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

Optimal. Leaf size=85 \[ \frac {2}{9} A b^3 x^{9/2}+\frac {2}{11} b^2 x^{11/2} (3 A c+b B)+\frac {2}{15} c^2 x^{15/2} (A c+3 b B)+\frac {6}{13} b c x^{13/2} (A c+b B)+\frac {2}{17} B c^3 x^{17/2} \]

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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 = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {765} \begin {gather*} \frac {2}{11} b^2 x^{11/2} (3 A c+b B)+\frac {2}{9} A b^3 x^{9/2}+\frac {2}{15} c^2 x^{15/2} (A c+3 b B)+\frac {6}{13} b c x^{13/2} (A c+b B)+\frac {2}{17} B c^3 x^{17/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*A*b^3*x^(9/2))/9 + (2*b^2*(b*B + 3*A*c)*x^(11/2))/11 + (6*b*c*(b*B + A*c)*x^(13/2))/13 + (2*c^2*(3*b*B + A*
c)*x^(15/2))/15 + (2*B*c^3*x^(17/2))/17

Rule 765

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand
Integrand[(e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, e, f, g, m}, x] && IntegerQ[p] && (
GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

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Mathematica [A]  time = 0.05, size = 70, normalized size = 0.82 \begin {gather*} \frac {2 \left (B x^{9/2} (b+c x)^4-\frac {x^{9/2} \left (715 b^3+1755 b^2 c x+1485 b c^2 x^2+429 c^3 x^3\right ) (9 b B-17 A c)}{6435}\right )}{17 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(B*x^(9/2)*(b + c*x)^4 - ((9*b*B - 17*A*c)*x^(9/2)*(715*b^3 + 1755*b^2*c*x + 1485*b*c^2*x^2 + 429*c^3*x^3))
/6435))/(17*c)

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IntegrateAlgebraic [A]  time = 0.05, size = 97, normalized size = 1.14 \begin {gather*} \frac {2 \left (12155 A b^3 x^{9/2}+29835 A b^2 c x^{11/2}+25245 A b c^2 x^{13/2}+7293 A c^3 x^{15/2}+9945 b^3 B x^{11/2}+25245 b^2 B c x^{13/2}+21879 b B c^2 x^{15/2}+6435 B c^3 x^{17/2}\right )}{109395} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(12155*A*b^3*x^(9/2) + 9945*b^3*B*x^(11/2) + 29835*A*b^2*c*x^(11/2) + 25245*b^2*B*c*x^(13/2) + 25245*A*b*c^
2*x^(13/2) + 21879*b*B*c^2*x^(15/2) + 7293*A*c^3*x^(15/2) + 6435*B*c^3*x^(17/2)))/109395

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fricas [A]  time = 0.38, size = 78, normalized size = 0.92 \begin {gather*} \frac {2}{109395} \, {\left (6435 \, B c^{3} x^{8} + 12155 \, A b^{3} x^{4} + 7293 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{7} + 25245 \, {\left (B b^{2} c + A b c^{2}\right )} x^{6} + 9945 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{5}\right )} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/109395*(6435*B*c^3*x^8 + 12155*A*b^3*x^4 + 7293*(3*B*b*c^2 + A*c^3)*x^7 + 25245*(B*b^2*c + A*b*c^2)*x^6 + 99
45*(B*b^3 + 3*A*b^2*c)*x^5)*sqrt(x)

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giac [A]  time = 0.16, size = 77, normalized size = 0.91 \begin {gather*} \frac {2}{17} \, B c^{3} x^{\frac {17}{2}} + \frac {2}{5} \, B b c^{2} x^{\frac {15}{2}} + \frac {2}{15} \, A c^{3} x^{\frac {15}{2}} + \frac {6}{13} \, B b^{2} c x^{\frac {13}{2}} + \frac {6}{13} \, A b c^{2} x^{\frac {13}{2}} + \frac {2}{11} \, B b^{3} x^{\frac {11}{2}} + \frac {6}{11} \, A b^{2} c x^{\frac {11}{2}} + \frac {2}{9} \, A b^{3} x^{\frac {9}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/17*B*c^3*x^(17/2) + 2/5*B*b*c^2*x^(15/2) + 2/15*A*c^3*x^(15/2) + 6/13*B*b^2*c*x^(13/2) + 6/13*A*b*c^2*x^(13/
2) + 2/11*B*b^3*x^(11/2) + 6/11*A*b^2*c*x^(11/2) + 2/9*A*b^3*x^(9/2)

