3.7.77 \(\int \sqrt {x} (A+B x) (a^2+2 a b x+b^2 x^2)^3 \, dx\)

Optimal. Leaf size=159 \[ \frac {2}{3} a^6 A x^{3/2}+\frac {2}{5} a^5 x^{5/2} (a B+6 A b)+\frac {6}{7} a^4 b x^{7/2} (2 a B+5 A b)+\frac {10}{9} a^3 b^2 x^{9/2} (3 a B+4 A b)+\frac {10}{11} a^2 b^3 x^{11/2} (4 a B+3 A b)+\frac {2}{15} b^5 x^{15/2} (6 a B+A b)+\frac {6}{13} a b^4 x^{13/2} (5 a B+2 A b)+\frac {2}{17} b^6 B x^{17/2} \]

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Rubi [A]  time = 0.08, antiderivative size = 159, 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}{11} a^2 b^3 x^{11/2} (4 a B+3 A b)+\frac {10}{9} a^3 b^2 x^{9/2} (3 a B+4 A b)+\frac {6}{7} a^4 b x^{7/2} (2 a B+5 A b)+\frac {2}{5} a^5 x^{5/2} (a B+6 A b)+\frac {2}{3} a^6 A x^{3/2}+\frac {2}{15} b^5 x^{15/2} (6 a B+A b)+\frac {6}{13} a b^4 x^{13/2} (5 a B+2 A b)+\frac {2}{17} b^6 B x^{17/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^6*A*x^(3/2))/3 + (2*a^5*(6*A*b + a*B)*x^(5/2))/5 + (6*a^4*b*(5*A*b + 2*a*B)*x^(7/2))/7 + (10*a^3*b^2*(4*A
*b + 3*a*B)*x^(9/2))/9 + (10*a^2*b^3*(3*A*b + 4*a*B)*x^(11/2))/11 + (6*a*b^4*(2*A*b + 5*a*B)*x^(13/2))/13 + (2
*b^5*(A*b + 6*a*B)*x^(15/2))/15 + (2*b^6*B*x^(17/2))/17

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 \sqrt {x} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int \sqrt {x} (a+b x)^6 (A+B x) \, dx\\ &=\int \left (a^6 A \sqrt {x}+a^5 (6 A b+a B) x^{3/2}+3 a^4 b (5 A b+2 a B) x^{5/2}+5 a^3 b^2 (4 A b+3 a B) x^{7/2}+5 a^2 b^3 (3 A b+4 a B) x^{9/2}+3 a b^4 (2 A b+5 a B) x^{11/2}+b^5 (A b+6 a B) x^{13/2}+b^6 B x^{15/2}\right ) \, dx\\ &=\frac {2}{3} a^6 A x^{3/2}+\frac {2}{5} a^5 (6 A b+a B) x^{5/2}+\frac {6}{7} a^4 b (5 A b+2 a B) x^{7/2}+\frac {10}{9} a^3 b^2 (4 A b+3 a B) x^{9/2}+\frac {10}{11} a^2 b^3 (3 A b+4 a B) x^{11/2}+\frac {6}{13} a b^4 (2 A b+5 a B) x^{13/2}+\frac {2}{15} b^5 (A b+6 a B) x^{15/2}+\frac {2}{17} b^6 B x^{17/2}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 103, normalized size = 0.65 \begin {gather*} \frac {2 \left (\frac {x^{3/2} \left (15015 a^6+54054 a^5 b x+96525 a^4 b^2 x^2+100100 a^3 b^3 x^3+61425 a^2 b^4 x^4+20790 a b^5 x^5+3003 b^6 x^6\right ) (17 A b-3 a B)}{45045}+B x^{3/2} (a+b x)^7\right )}{17 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(B*x^(3/2)*(a + b*x)^7 + ((17*A*b - 3*a*B)*x^(3/2)*(15015*a^6 + 54054*a^5*b*x + 96525*a^4*b^2*x^2 + 100100*
a^3*b^3*x^3 + 61425*a^2*b^4*x^4 + 20790*a*b^5*x^5 + 3003*b^6*x^6))/45045))/(17*b)

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IntegrateAlgebraic [A]  time = 0.07, size = 181, normalized size = 1.14 \begin {gather*} \frac {2 \left (255255 a^6 A x^{3/2}+153153 a^6 B x^{5/2}+918918 a^5 A b x^{5/2}+656370 a^5 b B x^{7/2}+1640925 a^4 A b^2 x^{7/2}+1276275 a^4 b^2 B x^{9/2}+1701700 a^3 A b^3 x^{9/2}+1392300 a^3 b^3 B x^{11/2}+1044225 a^2 A b^4 x^{11/2}+883575 a^2 b^4 B x^{13/2}+353430 a A b^5 x^{13/2}+306306 a b^5 B x^{15/2}+51051 A b^6 x^{15/2}+45045 b^6 B x^{17/2}\right )}{765765} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(255255*a^6*A*x^(3/2) + 918918*a^5*A*b*x^(5/2) + 153153*a^6*B*x^(5/2) + 1640925*a^4*A*b^2*x^(7/2) + 656370*
a^5*b*B*x^(7/2) + 1701700*a^3*A*b^3*x^(9/2) + 1276275*a^4*b^2*B*x^(9/2) + 1044225*a^2*A*b^4*x^(11/2) + 1392300
*a^3*b^3*B*x^(11/2) + 353430*a*A*b^5*x^(13/2) + 883575*a^2*b^4*B*x^(13/2) + 51051*A*b^6*x^(15/2) + 306306*a*b^
5*B*x^(15/2) + 45045*b^6*B*x^(17/2)))/765765

