a b^4 x^{7/2} (5 a B+2 A b)+\frac {2}{9} b^5 x^{9/2} (6 a B+A b)+\frac {2}{11} b^6 B x^{11/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^(5/2),x]

[Out]

(-2*a^6*A)/(3*x^(3/2)) - (2*a^5*(6*A*b + a*B))/Sqrt[x] + 6*a^4*b*(5*A*b + 2*a*B)*Sqrt[x] + (10*a^3*b^2*(4*A*b
+ 3*a*B)*x^(3/2))/3 + 2*a^2*b^3*(3*A*b + 4*a*B)*x^(5/2) + (6*a*b^4*(2*A*b + 5*a*B)*x^(7/2))/7 + (2*b^5*(A*b +
6*a*B)*x^(9/2))/9 + (2*b^6*B*x^(11/2))/11

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

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Mathematica [A]  time = 0.07, size = 97, normalized size = 0.63 \begin {gather*} \frac {2 \left (x \left (-231 a^6+1386 a^5 b x+1155 a^4 b^2 x^2+924 a^3 b^3 x^3+495 a^2 b^4 x^4+154 a b^5 x^5+21 b^6 x^6\right ) (3 a B+11 A b)-231 A (a+b x)^7\right )}{693 a x^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^(5/2),x]

[Out]

(2*(-231*A*(a + b*x)^7 + (11*A*b + 3*a*B)*x*(-231*a^6 + 1386*a^5*b*x + 1155*a^4*b^2*x^2 + 924*a^3*b^3*x^3 + 49
5*a^2*b^4*x^4 + 154*a*b^5*x^5 + 21*b^6*x^6)))/(693*a*x^(3/2))

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IntegrateAlgebraic [A]  time = 0.09, size = 151, normalized size = 0.99 \begin {gather*} \frac {2 \left (-231 a^6 A-693 a^6 B x-4158 a^5 A b x+4158 a^5 b B x^2+10395 a^4 A b^2 x^2+3465 a^4 b^2 B x^3+4620 a^3 A b^3 x^3+2772 a^3 b^3 B x^4+2079 a^2 A b^4 x^4+1485 a^2 b^4 B x^5+594 a A b^5 x^5+462 a b^5 B x^6+77 A b^6 x^6+63 b^6 B x^7\right )}{693 x^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^(5/2),x]

[Out]

(2*(-231*a^6*A - 4158*a^5*A*b*x - 693*a^6*B*x + 10395*a^4*A*b^2*x^2 + 4158*a^5*b*B*x^2 + 4620*a^3*A*b^3*x^3 +
3465*a^4*b^2*B*x^3 + 2079*a^2*A*b^4*x^4 + 2772*a^3*b^3*B*x^4 + 594*a*A*b^5*x^5 + 1485*a^2*b^4*B*x^5 + 77*A*b^6
*x^6 + 462*a*b^5*B*x^6 + 63*b^6*B*x^7))/(693*x^(3/2))

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fricas [A]  time = 0.40, size = 147, normalized size = 0.96 \begin {gather*} \frac {2 \, {\left (63 \, B b^{6} x^{7} - 231 \, A a^{6} + 77 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 297 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 693 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 1155 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 2079 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} - 693 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x\right )}}{693 \, x^{\frac {3}{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^(5/2),x, algorithm="fricas")

[Out]

2/693*(63*B*b^6*x^7 - 231*A*a^6 + 77*(6*B*a*b^5 + A*b^6)*x^6 + 297*(5*B*a^2*b^4 + 2*A*a*b^5)*x^5 + 693*(4*B*a^
3*b^3 + 3*A*a^2*b^4)*x^4 + 1155*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 + 2079*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 - 693*(B*
a^6 + 6*A*a^5*b)*x)/x^(3/2)

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giac [A]  time = 0.17, size = 147, normalized size = 0.96 \begin {gather*} \frac {2}{11} \, B b^{6} x^{\frac {11}{2}} + \frac {4}{3} \, B a b^{5} x^{\frac {9}{2}} + \frac {2}{9} \, A b^{6} x^{\frac {9}{2}} + \frac {30}{7} \, B a^{2} b^{4} x^{\frac {7}{2}} + \frac {12}{7} \, A a b^{5} x^{\frac {7}{2}} + 8 \, B a^{3} b^{3} x^{\frac {5}{2}} + 6 \, A a^{2} b^{4} x^{\frac {5}{2}} + 10 \, B a^{4} b^{2} x^{\frac {3}{2}} + \frac {40}{3} \, A a^{3} b^{3} x^{\frac {3}{2}} + 12 \, B a^{5} b \sqrt {x} + 30 \, A a^{4} b^{2} \sqrt {x} - \frac {2 \, {\left (3 \, B a^{6} x + 18 \, A a^{5} b x + A a^{6}\right )}}{3 \, x^{\frac {3}{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^(5/2),x, algorithm="giac")

[Out]