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maple [A]  time = 0.05, size = 76, normalized size = 0.89 \begin {gather*} \frac {2 \left (6435 B \,c^{3} x^{4}+7293 A \,c^{3} x^{3}+21879 B b \,c^{2} x^{3}+25245 A b \,c^{2} x^{2}+25245 B \,b^{2} c \,x^{2}+29835 A \,b^{2} c x +9945 B \,b^{3} x +12155 A \,b^{3}\right ) x^{\frac {9}{2}}}{109395} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/109395*x^(9/2)*(6435*B*c^3*x^4+7293*A*c^3*x^3+21879*B*b*c^2*x^3+25245*A*b*c^2*x^2+25245*B*b^2*c*x^2+29835*A*
b^2*c*x+9945*B*b^3*x+12155*A*b^3)

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maxima [A]  time = 0.68, size = 73, normalized size = 0.86 \begin {gather*} \frac {2}{17} \, B c^{3} x^{\frac {17}{2}} + \frac {2}{9} \, A b^{3} x^{\frac {9}{2}} + \frac {2}{15} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac {15}{2}} + \frac {6}{13} \, {\left (B b^{2} c + A b c^{2}\right )} x^{\frac {13}{2}} + \frac {2}{11} \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{\frac {11}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/17*B*c^3*x^(17/2) + 2/9*A*b^3*x^(9/2) + 2/15*(3*B*b*c^2 + A*c^3)*x^(15/2) + 6/13*(B*b^2*c + A*b*c^2)*x^(13/2
) + 2/11*(B*b^3 + 3*A*b^2*c)*x^(11/2)

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mupad [B]  time = 0.03, size = 69, normalized size = 0.81 \begin {gather*} x^{11/2}\,\left (\frac {2\,B\,b^3}{11}+\frac {6\,A\,c\,b^2}{11}\right )+x^{15/2}\,\left (\frac {2\,A\,c^3}{15}+\frac {2\,B\,b\,c^2}{5}\right )+\frac {2\,A\,b^3\,x^{9/2}}{9}+\frac {2\,B\,c^3\,x^{17/2}}{17}+\frac {6\,b\,c\,x^{13/2}\,\left (A\,c+B\,b\right )}{13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^(11/2)*((2*B*b^3)/11 + (6*A*b^2*c)/11) + x^(15/2)*((2*A*c^3)/15 + (2*B*b*c^2)/5) + (2*A*b^3*x^(9/2))/9 + (2*
B*c^3*x^(17/2))/17 + (6*b*c*x^(13/2)*(A*c + B*b))/13

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sympy [A]  time = 3.76, size = 95, normalized size = 1.12 \begin {gather*} \frac {2 A b^{3} x^{\frac {9}{2}}}{9} + \frac {2 B c^{3} x^{\frac {17}{2}}}{17} + \frac {2 x^{\frac {15}{2}} \left (A c^{3} + 3 B b c^{2}\right )}{15} + \frac {2 x^{\frac {13}{2}} \left (3 A b c^{2} + 3 B b^{2} c\right )}{13} + \frac {2 x^{\frac {11}{2}} \left (3 A b^{2} c + B b^{3}\right )}{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*A*b**3*x**(9/2)/9 + 2*B*c**3*x**(17/2)/17 + 2*x**(15/2)*(A*c**3 + 3*B*b*c**2)/15 + 2*x**(13/2)*(3*A*b*c**2 +
3*B*b**2*c)/13 + 2*x**(11/2)*(3*A*b**2*c + B*b**3)/11

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