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fricas [A]  time = 0.42, size = 150, normalized size = 0.94 \begin {gather*} \frac {2}{765765} \, {\left (45045 \, B b^{6} x^{8} + 255255 \, A a^{6} x + 51051 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{7} + 176715 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{6} + 348075 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{5} + 425425 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{4} + 328185 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{3} + 153153 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x^{2}\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^(1/2),x, algorithm="fricas")

[Out]

2/765765*(45045*B*b^6*x^8 + 255255*A*a^6*x + 51051*(6*B*a*b^5 + A*b^6)*x^7 + 176715*(5*B*a^2*b^4 + 2*A*a*b^5)*
x^6 + 348075*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x^5 + 425425*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^4 + 328185*(2*B*a^5*b + 5*
A*a^4*b^2)*x^3 + 153153*(B*a^6 + 6*A*a^5*b)*x^2)*sqrt(x)

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giac [A]  time = 0.17, size = 149, normalized size = 0.94 \begin {gather*} \frac {2}{17} \, B b^{6} x^{\frac {17}{2}} + \frac {4}{5} \, B a b^{5} x^{\frac {15}{2}} + \frac {2}{15} \, A b^{6} x^{\frac {15}{2}} + \frac {30}{13} \, B a^{2} b^{4} x^{\frac {13}{2}} + \frac {12}{13} \, A a b^{5} x^{\frac {13}{2}} + \frac {40}{11} \, B a^{3} b^{3} x^{\frac {11}{2}} + \frac {30}{11} \, A a^{2} b^{4} x^{\frac {11}{2}} + \frac {10}{3} \, B a^{4} b^{2} x^{\frac {9}{2}} + \frac {40}{9} \, A a^{3} b^{3} x^{\frac {9}{2}} + \frac {12}{7} \, B a^{5} b x^{\frac {7}{2}} + \frac {30}{7} \, A a^{4} b^{2} x^{\frac {7}{2}} + \frac {2}{5} \, B a^{6} x^{\frac {5}{2}} + \frac {12}{5} \, A a^{5} b x^{\frac {5}{2}} + \frac {2}{3} \, A a^{6} 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^(1/2),x, algorithm="giac")

[Out]

2/17*B*b^6*x^(17/2) + 4/5*B*a*b^5*x^(15/2) + 2/15*A*b^6*x^(15/2) + 30/13*B*a^2*b^4*x^(13/2) + 12/13*A*a*b^5*x^
(13/2) + 40/11*B*a^3*b^3*x^(11/2) + 30/11*A*a^2*b^4*x^(11/2) + 10/3*B*a^4*b^2*x^(9/2) + 40/9*A*a^3*b^3*x^(9/2)
+ 12/7*B*a^5*b*x^(7/2) + 30/7*A*a^4*b^2*x^(7/2) + 2/5*B*a^6*x^(5/2) + 12/5*A*a^5*b*x^(5/2) + 2/3*A*a^6*x^(3/2
)

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maple [A]  time = 0.05, size = 148, normalized size = 0.93 \begin {gather*} \frac {2 \left (45045 B \,b^{6} x^{7}+51051 A \,b^{6} x^{6}+306306 x^{6} B a \,b^{5}+353430 A a \,b^{5} x^{5}+883575 x^{5} B \,a^{2} b^{4}+1044225 A \,a^{2} b^{4} x^{4}+1392300 x^{4} B \,a^{3} b^{3}+1701700 A \,a^{3} b^{3} x^{3}+1276275 B \,a^{4} b^{2} x^{3}+1640925 A \,a^{4} b^{2} x^{2}+656370 x^{2} B \,a^{5} b +918918 A \,a^{5} b x +153153 x B \,a^{6}+255255 A \,a^{6}\right ) x^{\frac {3}{2}}}{765765} \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^(1/2),x)

[Out]

2/765765*x^(3/2)*(45045*B*b^6*x^7+51051*A*b^6*x^6+306306*B*a*b^5*x^6+353430*A*a*b^5*x^5+883575*B*a^2*b^4*x^5+1
044225*A*a^2*b^4*x^4+1392300*B*a^3*b^3*x^4+1701700*A*a^3*b^3*x^3+1276275*B*a^4*b^2*x^3+1640925*A*a^4*b^2*x^2+6
56370*B*a^5*b*x^2+918918*A*a^5*b*x+153153*B*a^6*x+255255*A*a^6)

