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

Optimal. Leaf size=151 \[ -\frac {2 a^6 A}{\sqrt {x}}+2 a^5 \sqrt {x} (a B+6 A b)+2 a^4 b x^{3/2} (2 a B+5 A b)+2 a^3 b^2 x^{5/2} (3 a B+4 A b)+\frac {10}{7} a^2 b^3 x^{7/2} (4 a B+3 A b)+\frac {2}{11} b^5 x^{11/2} (6 a B+A b)+\frac {2}{3} a b^4 x^{9/2} (5 a B+2 A b)+\frac {2}{13} b^6 B x^{13/2} \]

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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}{7} a^2 b^3 x^{7/2} (4 a B+3 A b)+2 a^3 b^2 x^{5/2} (3 a B+4 A b)+2 a^4 b x^{3/2} (2 a B+5 A b)+2 a^5 \sqrt {x} (a B+6 A b)-\frac {2 a^6 A}{\sqrt {x}}+\frac {2}{11} b^5 x^{11/2} (6 a B+A b)+\frac {2}{3} a b^4 x^{9/2} (5 a B+2 A b)+\frac {2}{13} b^6 B x^{13/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a^6*A)/Sqrt[x] + 2*a^5*(6*A*b + a*B)*Sqrt[x] + 2*a^4*b*(5*A*b + 2*a*B)*x^(3/2) + 2*a^3*b^2*(4*A*b + 3*a*B)
*x^(5/2) + (10*a^2*b^3*(3*A*b + 4*a*B)*x^(7/2))/7 + (2*a*b^4*(2*A*b + 5*a*B)*x^(9/2))/3 + (2*b^5*(A*b + 6*a*B)
*x^(11/2))/11 + (2*b^6*B*x^(13/2))/13

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^{3/2}} \, dx &=\int \frac {(a+b x)^6 (A+B x)}{x^{3/2}} \, dx\\ &=\int \left (\frac {a^6 A}{x^{3/2}}+\frac {a^5 (6 A b+a B)}{\sqrt {x}}+3 a^4 b (5 A b+2 a B) \sqrt {x}+5 a^3 b^2 (4 A b+3 a B) x^{3/2}+5 a^2 b^3 (3 A b+4 a B) x^{5/2}+3 a b^4 (2 A b+5 a B) x^{7/2}+b^5 (A b+6 a B) x^{9/2}+b^6 B x^{11/2}\right ) \, dx\\ &=-\frac {2 a^6 A}{\sqrt {x}}+2 a^5 (6 A b+a B) \sqrt {x}+2 a^4 b (5 A b+2 a B) x^{3/2}+2 a^3 b^2 (4 A b+3 a B) x^{5/2}+\frac {10}{7} a^2 b^3 (3 A b+4 a B) x^{7/2}+\frac {2}{3} a b^4 (2 A b+5 a B) x^{9/2}+\frac {2}{11} b^5 (A b+6 a B) x^{11/2}+\frac {2}{13} b^6 B x^{13/2}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 101, normalized size = 0.67 \begin {gather*} \frac {2 \left (\frac {\sqrt {x} \left (3003 a^6+6006 a^5 b x+9009 a^4 b^2 x^2+8580 a^3 b^3 x^3+5005 a^2 b^4 x^4+1638 a b^5 x^5+231 b^6 x^6\right ) (a B+13 A b)}{3003}-\frac {A (a+b x)^7}{\sqrt {x}}\right )}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-((A*(a + b*x)^7)/Sqrt[x]) + ((13*A*b + a*B)*Sqrt[x]*(3003*a^6 + 6006*a^5*b*x + 9009*a^4*b^2*x^2 + 8580*a^
3*b^3*x^3 + 5005*a^2*b^4*x^4 + 1638*a*b^5*x^5 + 231*b^6*x^6))/3003))/a

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IntegrateAlgebraic [A]  time = 0.08, size = 151, normalized size = 1.00 \begin {gather*} \frac {2 \left (-3003 a^6 A+3003 a^6 B x+18018 a^5 A b x+6006 a^5 b B x^2+15015 a^4 A b^2 x^2+9009 a^4 b^2 B x^3+12012 a^3 A b^3 x^3+8580 a^3 b^3 B x^4+6435 a^2 A b^4 x^4+5005 a^2 b^4 B x^5+2002 a A b^5 x^5+1638 a b^5 B x^6+273 A b^6 x^6+231 b^6 B x^7\right )}{3003 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-3003*a^6*A + 18018*a^5*A*b*x + 3003*a^6*B*x + 15015*a^4*A*b^2*x^2 + 6006*a^5*b*B*x^2 + 12012*a^3*A*b^3*x^
3 + 9009*a^4*b^2*B*x^3 + 6435*a^2*A*b^4*x^4 + 8580*a^3*b^3*B*x^4 + 2002*a*A*b^5*x^5 + 5005*a^2*b^4*B*x^5 + 273
*A*b^6*x^6 + 1638*a*b^5*B*x^6 + 231*b^6*B*x^7))/(3003*Sqrt[x])

