3.10.32 \(\int \frac {(A+B x) (a+b x+c x^2)^3}{x^{9/2}} \, dx\)

Optimal. Leaf size=174 \[ -\frac {2 a^3 A}{7 x^{7/2}}-\frac {2 a^2 (a B+3 A b)}{5 x^{5/2}}+2 c x^{3/2} \left (a B c+A b c+b^2 B\right )-\frac {2 a \left (A \left (a c+b^2\right )+a b B\right )}{x^{3/2}}+2 \sqrt {x} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )-\frac {2 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )}{\sqrt {x}}+\frac {2}{5} c^2 x^{5/2} (A c+3 b B)+\frac {2}{7} B c^3 x^{7/2} \]

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Rubi [A]  time = 0.12, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {765} \begin {gather*} -\frac {2 a^2 (a B+3 A b)}{5 x^{5/2}}-\frac {2 a^3 A}{7 x^{7/2}}+2 \sqrt {x} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+2 c x^{3/2} \left (a B c+A b c+b^2 B\right )-\frac {2 a \left (A \left (a c+b^2\right )+a b B\right )}{x^{3/2}}-\frac {2 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )}{\sqrt {x}}+\frac {2}{5} c^2 x^{5/2} (A c+3 b B)+\frac {2}{7} B c^3 x^{7/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + b*x + c*x^2)^3)/x^(9/2),x]

[Out]

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

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

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Mathematica [A]  time = 0.19, size = 168, normalized size = 0.97 \begin {gather*} \frac {2 \left (-\left (a^3 (5 A+7 B x)\right )-7 a^2 x (A (3 b+5 c x)+5 B x (b+3 c x))-35 a x^2 \left (A \left (b^2+6 b c x-3 c^2 x^2\right )-B x \left (-3 b^2+6 b c x+c^2 x^2\right )\right )+x^3 \left (7 A \left (-5 b^3+15 b^2 c x+5 b c^2 x^2+c^3 x^3\right )+B x \left (35 b^3+35 b^2 c x+21 b c^2 x^2+5 c^3 x^3\right )\right )\right )}{35 x^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + b*x + c*x^2)^3)/x^(9/2),x]

[Out]

(2*(-(a^3*(5*A + 7*B*x)) - 7*a^2*x*(5*B*x*(b + 3*c*x) + A*(3*b + 5*c*x)) - 35*a*x^2*(A*(b^2 + 6*b*c*x - 3*c^2*
x^2) - B*x*(-3*b^2 + 6*b*c*x + c^2*x^2)) + x^3*(7*A*(-5*b^3 + 15*b^2*c*x + 5*b*c^2*x^2 + c^3*x^3) + B*x*(35*b^
3 + 35*b^2*c*x + 21*b*c^2*x^2 + 5*c^3*x^3))))/(35*x^(7/2))

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IntegrateAlgebraic [A]  time = 0.16, size = 195, normalized size = 1.12 \begin {gather*} \frac {2 \left (-5 a^3 A-7 a^3 B x-21 a^2 A b x-35 a^2 A c x^2-35 a^2 b B x^2-105 a^2 B c x^3-35 a A b^2 x^2-210 a A b c x^3+105 a A c^2 x^4-105 a b^2 B x^3+210 a b B c x^4+35 a B c^2 x^5-35 A b^3 x^3+105 A b^2 c x^4+35 A b c^2 x^5+7 A c^3 x^6+35 b^3 B x^4+35 b^2 B c x^5+21 b B c^2 x^6+5 B c^3 x^7\right )}{35 x^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*(a + b*x + c*x^2)^3)/x^(9/2),x]

[Out]

(2*(-5*a^3*A - 21*a^2*A*b*x - 7*a^3*B*x - 35*a*A*b^2*x^2 - 35*a^2*b*B*x^2 - 35*a^2*A*c*x^2 - 35*A*b^3*x^3 - 10
5*a*b^2*B*x^3 - 210*a*A*b*c*x^3 - 105*a^2*B*c*x^3 + 35*b^3*B*x^4 + 105*A*b^2*c*x^4 + 210*a*b*B*c*x^4 + 105*a*A
*c^2*x^4 + 35*b^2*B*c*x^5 + 35*A*b*c^2*x^5 + 35*a*B*c^2*x^5 + 21*b*B*c^2*x^6 + 7*A*c^3*x^6 + 5*B*c^3*x^7))/(35
*x^(7/2))

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fricas [A]  time = 0.43, size = 166, normalized size = 0.95 \begin {gather*} \frac {2 \, {\left (5 \, B c^{3} x^{7} + 7 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 35 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{5} + 35 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} - 5 \, A a^{3} - 35 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} - 35 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} - 7 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )}}{35 \, x^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)^3/x^(9/2),x, algorithm="fricas")

[Out]

