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

Optimal. Leaf size=85 \[ \frac {2}{5} a^3 A x^{5/2}+\frac {2}{7} a^2 x^{7/2} (a B+3 A b)+\frac {2}{11} b^2 x^{11/2} (3 a B+A b)+\frac {2}{3} a b x^{9/2} (a B+A b)+\frac {2}{13} b^3 B x^{13/2} \]

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Rubi [A]  time = 0.04, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {76} \begin {gather*} \frac {2}{7} a^2 x^{7/2} (a B+3 A b)+\frac {2}{5} a^3 A x^{5/2}+\frac {2}{11} b^2 x^{11/2} (3 a B+A b)+\frac {2}{3} a b x^{9/2} (a B+A b)+\frac {2}{13} b^3 B x^{13/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(3/2)*(a + b*x)^3*(A + B*x),x]

[Out]

(2*a^3*A*x^(5/2))/5 + (2*a^2*(3*A*b + a*B)*x^(7/2))/7 + (2*a*b*(A*b + a*B)*x^(9/2))/3 + (2*b^2*(A*b + 3*a*B)*x
^(11/2))/11 + (2*b^3*B*x^(13/2))/13

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

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Mathematica [A]  time = 0.03, size = 71, normalized size = 0.84 \begin {gather*} \frac {2 x^{5/2} \left (429 a^3 (7 A+5 B x)+715 a^2 b x (9 A+7 B x)+455 a b^2 x^2 (11 A+9 B x)+105 b^3 x^3 (13 A+11 B x)\right )}{15015} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(5/2)*(429*a^3*(7*A + 5*B*x) + 715*a^2*b*x*(9*A + 7*B*x) + 455*a*b^2*x^2*(11*A + 9*B*x) + 105*b^3*x^3*(13
*A + 11*B*x)))/15015

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IntegrateAlgebraic [A]  time = 0.04, size = 97, normalized size = 1.14 \begin {gather*} \frac {2 \left (3003 a^3 A x^{5/2}+2145 a^3 B x^{7/2}+6435 a^2 A b x^{7/2}+5005 a^2 b B x^{9/2}+5005 a A b^2 x^{9/2}+4095 a b^2 B x^{11/2}+1365 A b^3 x^{11/2}+1155 b^3 B x^{13/2}\right )}{15015} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(3003*a^3*A*x^(5/2) + 6435*a^2*A*b*x^(7/2) + 2145*a^3*B*x^(7/2) + 5005*a*A*b^2*x^(9/2) + 5005*a^2*b*B*x^(9/
2) + 1365*A*b^3*x^(11/2) + 4095*a*b^2*B*x^(11/2) + 1155*b^3*B*x^(13/2)))/15015

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fricas [A]  time = 1.02, size = 78, normalized size = 0.92 \begin {gather*} \frac {2}{15015} \, {\left (1155 \, B b^{3} x^{6} + 3003 \, A a^{3} x^{2} + 1365 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{5} + 5005 \, {\left (B a^{2} b + A a b^{2}\right )} x^{4} + 2145 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{3}\right )} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15015*(1155*B*b^3*x^6 + 3003*A*a^3*x^2 + 1365*(3*B*a*b^2 + A*b^3)*x^5 + 5005*(B*a^2*b + A*a*b^2)*x^4 + 2145*
(B*a^3 + 3*A*a^2*b)*x^3)*sqrt(x)

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giac [A]  time = 1.21, size = 77, normalized size = 0.91 \begin {gather*} \frac {2}{13} \, B b^{3} x^{\frac {13}{2}} + \frac {6}{11} \, B a b^{2} x^{\frac {11}{2}} + \frac {2}{11} \, A b^{3} x^{\frac {11}{2}} + \frac {2}{3} \, B a^{2} b x^{\frac {9}{2}} + \frac {2}{3} \, A a b^{2} x^{\frac {9}{2}} + \frac {2}{7} \, B a^{3} x^{\frac {7}{2}} + \frac {6}{7} \, A a^{2} b x^{\frac {7}{2}} + \frac {2}{5} \, A a^{3} x^{\frac {5}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 76, normalized size = 0.89 \begin {gather*} \frac {2 \left (1155 B \,b^{3} x^{4}+1365 A \,b^{3} x^{3}+4095 B a \,b^{2} x^{3}+5005 A a \,b^{2} x^{2}+5005 B \,a^{2} b \,x^{2}+6435 A \,a^{2} b x +2145 B \,a^{3} x +3003 a^{3} A \right ) x^{\frac {5}{2}}}{15015} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15015*x^(5/2)*(1155*B*b^3*x^4+1365*A*b^3*x^3+4095*B*a*b^2*x^3+5005*A*a*b^2*x^2+5005*B*a^2*b*x^2+6435*A*a^2*b
*x+2145*B*a^3*x+3003*A*a^3)

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maxima [A]  time = 0.85, size = 73, normalized size = 0.86 \begin {gather*} \frac {2}{13} \, B b^{3} x^{\frac {13}{2}} + \frac {2}{5} \, A a^{3} x^{\frac {5}{2}} + \frac {2}{11} \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac {11}{2}} + \frac {2}{3} \, {\left (B a^{2} b + A a b^{2}\right )} x^{\frac {9}{2}} + \frac {2}{7} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac {7}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/13*B*b^3*x^(13/2) + 2/5*A*a^3*x^(5/2) + 2/11*(3*B*a*b^2 + A*b^3)*x^(11/2) + 2/3*(B*a^2*b + A*a*b^2)*x^(9/2)
+ 2/7*(B*a^3 + 3*A*a^2*b)*x^(7/2)

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mupad [B]  time = 0.03, size = 69, normalized size = 0.81 \begin {gather*} x^{7/2}\,\left (\frac {2\,B\,a^3}{7}+\frac {6\,A\,b\,a^2}{7}\right )+x^{11/2}\,\left (\frac {2\,A\,b^3}{11}+\frac {6\,B\,a\,b^2}{11}\right )+\frac {2\,A\,a^3\,x^{5/2}}{5}+\frac {2\,B\,b^3\,x^{13/2}}{13}+\frac {2\,a\,b\,x^{9/2}\,\left (A\,b+B\,a\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(3/2)*(A + B*x)*(a + b*x)^3,x)

[Out]

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

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sympy [A]  time = 3.04, size = 114, normalized size = 1.34 \begin {gather*} \frac {2 A a^{3} x^{\frac {5}{2}}}{5} + \frac {6 A a^{2} b x^{\frac {7}{2}}}{7} + \frac {2 A a b^{2} x^{\frac {9}{2}}}{3} + \frac {2 A b^{3} x^{\frac {11}{2}}}{11} + \frac {2 B a^{3} x^{\frac {7}{2}}}{7} + \frac {2 B a^{2} b x^{\frac {9}{2}}}{3} + \frac {6 B a b^{2} x^{\frac {11}{2}}}{11} + \frac {2 B b^{3} x^{\frac {13}{2}}}{13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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