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

Optimal. Leaf size=83 \[ 2 A b^3 \sqrt {x}+\frac {2}{3} b^2 x^{3/2} (3 A c+b B)+\frac {2}{7} c^2 x^{7/2} (A c+3 b B)+\frac {6}{5} b c x^{5/2} (A c+b B)+\frac {2}{9} B c^3 x^{9/2} \]

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Rubi [A]  time = 0.04, antiderivative size = 83, 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}{3} b^2 x^{3/2} (3 A c+b B)+2 A b^3 \sqrt {x}+\frac {2}{7} c^2 x^{7/2} (A c+3 b B)+\frac {6}{5} b c x^{5/2} (A c+b B)+\frac {2}{9} B c^3 x^{9/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*A*b^3*Sqrt[x] + (2*b^2*(b*B + 3*A*c)*x^(3/2))/3 + (6*b*c*(b*B + A*c)*x^(5/2))/5 + (2*c^2*(3*b*B + A*c)*x^(7/
2))/7 + (2*B*c^3*x^(9/2))/9

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

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Mathematica [A]  time = 0.04, size = 63, normalized size = 0.76 \begin {gather*} \frac {2 \sqrt {x} \left (35 B (b+c x)^4-\left (35 b^3+35 b^2 c x+21 b c^2 x^2+5 c^3 x^3\right ) (b B-9 A c)\right )}{315 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[x]*(35*B*(b + c*x)^4 - (b*B - 9*A*c)*(35*b^3 + 35*b^2*c*x + 21*b*c^2*x^2 + 5*c^3*x^3)))/(315*c)

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IntegrateAlgebraic [A]  time = 0.04, size = 97, normalized size = 1.17 \begin {gather*} \frac {2}{315} \left (315 A b^3 \sqrt {x}+315 A b^2 c x^{3/2}+189 A b c^2 x^{5/2}+45 A c^3 x^{7/2}+105 b^3 B x^{3/2}+189 b^2 B c x^{5/2}+135 b B c^2 x^{7/2}+35 B c^3 x^{9/2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(315*A*b^3*Sqrt[x] + 105*b^3*B*x^(3/2) + 315*A*b^2*c*x^(3/2) + 189*b^2*B*c*x^(5/2) + 189*A*b*c^2*x^(5/2) +
135*b*B*c^2*x^(7/2) + 45*A*c^3*x^(7/2) + 35*B*c^3*x^(9/2)))/315

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fricas [A]  time = 0.40, size = 73, normalized size = 0.88 \begin {gather*} \frac {2}{315} \, {\left (35 \, B c^{3} x^{4} + 315 \, A b^{3} + 45 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{3} + 189 \, {\left (B b^{2} c + A b c^{2}\right )} x^{2} + 105 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x\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^(7/2),x, algorithm="fricas")

[Out]

2/315*(35*B*c^3*x^4 + 315*A*b^3 + 45*(3*B*b*c^2 + A*c^3)*x^3 + 189*(B*b^2*c + A*b*c^2)*x^2 + 105*(B*b^3 + 3*A*
b^2*c)*x)*sqrt(x)

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giac [A]  time = 0.16, size = 77, normalized size = 0.93 \begin {gather*} \frac {2}{9} \, B c^{3} x^{\frac {9}{2}} + \frac {6}{7} \, B b c^{2} x^{\frac {7}{2}} + \frac {2}{7} \, A c^{3} x^{\frac {7}{2}} + \frac {6}{5} \, B b^{2} c x^{\frac {5}{2}} + \frac {6}{5} \, A b c^{2} x^{\frac {5}{2}} + \frac {2}{3} \, B b^{3} x^{\frac {3}{2}} + 2 \, A b^{2} c x^{\frac {3}{2}} + 2 \, A b^{3} \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^(7/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.05, size = 76, normalized size = 0.92 \begin {gather*} \frac {2 \left (35 B \,c^{3} x^{4}+45 A \,c^{3} x^{3}+135 B b \,c^{2} x^{3}+189 A b \,c^{2} x^{2}+189 B \,b^{2} c \,x^{2}+315 A \,b^{2} c x +105 B \,b^{3} x +315 A \,b^{3}\right ) \sqrt {x}}{315} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*x^(1/2)*(35*B*c^3*x^4+45*A*c^3*x^3+135*B*b*c^2*x^3+189*A*b*c^2*x^2+189*B*b^2*c*x^2+315*A*b^2*c*x+105*B*b
^3*x+315*A*b^3)

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maxima [A]  time = 0.49, size = 73, normalized size = 0.88 \begin {gather*} \frac {2}{9} \, B c^{3} x^{\frac {9}{2}} + 2 \, A b^{3} \sqrt {x} + \frac {2}{7} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac {7}{2}} + \frac {6}{5} \, {\left (B b^{2} c + A b c^{2}\right )} x^{\frac {5}{2}} + \frac {2}{3} \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{\frac {3}{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^(7/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.03, size = 69, normalized size = 0.83 \begin {gather*} x^{3/2}\,\left (\frac {2\,B\,b^3}{3}+2\,A\,c\,b^2\right )+x^{7/2}\,\left (\frac {2\,A\,c^3}{7}+\frac {6\,B\,b\,c^2}{7}\right )+2\,A\,b^3\,\sqrt {x}+\frac {2\,B\,c^3\,x^{9/2}}{9}+\frac {6\,b\,c\,x^{5/2}\,\left (A\,c+B\,b\right )}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^(3/2)*((2*B*b^3)/3 + 2*A*b^2*c) + x^(7/2)*((2*A*c^3)/7 + (6*B*b*c^2)/7) + 2*A*b^3*x^(1/2) + (2*B*c^3*x^(9/2)
)/9 + (6*b*c*x^(5/2)*(A*c + B*b))/5

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sympy [A]  time = 5.97, size = 110, normalized size = 1.33 \begin {gather*} 2 A b^{3} \sqrt {x} + 2 A b^{2} c x^{\frac {3}{2}} + \frac {6 A b c^{2} x^{\frac {5}{2}}}{5} + \frac {2 A c^{3} x^{\frac {7}{2}}}{7} + \frac {2 B b^{3} x^{\frac {3}{2}}}{3} + \frac {6 B b^{2} c x^{\frac {5}{2}}}{5} + \frac {6 B b c^{2} x^{\frac {7}{2}}}{7} + \frac {2 B c^{3} x^{\frac {9}{2}}}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*A*b**3*sqrt(x) + 2*A*b**2*c*x**(3/2) + 6*A*b*c**2*x**(5/2)/5 + 2*A*c**3*x**(7/2)/7 + 2*B*b**3*x**(3/2)/3 + 6
*B*b**2*c*x**(5/2)/5 + 6*B*b*c**2*x**(7/2)/7 + 2*B*c**3*x**(9/2)/9

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