2/11*B*b^6*x^(11/2) + 4/3*B*a*b^5*x^(9/2) + 2/9*A*b^6*x^(9/2) + 30/7*B*a^2*b^4*x^(7/2) + 12/7*A*a*b^5*x^(7/2)
+ 8*B*a^3*b^3*x^(5/2) + 6*A*a^2*b^4*x^(5/2) + 10*B*a^4*b^2*x^(3/2) + 40/3*A*a^3*b^3*x^(3/2) + 12*B*a^5*b*sqrt(
x) + 30*A*a^4*b^2*sqrt(x) - 2/3*(3*B*a^6*x + 18*A*a^5*b*x + A*a^6)/x^(3/2)

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maple [A]  time = 0.06, size = 148, normalized size = 0.97 \begin {gather*} -\frac {2 \left (-63 B \,b^{6} x^{7}-77 A \,b^{6} x^{6}-462 x^{6} B a \,b^{5}-594 A a \,b^{5} x^{5}-1485 x^{5} B \,a^{2} b^{4}-2079 A \,a^{2} b^{4} x^{4}-2772 x^{4} B \,a^{3} b^{3}-4620 A \,a^{3} b^{3} x^{3}-3465 B \,a^{4} b^{2} x^{3}-10395 A \,a^{4} b^{2} x^{2}-4158 x^{2} B \,a^{5} b +4158 A \,a^{5} b x +693 x B \,a^{6}+231 A \,a^{6}\right )}{693 x^{\frac {3}{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^(5/2),x)

[Out]

-2/693*(-63*B*b^6*x^7-77*A*b^6*x^6-462*B*a*b^5*x^6-594*A*a*b^5*x^5-1485*B*a^2*b^4*x^5-2079*A*a^2*b^4*x^4-2772*
B*a^3*b^3*x^4-4620*A*a^3*b^3*x^3-3465*B*a^4*b^2*x^3-10395*A*a^4*b^2*x^2-4158*B*a^5*b*x^2+4158*A*a^5*b*x+693*B*
a^6*x+231*A*a^6)/x^(3/2)

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maxima [A]  time = 0.66, size = 147, normalized size = 0.96 \begin {gather*} \frac {2}{11} \, B b^{6} x^{\frac {11}{2}} + \frac {2}{9} \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{\frac {9}{2}} + \frac {6}{7} \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{\frac {7}{2}} + 2 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{\frac {5}{2}} + \frac {10}{3} \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{\frac {3}{2}} + 6 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} \sqrt {x} - \frac {2 \, {\left (A a^{6} + 3 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x\right )}}{3 \, x^{\frac {3}{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^(5/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.05, size = 132, normalized size = 0.86 \begin {gather*} x^{9/2}\,\left (\frac {2\,A\,b^6}{9}+\frac {4\,B\,a\,b^5}{3}\right )-\frac {x\,\left (2\,B\,a^6+12\,A\,b\,a^5\right )+\frac {2\,A\,a^6}{3}}{x^{3/2}}+\frac {2\,B\,b^6\,x^{11/2}}{11}+\frac {10\,a^3\,b^2\,x^{3/2}\,\left (4\,A\,b+3\,B\,a\right )}{3}+2\,a^2\,b^3\,x^{5/2}\,\left (3\,A\,b+4\,B\,a\right )+6\,a^4\,b\,\sqrt {x}\,\left (5\,A\,b+2\,B\,a\right )+\frac {6\,a\,b^4\,x^{7/2}\,\left (2\,A\,b+5\,B\,a\right )}{7} \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^(5/2),x)

[Out]

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

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sympy [A]  time = 5.44, size = 204, normalized size = 1.33 \begin {gather*} - \frac {2 A a^{6}}{3 x^{\frac {3}{2}}} - \frac {12 A a^{5} b}{\sqrt {x}} + 30 A a^{4} b^{2} \sqrt {x} + \frac {40 A a^{3} b^{3} x^{\frac {3}{2}}}{3} + 6 A a^{2} b^{4} x^{\frac {5}{2}} + \frac {12 A a b^{5} x^{\frac {7}{2}}}{7} + \frac {2 A b^{6} x^{\frac {9}{2}}}{9} - \frac {2 B a^{6}}{\sqrt {x}} + 12 B a^{5} b \sqrt {x} + 10 B a^{4} b^{2} x^{\frac {3}{2}} + 8 B a^{3} b^{3} x^{\frac {5}{2}} + \frac {30 B a^{2} b^{4} x^{\frac {7}{2}}}{7} + \frac {4 B a b^{5} x^{\frac {9}{2}}}{3} + \frac {2 B b^{6} x^{\frac {11}{2}}}{11} \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**(5/2),x)

[Out]

-2*A*a**6/(3*x**(3/2)) - 12*A*a**5*b/sqrt(x) + 30*A*a**4*b**2*sqrt(x) + 40*A*a**3*b**3*x**(3/2)/3 + 6*A*a**2*b
**4*x**(5/2) + 12*A*a*b**5*x**(7/2)/7 + 2*A*b**6*x**(9/2)/9 - 2*B*a**6/sqrt(x) + 12*B*a**5*b*sqrt(x) + 10*B*a*
*4*b**2*x**(3/2) + 8*B*a**3*b**3*x**(5/2) + 30*B*a**2*b**4*x**(7/2)/7 + 4*B*a*b**5*x**(9/2)/3 + 2*B*b**6*x**(1
1/2)/11

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