2/35*(5*B*c^3*x^7 + 7*(3*B*b*c^2 + A*c^3)*x^6 + 35*(B*b^2*c + (B*a + A*b)*c^2)*x^5 + 35*(B*b^3 + 3*A*a*c^2 + 3
*(2*B*a*b + A*b^2)*c)*x^4 - 5*A*a^3 - 35*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^3 - 35*(B*a^2*b + A*a*b
^2 + A*a^2*c)*x^2 - 7*(B*a^3 + 3*A*a^2*b)*x)/x^(7/2)

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giac [A]  time = 0.17, size = 192, normalized size = 1.10 \begin {gather*} \frac {2}{7} \, B c^{3} x^{\frac {7}{2}} + \frac {6}{5} \, B b c^{2} x^{\frac {5}{2}} + \frac {2}{5} \, A c^{3} x^{\frac {5}{2}} + 2 \, B b^{2} c x^{\frac {3}{2}} + 2 \, B a c^{2} x^{\frac {3}{2}} + 2 \, A b c^{2} x^{\frac {3}{2}} + 2 \, B b^{3} \sqrt {x} + 12 \, B a b c \sqrt {x} + 6 \, A b^{2} c \sqrt {x} + 6 \, A a c^{2} \sqrt {x} - \frac {2 \, {\left (105 \, B a b^{2} x^{3} + 35 \, A b^{3} x^{3} + 105 \, B a^{2} c x^{3} + 210 \, A a b c x^{3} + 35 \, B a^{2} b x^{2} + 35 \, A a b^{2} x^{2} + 35 \, A a^{2} c x^{2} + 7 \, B a^{3} x + 21 \, A a^{2} b x + 5 \, A a^{3}\right )}}{35 \, x^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)^3/x^(9/2),x, algorithm="giac")

[Out]

2/7*B*c^3*x^(7/2) + 6/5*B*b*c^2*x^(5/2) + 2/5*A*c^3*x^(5/2) + 2*B*b^2*c*x^(3/2) + 2*B*a*c^2*x^(3/2) + 2*A*b*c^
2*x^(3/2) + 2*B*b^3*sqrt(x) + 12*B*a*b*c*sqrt(x) + 6*A*b^2*c*sqrt(x) + 6*A*a*c^2*sqrt(x) - 2/35*(105*B*a*b^2*x
^3 + 35*A*b^3*x^3 + 105*B*a^2*c*x^3 + 210*A*a*b*c*x^3 + 35*B*a^2*b*x^2 + 35*A*a*b^2*x^2 + 35*A*a^2*c*x^2 + 7*B
*a^3*x + 21*A*a^2*b*x + 5*A*a^3)/x^(7/2)

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maple [A]  time = 0.06, size = 192, normalized size = 1.10 \begin {gather*} -\frac {2 \left (-5 B \,c^{3} x^{7}-7 A \,c^{3} x^{6}-21 x^{6} B b \,c^{2}-35 x^{5} A b \,c^{2}-35 B a \,c^{2} x^{5}-35 x^{5} B \,b^{2} c -105 A a \,c^{2} x^{4}-105 x^{4} A \,b^{2} c -210 x^{4} a b B c -35 x^{4} b^{3} B +210 x^{3} A a b c +35 A \,b^{3} x^{3}+105 B \,a^{2} c \,x^{3}+105 x^{3} B a \,b^{2}+35 A \,a^{2} c \,x^{2}+35 x^{2} A a \,b^{2}+35 B \,a^{2} b \,x^{2}+21 x A \,a^{2} b +7 B \,a^{3} x +5 A \,a^{3}\right )}{35 x^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x+a)^3/x^(9/2),x)

[Out]

-2/35*(-5*B*c^3*x^7-7*A*c^3*x^6-21*B*b*c^2*x^6-35*A*b*c^2*x^5-35*B*a*c^2*x^5-35*B*b^2*c*x^5-105*A*a*c^2*x^4-10
5*A*b^2*c*x^4-210*B*a*b*c*x^4-35*B*b^3*x^4+210*A*a*b*c*x^3+35*A*b^3*x^3+105*B*a^2*c*x^3+105*B*a*b^2*x^3+35*A*a
^2*c*x^2+35*A*a*b^2*x^2+35*B*a^2*b*x^2+21*A*a^2*b*x+7*B*a^3*x+5*A*a^3)/x^(7/2)

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maxima [A]  time = 0.52, size = 167, normalized size = 0.96 \begin {gather*} \frac {2}{7} \, B c^{3} x^{\frac {7}{2}} + \frac {2}{5} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac {5}{2}} + 2 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{\frac {3}{2}} + 2 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} \sqrt {x} - \frac {2 \, {\left (5 \, A a^{3} + 35 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} + 35 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 7 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )}}{35 \, x^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)^3/x^(9/2),x, algorithm="maxima")

[Out]

