3.158 \(\int x^{7/2} (A+B x) (b x+c x^2)^3 \, dx\)

Optimal. Leaf size=85 \[ \frac{2}{17} b^2 x^{17/2} (3 A c+b B)+\frac{2}{15} A b^3 x^{15/2}+\frac{2}{21} c^2 x^{21/2} (A c+3 b B)+\frac{6}{19} b c x^{19/2} (A c+b B)+\frac{2}{23} B c^3 x^{23/2} \]

[Out]

(2*A*b^3*x^(15/2))/15 + (2*b^2*(b*B + 3*A*c)*x^(17/2))/17 + (6*b*c*(b*B + A*c)*x^(19/2))/19 + (2*c^2*(3*b*B +
A*c)*x^(21/2))/21 + (2*B*c^3*x^(23/2))/23

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Rubi [A]  time = 0.0475405, antiderivative size = 85, normalized size of antiderivative = 1., 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} \[ \frac{2}{17} b^2 x^{17/2} (3 A c+b B)+\frac{2}{15} A b^3 x^{15/2}+\frac{2}{21} c^2 x^{21/2} (A c+3 b B)+\frac{6}{19} b c x^{19/2} (A c+b B)+\frac{2}{23} B c^3 x^{23/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(7/2)*(A + B*x)*(b*x + c*x^2)^3,x]

[Out]

(2*A*b^3*x^(15/2))/15 + (2*b^2*(b*B + 3*A*c)*x^(17/2))/17 + (6*b*c*(b*B + A*c)*x^(19/2))/19 + (2*c^2*(3*b*B +
A*c)*x^(21/2))/21 + (2*B*c^3*x^(23/2))/23

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 x^{7/2} (A+B x) \left (b x+c x^2\right )^3 \, dx &=\int \left (A b^3 x^{13/2}+b^2 (b B+3 A c) x^{15/2}+3 b c (b B+A c) x^{17/2}+c^2 (3 b B+A c) x^{19/2}+B c^3 x^{21/2}\right ) \, dx\\ &=\frac{2}{15} A b^3 x^{15/2}+\frac{2}{17} b^2 (b B+3 A c) x^{17/2}+\frac{6}{19} b c (b B+A c) x^{19/2}+\frac{2}{21} c^2 (3 b B+A c) x^{21/2}+\frac{2}{23} B c^3 x^{23/2}\\ \end{align*}

Mathematica [A]  time = 0.0472709, size = 70, normalized size = 0.82 \[ \frac{2 \left (B x^{15/2} (b+c x)^4-\frac{x^{15/2} \left (5985 b^2 c x+2261 b^3+5355 b c^2 x^2+1615 c^3 x^3\right ) (15 b B-23 A c)}{33915}\right )}{23 c} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(7/2)*(A + B*x)*(b*x + c*x^2)^3,x]

[Out]

(2*(B*x^(15/2)*(b + c*x)^4 - ((15*b*B - 23*A*c)*x^(15/2)*(2261*b^3 + 5985*b^2*c*x + 5355*b*c^2*x^2 + 1615*c^3*
x^3))/33915))/(23*c)

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Maple [A]  time = 0.005, size = 76, normalized size = 0.9 \begin{align*}{\frac{67830\,B{c}^{3}{x}^{4}+74290\,A{x}^{3}{c}^{3}+222870\,B{x}^{3}b{c}^{2}+246330\,A{x}^{2}b{c}^{2}+246330\,B{x}^{2}{b}^{2}c+275310\,A{b}^{2}cx+91770\,{b}^{3}Bx+104006\,A{b}^{3}}{780045}{x}^{{\frac{15}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)*(B*x+A)*(c*x^2+b*x)^3,x)

[Out]

2/780045*x^(15/2)*(33915*B*c^3*x^4+37145*A*c^3*x^3+111435*B*b*c^2*x^3+123165*A*b*c^2*x^2+123165*B*b^2*c*x^2+13
7655*A*b^2*c*x+45885*B*b^3*x+52003*A*b^3)