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fricas [A]  time = 0.40, size = 147, normalized size = 0.97 \begin {gather*} \frac {2 \, {\left (231 \, B b^{6} x^{7} - 3003 \, A a^{6} + 273 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 1001 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 2145 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 3003 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 3003 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 3003 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x\right )}}{3003 \, \sqrt {x}} \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^(3/2),x, algorithm="fricas")

[Out]

2/3003*(231*B*b^6*x^7 - 3003*A*a^6 + 273*(6*B*a*b^5 + A*b^6)*x^6 + 1001*(5*B*a^2*b^4 + 2*A*a*b^5)*x^5 + 2145*(
4*B*a^3*b^3 + 3*A*a^2*b^4)*x^4 + 3003*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 + 3003*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 + 3
003*(B*a^6 + 6*A*a^5*b)*x)/sqrt(x)

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giac [A]  time = 0.16, size = 149, normalized size = 0.99 \begin {gather*} \frac {2}{13} \, B b^{6} x^{\frac {13}{2}} + \frac {12}{11} \, B a b^{5} x^{\frac {11}{2}} + \frac {2}{11} \, A b^{6} x^{\frac {11}{2}} + \frac {10}{3} \, B a^{2} b^{4} x^{\frac {9}{2}} + \frac {4}{3} \, A a b^{5} x^{\frac {9}{2}} + \frac {40}{7} \, B a^{3} b^{3} x^{\frac {7}{2}} + \frac {30}{7} \, A a^{2} b^{4} x^{\frac {7}{2}} + 6 \, B a^{4} b^{2} x^{\frac {5}{2}} + 8 \, A a^{3} b^{3} x^{\frac {5}{2}} + 4 \, B a^{5} b x^{\frac {3}{2}} + 10 \, A a^{4} b^{2} x^{\frac {3}{2}} + 2 \, B a^{6} \sqrt {x} + 12 \, A a^{5} b \sqrt {x} - \frac {2 \, A a^{6}}{\sqrt {x}} \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^(3/2),x, algorithm="giac")

[Out]

2/13*B*b^6*x^(13/2) + 12/11*B*a*b^5*x^(11/2) + 2/11*A*b^6*x^(11/2) + 10/3*B*a^2*b^4*x^(9/2) + 4/3*A*a*b^5*x^(9
/2) + 40/7*B*a^3*b^3*x^(7/2) + 30/7*A*a^2*b^4*x^(7/2) + 6*B*a^4*b^2*x^(5/2) + 8*A*a^3*b^3*x^(5/2) + 4*B*a^5*b*
x^(3/2) + 10*A*a^4*b^2*x^(3/2) + 2*B*a^6*sqrt(x) + 12*A*a^5*b*sqrt(x) - 2*A*a^6/sqrt(x)

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maple [A]  time = 0.05, size = 148, normalized size = 0.98 \begin {gather*} -\frac {2 \left (-231 B \,b^{6} x^{7}-273 A \,b^{6} x^{6}-1638 x^{6} B a \,b^{5}-2002 A a \,b^{5} x^{5}-5005 x^{5} B \,a^{2} b^{4}-6435 A \,a^{2} b^{4} x^{4}-8580 x^{4} B \,a^{3} b^{3}-12012 A \,a^{3} b^{3} x^{3}-9009 B \,a^{4} b^{2} x^{3}-15015 A \,a^{4} b^{2} x^{2}-6006 x^{2} B \,a^{5} b -18018 A \,a^{5} b x -3003 x B \,a^{6}+3003 A \,a^{6}\right )}{3003 \sqrt {x}} \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^(3/2),x)

[Out]

-2/3003*(-231*B*b^6*x^7-273*A*b^6*x^6-1638*B*a*b^5*x^6-2002*A*a*b^5*x^5-5005*B*a^2*b^4*x^5-6435*A*a^2*b^4*x^4-
8580*B*a^3*b^3*x^4-12012*A*a^3*b^3*x^3-9009*B*a^4*b^2*x^3-15015*A*a^4*b^2*x^2-6006*B*a^5*b*x^2-18018*A*a^5*b*x
-3003*B*a^6*x+3003*A*a^6)/x^(1/2)

