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

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

[Out]

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

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Rubi [A]  time = 0.05, 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} \[ \frac {2}{17} b^2 x^{17/2} (3 A c+b B)+\frac {2}{15} A b^3 x^{15/2}+\frac {2}{21} c^2 x^{21/2} (A c+3 b B)+\frac {6}{19} b c x^{19/2} (A c+b B)+\frac {2}{23} B c^3 x^{23/2} \]

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*}

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Mathematica [A]  time = 0.05, size = 70, normalized size = 0.82 \[ \frac {2 \left (B x^{15/2} (b+c x)^4-\frac {x^{15/2} \left (2261 b^3+5985 b^2 c x+5355 b c^2 x^2+1615 c^3 x^3\right ) (15 b B-23 A c)}{33915}\right )}{23 c} \]

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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fricas [A]  time = 0.99, size = 78, normalized size = 0.92 \[ \frac {2}{780045} \, {\left (33915 \, B c^{3} x^{11} + 52003 \, A b^{3} x^{7} + 37145 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{10} + 123165 \, {\left (B b^{2} c + A b c^{2}\right )} x^{9} + 45885 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{8}\right )} \sqrt {x} \]

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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giac [A]  time = 0.16, size = 77, normalized size = 0.91 \[ \frac {2}{23} \, B c^{3} x^{\frac {23}{2}} + \frac {2}{7} \, B b c^{2} x^{\frac {21}{2}} + \frac {2}{21} \, A c^{3} x^{\frac {21}{2}} + \frac {6}{19} \, B b^{2} c x^{\frac {19}{2}} + \frac {6}{19} \, A b c^{2} x^{\frac {19}{2}} + \frac {2}{17} \, B b^{3} x^{\frac {17}{2}} + \frac {6}{17} \, A b^{2} c x^{\frac {17}{2}} + \frac {2}{15} \, A b^{3} x^{\frac {15}{2}} \]

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)

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maple [A]  time = 0.05, size = 76, normalized size = 0.89 \[ \frac {2 \left (33915 B \,c^{3} x^{4}+37145 A \,c^{3} x^{3}+111435 B b \,c^{2} x^{3}+123165 A b \,c^{2} x^{2}+123165 B \,b^{2} c \,x^{2}+137655 A \,b^{2} c x +45885 B \,b^{3} x +52003 A \,b^{3}\right ) x^{\frac {15}{2}}}{780045} \]

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 = 0.47, size = 73, normalized size = 0.86 \[ \frac {2}{23} \, B c^{3} x^{\frac {23}{2}} + \frac {2}{15} \, A b^{3} x^{\frac {15}{2}} + \frac {2}{21} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac {21}{2}} + \frac {6}{19} \, {\left (B b^{2} c + A b c^{2}\right )} x^{\frac {19}{2}} + \frac {2}{17} \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{\frac {17}{2}} \]

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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mupad [B]  time = 1.01, size = 69, normalized size = 0.81 \[ x^{17/2}\,\left (\frac {2\,B\,b^3}{17}+\frac {6\,A\,c\,b^2}{17}\right )+x^{21/2}\,\left (\frac {2\,A\,c^3}{21}+\frac {2\,B\,b\,c^2}{7}\right )+\frac {2\,A\,b^3\,x^{15/2}}{15}+\frac {2\,B\,c^3\,x^{23/2}}{23}+\frac {6\,b\,c\,x^{19/2}\,\left (A\,c+B\,b\right )}{19} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 26.59, size = 114, normalized size = 1.34 \[ \frac {2 A b^{3} x^{\frac {15}{2}}}{15} + \frac {6 A b^{2} c x^{\frac {17}{2}}}{17} + \frac {6 A b c^{2} x^{\frac {19}{2}}}{19} + \frac {2 A c^{3} x^{\frac {21}{2}}}{21} + \frac {2 B b^{3} x^{\frac {17}{2}}}{17} + \frac {6 B b^{2} c x^{\frac {19}{2}}}{19} + \frac {2 B b c^{2} x^{\frac {21}{2}}}{7} + \frac {2 B c^{3} x^{\frac {23}{2}}}{23} \]

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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