3.2.64 \(\int \frac {(A+B x) (b x+c x^2)^3}{x^{5/2}} \, dx\)

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

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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 = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {765} \begin {gather*} \frac {2}{5} b^2 x^{5/2} (3 A c+b B)+\frac {2}{3} A b^3 x^{3/2}+\frac {2}{9} c^2 x^{9/2} (A c+3 b B)+\frac {6}{7} b c x^{7/2} (A c+b B)+\frac {2}{11} B c^3 x^{11/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2)^3)/x^(5/2),x]

[Out]

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

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

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Mathematica [A]  time = 0.04, size = 70, normalized size = 0.82 \begin {gather*} \frac {2 \left (B x^{3/2} (b+c x)^4-\frac {1}{315} x^{3/2} \left (105 b^3+189 b^2 c x+135 b c^2 x^2+35 c^3 x^3\right ) (3 b B-11 A c)\right )}{11 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(b*x + c*x^2)^3)/x^(5/2),x]

[Out]

(2*(B*x^(3/2)*(b + c*x)^4 - ((3*b*B - 11*A*c)*x^(3/2)*(105*b^3 + 189*b^2*c*x + 135*b*c^2*x^2 + 35*c^3*x^3))/31
5))/(11*c)

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

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*(b*x + c*x^2)^3)/x^(5/2),x]

[Out]

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

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fricas [A]  time = 0.40, size = 76, normalized size = 0.89 \begin {gather*} \frac {2}{3465} \, {\left (315 \, B c^{3} x^{5} + 1155 \, A b^{3} x + 385 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{4} + 1485 \, {\left (B b^{2} c + A b c^{2}\right )} x^{3} + 693 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{2}\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^(5/2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.15, size = 77, normalized size = 0.91 \begin {gather*} \frac {2}{11} \, B c^{3} x^{\frac {11}{2}} + \frac {2}{3} \, B b c^{2} x^{\frac {9}{2}} + \frac {2}{9} \, A c^{3} x^{\frac {9}{2}} + \frac {6}{7} \, B b^{2} c x^{\frac {7}{2}} + \frac {6}{7} \, A b c^{2} x^{\frac {7}{2}} + \frac {2}{5} \, B b^{3} x^{\frac {5}{2}} + \frac {6}{5} \, A b^{2} c x^{\frac {5}{2}} + \frac {2}{3} \, A b^{3} 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^(5/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.04, size = 76, normalized size = 0.89 \begin {gather*} \frac {2 \left (315 B \,c^{3} x^{4}+385 A \,c^{3} x^{3}+1155 B b \,c^{2} x^{3}+1485 A b \,c^{2} x^{2}+1485 B \,b^{2} c \,x^{2}+2079 A \,b^{2} c x +693 B \,b^{3} x +1155 A \,b^{3}\right ) x^{\frac {3}{2}}}{3465} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x)^3/x^(5/2),x)

[Out]

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

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maxima [A]  time = 0.65, size = 73, normalized size = 0.86 \begin {gather*} \frac {2}{11} \, B c^{3} x^{\frac {11}{2}} + \frac {2}{3} \, A b^{3} x^{\frac {3}{2}} + \frac {2}{9} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac {9}{2}} + \frac {6}{7} \, {\left (B b^{2} c + A b c^{2}\right )} x^{\frac {7}{2}} + \frac {2}{5} \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{\frac {5}{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^(5/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.03, size = 69, normalized size = 0.81 \begin {gather*} x^{5/2}\,\left (\frac {2\,B\,b^3}{5}+\frac {6\,A\,c\,b^2}{5}\right )+x^{9/2}\,\left (\frac {2\,A\,c^3}{9}+\frac {2\,B\,b\,c^2}{3}\right )+\frac {2\,A\,b^3\,x^{3/2}}{3}+\frac {2\,B\,c^3\,x^{11/2}}{11}+\frac {6\,b\,c\,x^{7/2}\,\left (A\,c+B\,b\right )}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b*x + c*x^2)^3*(A + B*x))/x^(5/2),x)

[Out]

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

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sympy [A]  time = 4.35, size = 114, normalized size = 1.34 \begin {gather*} \frac {2 A b^{3} x^{\frac {3}{2}}}{3} + \frac {6 A b^{2} c x^{\frac {5}{2}}}{5} + \frac {6 A b c^{2} x^{\frac {7}{2}}}{7} + \frac {2 A c^{3} x^{\frac {9}{2}}}{9} + \frac {2 B b^{3} x^{\frac {5}{2}}}{5} + \frac {6 B b^{2} c x^{\frac {7}{2}}}{7} + \frac {2 B b c^{2} x^{\frac {9}{2}}}{3} + \frac {2 B c^{3} x^{\frac {11}{2}}}{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**(5/2),x)

[Out]

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

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