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maxima [A]  time = 0.56, size = 147, normalized size = 0.97 \begin {gather*} \frac {2}{13} \, B b^{6} x^{\frac {13}{2}} - \frac {2 \, A a^{6}}{\sqrt {x}} + \frac {2}{11} \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{\frac {11}{2}} + \frac {2}{3} \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{\frac {9}{2}} + \frac {10}{7} \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{\frac {7}{2}} + 2 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{\frac {5}{2}} + 2 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{\frac {3}{2}} + 2 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} \sqrt {x} \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^(3/2),x, algorithm="maxima")

[Out]

2/13*B*b^6*x^(13/2) - 2*A*a^6/sqrt(x) + 2/11*(6*B*a*b^5 + A*b^6)*x^(11/2) + 2/3*(5*B*a^2*b^4 + 2*A*a*b^5)*x^(9
/2) + 10/7*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x^(7/2) + 2*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^(5/2) + 2*(2*B*a^5*b + 5*A*a^
4*b^2)*x^(3/2) + 2*(B*a^6 + 6*A*a^5*b)*sqrt(x)

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mupad [B]  time = 0.05, size = 131, normalized size = 0.87 \begin {gather*} \sqrt {x}\,\left (2\,B\,a^6+12\,A\,b\,a^5\right )+x^{11/2}\,\left (\frac {2\,A\,b^6}{11}+\frac {12\,B\,a\,b^5}{11}\right )-\frac {2\,A\,a^6}{\sqrt {x}}+\frac {2\,B\,b^6\,x^{13/2}}{13}+2\,a^3\,b^2\,x^{5/2}\,\left (4\,A\,b+3\,B\,a\right )+\frac {10\,a^2\,b^3\,x^{7/2}\,\left (3\,A\,b+4\,B\,a\right )}{7}+2\,a^4\,b\,x^{3/2}\,\left (5\,A\,b+2\,B\,a\right )+\frac {2\,a\,b^4\,x^{9/2}\,\left (2\,A\,b+5\,B\,a\right )}{3} \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^(3/2),x)

[Out]

x^(1/2)*(2*B*a^6 + 12*A*a^5*b) + x^(11/2)*((2*A*b^6)/11 + (12*B*a*b^5)/11) - (2*A*a^6)/x^(1/2) + (2*B*b^6*x^(1
3/2))/13 + 2*a^3*b^2*x^(5/2)*(4*A*b + 3*B*a) + (10*a^2*b^3*x^(7/2)*(3*A*b + 4*B*a))/7 + 2*a^4*b*x^(3/2)*(5*A*b
+ 2*B*a) + (2*a*b^4*x^(9/2)*(2*A*b + 5*B*a))/3

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sympy [A]  time = 4.60, size = 204, normalized size = 1.35 \begin {gather*} - \frac {2 A a^{6}}{\sqrt {x}} + 12 A a^{5} b \sqrt {x} + 10 A a^{4} b^{2} x^{\frac {3}{2}} + 8 A a^{3} b^{3} x^{\frac {5}{2}} + \frac {30 A a^{2} b^{4} x^{\frac {7}{2}}}{7} + \frac {4 A a b^{5} x^{\frac {9}{2}}}{3} + \frac {2 A b^{6} x^{\frac {11}{2}}}{11} + 2 B a^{6} \sqrt {x} + 4 B a^{5} b x^{\frac {3}{2}} + 6 B a^{4} b^{2} x^{\frac {5}{2}} + \frac {40 B a^{3} b^{3} x^{\frac {7}{2}}}{7} + \frac {10 B a^{2} b^{4} x^{\frac {9}{2}}}{3} + \frac {12 B a b^{5} x^{\frac {11}{2}}}{11} + \frac {2 B b^{6} x^{\frac {13}{2}}}{13} \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**(3/2),x)

[Out]

-2*A*a**6/sqrt(x) + 12*A*a**5*b*sqrt(x) + 10*A*a**4*b**2*x**(3/2) + 8*A*a**3*b**3*x**(5/2) + 30*A*a**2*b**4*x*
*(7/2)/7 + 4*A*a*b**5*x**(9/2)/3 + 2*A*b**6*x**(11/2)/11 + 2*B*a**6*sqrt(x) + 4*B*a**5*b*x**(3/2) + 6*B*a**4*b
**2*x**(5/2) + 40*B*a**3*b**3*x**(7/2)/7 + 10*B*a**2*b**4*x**(9/2)/3 + 12*B*a*b**5*x**(11/2)/11 + 2*B*b**6*x**
(13/2)/13

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