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maxima [A]  time = 0.50, size = 147, normalized size = 0.92 \begin {gather*} \frac {2}{17} \, B b^{6} x^{\frac {17}{2}} + \frac {2}{3} \, A a^{6} x^{\frac {3}{2}} + \frac {2}{15} \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{\frac {15}{2}} + \frac {6}{13} \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{\frac {13}{2}} + \frac {10}{11} \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{\frac {11}{2}} + \frac {10}{9} \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{\frac {9}{2}} + \frac {6}{7} \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{\frac {7}{2}} + \frac {2}{5} \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x^{\frac {5}{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^(1/2),x, algorithm="maxima")

[Out]

2/17*B*b^6*x^(17/2) + 2/3*A*a^6*x^(3/2) + 2/15*(6*B*a*b^5 + A*b^6)*x^(15/2) + 6/13*(5*B*a^2*b^4 + 2*A*a*b^5)*x
^(13/2) + 10/11*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x^(11/2) + 10/9*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^(9/2) + 6/7*(2*B*a^5
*b + 5*A*a^4*b^2)*x^(7/2) + 2/5*(B*a^6 + 6*A*a^5*b)*x^(5/2)

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mupad [B]  time = 0.05, size = 131, normalized size = 0.82 \begin {gather*} x^{5/2}\,\left (\frac {2\,B\,a^6}{5}+\frac {12\,A\,b\,a^5}{5}\right )+x^{15/2}\,\left (\frac {2\,A\,b^6}{15}+\frac {4\,B\,a\,b^5}{5}\right )+\frac {2\,A\,a^6\,x^{3/2}}{3}+\frac {2\,B\,b^6\,x^{17/2}}{17}+\frac {10\,a^3\,b^2\,x^{9/2}\,\left (4\,A\,b+3\,B\,a\right )}{9}+\frac {10\,a^2\,b^3\,x^{11/2}\,\left (3\,A\,b+4\,B\,a\right )}{11}+\frac {6\,a^4\,b\,x^{7/2}\,\left (5\,A\,b+2\,B\,a\right )}{7}+\frac {6\,a\,b^4\,x^{13/2}\,\left (2\,A\,b+5\,B\,a\right )}{13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)*(A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^3,x)

[Out]

x^(5/2)*((2*B*a^6)/5 + (12*A*a^5*b)/5) + x^(15/2)*((2*A*b^6)/15 + (4*B*a*b^5)/5) + (2*A*a^6*x^(3/2))/3 + (2*B*
b^6*x^(17/2))/17 + (10*a^3*b^2*x^(9/2)*(4*A*b + 3*B*a))/9 + (10*a^2*b^3*x^(11/2)*(3*A*b + 4*B*a))/11 + (6*a^4*
b*x^(7/2)*(5*A*b + 2*B*a))/7 + (6*a*b^4*x^(13/2)*(2*A*b + 5*B*a))/13

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sympy [A]  time = 5.91, size = 182, normalized size = 1.14 \begin {gather*} \frac {2 A a^{6} x^{\frac {3}{2}}}{3} + \frac {2 B b^{6} x^{\frac {17}{2}}}{17} + \frac {2 x^{\frac {15}{2}} \left (A b^{6} + 6 B a b^{5}\right )}{15} + \frac {2 x^{\frac {13}{2}} \left (6 A a b^{5} + 15 B a^{2} b^{4}\right )}{13} + \frac {2 x^{\frac {11}{2}} \left (15 A a^{2} b^{4} + 20 B a^{3} b^{3}\right )}{11} + \frac {2 x^{\frac {9}{2}} \left (20 A a^{3} b^{3} + 15 B a^{4} b^{2}\right )}{9} + \frac {2 x^{\frac {7}{2}} \left (15 A a^{4} b^{2} + 6 B a^{5} b\right )}{7} + \frac {2 x^{\frac {5}{2}} \left (6 A a^{5} b + B a^{6}\right )}{5} \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**(1/2),x)

[Out]

2*A*a**6*x**(3/2)/3 + 2*B*b**6*x**(17/2)/17 + 2*x**(15/2)*(A*b**6 + 6*B*a*b**5)/15 + 2*x**(13/2)*(6*A*a*b**5 +
15*B*a**2*b**4)/13 + 2*x**(11/2)*(15*A*a**2*b**4 + 20*B*a**3*b**3)/11 + 2*x**(9/2)*(20*A*a**3*b**3 + 15*B*a**
4*b**2)/9 + 2*x**(7/2)*(15*A*a**4*b**2 + 6*B*a**5*b)/7 + 2*x**(5/2)*(6*A*a**5*b + B*a**6)/5

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