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Maxima [A]  time = 1.09783, size = 99, normalized size = 1.16 \begin{align*} \frac{2}{23} \, B c^{3} x^{\frac{23}{2}} + \frac{2}{15} \, A b^{3} x^{\frac{15}{2}} + \frac{2}{21} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac{21}{2}} + \frac{6}{19} \,{\left (B b^{2} c + A b c^{2}\right )} x^{\frac{19}{2}} + \frac{2}{17} \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x^{\frac{17}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(B*x+A)*(c*x^2+b*x)^3,x, algorithm="maxima")

[Out]

2/23*B*c^3*x^(23/2) + 2/15*A*b^3*x^(15/2) + 2/21*(3*B*b*c^2 + A*c^3)*x^(21/2) + 6/19*(B*b^2*c + A*b*c^2)*x^(19
/2) + 2/17*(B*b^3 + 3*A*b^2*c)*x^(17/2)

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Fricas [A]  time = 1.8574, size = 207, normalized size = 2.44 \begin{align*} \frac{2}{780045} \,{\left (33915 \, B c^{3} x^{11} + 52003 \, A b^{3} x^{7} + 37145 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{10} + 123165 \,{\left (B b^{2} c + A b c^{2}\right )} x^{9} + 45885 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x^{8}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(B*x+A)*(c*x^2+b*x)^3,x, algorithm="fricas")

[Out]

2/780045*(33915*B*c^3*x^11 + 52003*A*b^3*x^7 + 37145*(3*B*b*c^2 + A*c^3)*x^10 + 123165*(B*b^2*c + A*b*c^2)*x^9
+ 45885*(B*b^3 + 3*A*b^2*c)*x^8)*sqrt(x)

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Sympy [A]  time = 30.9051, size = 114, normalized size = 1.34 \begin{align*} \frac{2 A b^{3} x^{\frac{15}{2}}}{15} + \frac{6 A b^{2} c x^{\frac{17}{2}}}{17} + \frac{6 A b c^{2} x^{\frac{19}{2}}}{19} + \frac{2 A c^{3} x^{\frac{21}{2}}}{21} + \frac{2 B b^{3} x^{\frac{17}{2}}}{17} + \frac{6 B b^{2} c x^{\frac{19}{2}}}{19} + \frac{2 B b c^{2} x^{\frac{21}{2}}}{7} + \frac{2 B c^{3} x^{\frac{23}{2}}}{23} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(7/2)*(B*x+A)*(c*x**2+b*x)**3,x)

[Out]

2*A*b**3*x**(15/2)/15 + 6*A*b**2*c*x**(17/2)/17 + 6*A*b*c**2*x**(19/2)/19 + 2*A*c**3*x**(21/2)/21 + 2*B*b**3*x
**(17/2)/17 + 6*B*b**2*c*x**(19/2)/19 + 2*B*b*c**2*x**(21/2)/7 + 2*B*c**3*x**(23/2)/23

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Giac [A]  time = 1.12623, size = 104, normalized size = 1.22 \begin{align*} \frac{2}{23} \, B c^{3} x^{\frac{23}{2}} + \frac{2}{7} \, B b c^{2} x^{\frac{21}{2}} + \frac{2}{21} \, A c^{3} x^{\frac{21}{2}} + \frac{6}{19} \, B b^{2} c x^{\frac{19}{2}} + \frac{6}{19} \, A b c^{2} x^{\frac{19}{2}} + \frac{2}{17} \, B b^{3} x^{\frac{17}{2}} + \frac{6}{17} \, A b^{2} c x^{\frac{17}{2}} + \frac{2}{15} \, A b^{3} x^{\frac{15}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(B*x+A)*(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

2/23*B*c^3*x^(23/2) + 2/7*B*b*c^2*x^(21/2) + 2/21*A*c^3*x^(21/2) + 6/19*B*b^2*c*x^(19/2) + 6/19*A*b*c^2*x^(19/
2) + 2/17*B*b^3*x^(17/2) + 6/17*A*b^2*c*x^(17/2) + 2/15*A*b^3*x^(15/2)