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

Optimal. Leaf size=85 \[ \frac {2}{3} a^3 A x^{3/2}+\frac {2}{5} a^2 x^{5/2} (a B+3 A b)+\frac {2}{9} b^2 x^{9/2} (3 a B+A b)+\frac {6}{7} a b x^{7/2} (a B+A b)+\frac {2}{11} b^3 B x^{11/2} \]

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

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]*(a + b*x)^3*(A + B*x),x]

[Out]

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

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

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Mathematica [A]  time = 0.03, size = 71, normalized size = 0.84 \begin {gather*} \frac {2 x^{3/2} \left (231 a^3 (5 A+3 B x)+297 a^2 b x (7 A+5 B x)+165 a b^2 x^2 (9 A+7 B x)+35 b^3 x^3 (11 A+9 B x)\right )}{3465} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(3/2)*(231*a^3*(5*A + 3*B*x) + 297*a^2*b*x*(7*A + 5*B*x) + 165*a*b^2*x^2*(9*A + 7*B*x) + 35*b^3*x^3*(11*A
+ 9*B*x)))/3465

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IntegrateAlgebraic [A]  time = 0.04, size = 97, normalized size = 1.14 \begin {gather*} \frac {2 \left (1155 a^3 A x^{3/2}+693 a^3 B x^{5/2}+2079 a^2 A b x^{5/2}+1485 a^2 b B x^{7/2}+1485 a A b^2 x^{7/2}+1155 a b^2 B x^{9/2}+385 A b^3 x^{9/2}+315 b^3 B x^{11/2}\right )}{3465} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(1155*a^3*A*x^(3/2) + 2079*a^2*A*b*x^(5/2) + 693*a^3*B*x^(5/2) + 1485*a*A*b^2*x^(7/2) + 1485*a^2*b*B*x^(7/2
) + 385*A*b^3*x^(9/2) + 1155*a*b^2*B*x^(9/2) + 315*b^3*B*x^(11/2)))/3465

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fricas [A]  time = 0.87, size = 76, normalized size = 0.89 \begin {gather*} \frac {2}{3465} \, {\left (315 \, B b^{3} x^{5} + 1155 \, A a^{3} x + 385 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{4} + 1485 \, {\left (B a^{2} b + A a b^{2}\right )} x^{3} + 693 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{2}\right )} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)*x^(1/2),x, algorithm="fricas")

[Out]

2/3465*(315*B*b^3*x^5 + 1155*A*a^3*x + 385*(3*B*a*b^2 + A*b^3)*x^4 + 1485*(B*a^2*b + A*a*b^2)*x^3 + 693*(B*a^3
+ 3*A*a^2*b)*x^2)*sqrt(x)

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giac [A]  time = 1.22, size = 77, normalized size = 0.91 \begin {gather*} \frac {2}{11} \, B b^{3} x^{\frac {11}{2}} + \frac {2}{3} \, B a b^{2} x^{\frac {9}{2}} + \frac {2}{9} \, A b^{3} x^{\frac {9}{2}} + \frac {6}{7} \, B a^{2} b x^{\frac {7}{2}} + \frac {6}{7} \, A a b^{2} x^{\frac {7}{2}} + \frac {2}{5} \, B a^{3} x^{\frac {5}{2}} + \frac {6}{5} \, A a^{2} b x^{\frac {5}{2}} + \frac {2}{3} \, A a^{3} x^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)*x^(1/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.01, size = 76, normalized size = 0.89 \begin {gather*} \frac {2 \left (315 B \,b^{3} x^{4}+385 A \,b^{3} x^{3}+1155 B a \,b^{2} x^{3}+1485 A a \,b^{2} x^{2}+1485 B \,a^{2} b \,x^{2}+2079 A \,a^{2} b x +693 B \,a^{3} x +1155 a^{3} A \right ) x^{\frac {3}{2}}}{3465} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*x^(3/2)*(315*B*b^3*x^4+385*A*b^3*x^3+1155*B*a*b^2*x^3+1485*A*a*b^2*x^2+1485*B*a^2*b*x^2+2079*A*a^2*b*x+
693*B*a^3*x+1155*A*a^3)

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maxima [A]  time = 0.83, size = 73, normalized size = 0.86 \begin {gather*} \frac {2}{11} \, B b^{3} x^{\frac {11}{2}} + \frac {2}{3} \, A a^{3} x^{\frac {3}{2}} + \frac {2}{9} \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac {9}{2}} + \frac {6}{7} \, {\left (B a^{2} b + A a b^{2}\right )} x^{\frac {7}{2}} + \frac {2}{5} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac {5}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)*x^(1/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.04, size = 69, normalized size = 0.81 \begin {gather*} x^{5/2}\,\left (\frac {2\,B\,a^3}{5}+\frac {6\,A\,b\,a^2}{5}\right )+x^{9/2}\,\left (\frac {2\,A\,b^3}{9}+\frac {2\,B\,a\,b^2}{3}\right )+\frac {2\,A\,a^3\,x^{3/2}}{3}+\frac {2\,B\,b^3\,x^{11/2}}{11}+\frac {6\,a\,b\,x^{7/2}\,\left (A\,b+B\,a\right )}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)*(A + B*x)*(a + b*x)^3,x)

[Out]

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

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sympy [A]  time = 3.65, size = 95, normalized size = 1.12 \begin {gather*} \frac {2 A a^{3} x^{\frac {3}{2}}}{3} + \frac {2 B b^{3} x^{\frac {11}{2}}}{11} + \frac {2 x^{\frac {9}{2}} \left (A b^{3} + 3 B a b^{2}\right )}{9} + \frac {2 x^{\frac {7}{2}} \left (3 A a b^{2} + 3 B a^{2} b\right )}{7} + \frac {2 x^{\frac {5}{2}} \left (3 A a^{2} b + B a^{3}\right )}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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