2/7*B*c^3*x^(7/2) + 2/5*(3*B*b*c^2 + A*c^3)*x^(5/2) + 2*(B*b^2*c + (B*a + A*b)*c^2)*x^(3/2) + 2*(B*b^3 + 3*A*a
*c^2 + 3*(2*B*a*b + A*b^2)*c)*sqrt(x) - 2/35*(5*A*a^3 + 35*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^3 + 3
5*(B*a^2*b + A*a*b^2 + A*a^2*c)*x^2 + 7*(B*a^3 + 3*A*a^2*b)*x)/x^(7/2)

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mupad [B]  time = 0.06, size = 170, normalized size = 0.98 \begin {gather*} \sqrt {x}\,\left (2\,B\,b^3+6\,A\,b^2\,c+12\,B\,a\,b\,c+6\,A\,a\,c^2\right )-\frac {x^3\,\left (6\,B\,c\,a^2+6\,B\,a\,b^2+12\,A\,c\,a\,b+2\,A\,b^3\right )+x\,\left (\frac {2\,B\,a^3}{5}+\frac {6\,A\,b\,a^2}{5}\right )+\frac {2\,A\,a^3}{7}+x^2\,\left (2\,B\,a^2\,b+2\,A\,c\,a^2+2\,A\,a\,b^2\right )}{x^{7/2}}+x^{5/2}\,\left (\frac {2\,A\,c^3}{5}+\frac {6\,B\,b\,c^2}{5}\right )+x^{3/2}\,\left (2\,B\,b^2\,c+2\,A\,b\,c^2+2\,B\,a\,c^2\right )+\frac {2\,B\,c^3\,x^{7/2}}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x + c*x^2)^3)/x^(9/2),x)

[Out]

x^(1/2)*(2*B*b^3 + 6*A*a*c^2 + 6*A*b^2*c + 12*B*a*b*c) - (x^3*(2*A*b^3 + 6*B*a*b^2 + 6*B*a^2*c + 12*A*a*b*c) +
 x*((2*B*a^3)/5 + (6*A*a^2*b)/5) + (2*A*a^3)/7 + x^2*(2*A*a*b^2 + 2*A*a^2*c + 2*B*a^2*b))/x^(7/2) + x^(5/2)*((
2*A*c^3)/5 + (6*B*b*c^2)/5) + x^(3/2)*(2*A*b*c^2 + 2*B*a*c^2 + 2*B*b^2*c) + (2*B*c^3*x^(7/2))/7

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sympy [A]  time = 12.90, size = 270, normalized size = 1.55 \begin {gather*} - \frac {2 A a^{3}}{7 x^{\frac {7}{2}}} - \frac {6 A a^{2} b}{5 x^{\frac {5}{2}}} - \frac {2 A a^{2} c}{x^{\frac {3}{2}}} - \frac {2 A a b^{2}}{x^{\frac {3}{2}}} - \frac {12 A a b c}{\sqrt {x}} + 6 A a c^{2} \sqrt {x} - \frac {2 A b^{3}}{\sqrt {x}} + 6 A b^{2} c \sqrt {x} + 2 A b c^{2} x^{\frac {3}{2}} + \frac {2 A c^{3} x^{\frac {5}{2}}}{5} - \frac {2 B a^{3}}{5 x^{\frac {5}{2}}} - \frac {2 B a^{2} b}{x^{\frac {3}{2}}} - \frac {6 B a^{2} c}{\sqrt {x}} - \frac {6 B a b^{2}}{\sqrt {x}} + 12 B a b c \sqrt {x} + 2 B a c^{2} x^{\frac {3}{2}} + 2 B b^{3} \sqrt {x} + 2 B b^{2} c x^{\frac {3}{2}} + \frac {6 B b c^{2} x^{\frac {5}{2}}}{5} + \frac {2 B c^{3} x^{\frac {7}{2}}}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x+a)**3/x**(9/2),x)

[Out]

-2*A*a**3/(7*x**(7/2)) - 6*A*a**2*b/(5*x**(5/2)) - 2*A*a**2*c/x**(3/2) - 2*A*a*b**2/x**(3/2) - 12*A*a*b*c/sqrt
(x) + 6*A*a*c**2*sqrt(x) - 2*A*b**3/sqrt(x) + 6*A*b**2*c*sqrt(x) + 2*A*b*c**2*x**(3/2) + 2*A*c**3*x**(5/2)/5 -
 2*B*a**3/(5*x**(5/2)) - 2*B*a**2*b/x**(3/2) - 6*B*a**2*c/sqrt(x) - 6*B*a*b**2/sqrt(x) + 12*B*a*b*c*sqrt(x) +
2*B*a*c**2*x**(3/2) + 2*B*b**3*sqrt(x) + 2*B*b**2*c*x**(3/2) + 6*B*b*c**2*x**(5/2)/5 + 2*B*c**3*x**(7/2)/7

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