3.1007 \(\int \frac{(A+B x) \left (a+b x+c x^2\right )^3}{x^{9/2}} \, dx\)

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

[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

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Rubi [A]  time = 0.268429, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ -\frac{2 a^3 A}{7 x^{7/2}}-\frac{2 a^2 (a B+3 A b)}{5 x^{5/2}}+2 c x^{3/2} \left (a B c+A b c+b^2 B\right )-\frac{2 a \left (A \left (a c+b^2\right )+a b B\right )}{x^{3/2}}+2 \sqrt{x} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )-\frac{2 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )}{\sqrt{x}}+\frac{2}{5} c^2 x^{5/2} (A c+3 b B)+\frac{2}{7} B c^3 x^{7/2} \]

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

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Rubi in Sympy [A]  time = 38.7651, size = 189, normalized size = 1.09 \[ - \frac{2 A a^{3}}{7 x^{\frac{7}{2}}} + \frac{2 B c^{3} x^{\frac{7}{2}}}{7} - \frac{2 a^{2} \left (3 A b + B a\right )}{5 x^{\frac{5}{2}}} - \frac{2 a \left (A a c + A b^{2} + B a b\right )}{x^{\frac{3}{2}}} + \frac{2 c^{2} x^{\frac{5}{2}} \left (A c + 3 B b\right )}{5} + 2 c x^{\frac{3}{2}} \left (A b c + B a c + B b^{2}\right ) + \sqrt{x} \left (6 A a c^{2} + 6 A b^{2} c + 12 B a b c + 2 B b^{3}\right ) - \frac{12 A a b c + 2 A b^{3} + 6 B a^{2} c + 6 B a b^{2}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.128835, size = 168, normalized size = 0.97 \[ \frac{2 \left (a^3 (-(5 A+7 B x))-7 a^2 x (A (3 b+5 c x)+5 B x (b+3 c x))-35 a x^2 \left (A \left (b^2+6 b c x-3 c^2 x^2\right )-B x \left (-3 b^2+6 b c x+c^2 x^2\right )\right )+x^3 \left (7 A \left (-5 b^3+15 b^2 c x+5 b c^2 x^2+c^3 x^3\right )+B x \left (35 b^3+35 b^2 c x+21 b c^2 x^2+5 c^3 x^3\right )\right )\right )}{35 x^{7/2}} \]

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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Maple [A]  time = 0.011, size = 192, normalized size = 1.1 \[ -{\frac{-10\,B{c}^{3}{x}^{7}-14\,A{c}^{3}{x}^{6}-42\,B{x}^{6}b{c}^{2}-70\,A{x}^{5}b{c}^{2}-70\,aB{c}^{2}{x}^{5}-70\,B{x}^{5}{b}^{2}c-210\,aA{c}^{2}{x}^{4}-210\,A{x}^{4}{b}^{2}c-420\,B{x}^{4}abc-70\,B{x}^{4}{b}^{3}+420\,A{x}^{3}abc+70\,A{b}^{3}{x}^{3}+210\,{a}^{2}Bc{x}^{3}+210\,B{x}^{3}a{b}^{2}+70\,{a}^{2}Ac{x}^{2}+70\,A{x}^{2}a{b}^{2}+70\,B{x}^{2}{a}^{2}b+42\,A{a}^{2}bx+14\,{a}^{3}Bx+10\,A{a}^{3}}{35}{x}^{-{\frac{7}{2}}}} \]

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^4-105*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 = 0.724393, size = 225, normalized size = 1.29 \[ \frac{2}{7} \, B c^{3} x^{\frac{7}{2}} + \frac{2}{5} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac{5}{2}} + 2 \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{\frac{3}{2}} + 2 \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} \sqrt{x} - \frac{2 \,{\left (5 \, A a^{3} + 35 \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} + 35 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 7 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )}}{35 \, x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^3*(B*x + A)/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 + 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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Fricas [A]  time = 0.279198, size = 224, normalized size = 1.29 \[ \frac{2 \,{\left (5 \, B c^{3} x^{7} + 7 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 35 \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{5} + 35 \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} - 5 \, A a^{3} - 35 \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} - 35 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} - 7 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )}}{35 \, x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^3*(B*x + A)/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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Sympy [A]  time = 28.9885, size = 270, normalized size = 1.55 \[ - \frac{2 A a^{3}}{7 x^{\frac{7}{2}}} - \frac{6 A a^{2} b}{5 x^{\frac{5}{2}}} - \frac{2 A a^{2} c}{x^{\frac{3}{2}}} - \frac{2 A a b^{2}}{x^{\frac{3}{2}}} - \frac{12 A a b c}{\sqrt{x}} + 6 A a c^{2} \sqrt{x} - \frac{2 A b^{3}}{\sqrt{x}} + 6 A b^{2} c \sqrt{x} + 2 A b c^{2} x^{\frac{3}{2}} + \frac{2 A c^{3} x^{\frac{5}{2}}}{5} - \frac{2 B a^{3}}{5 x^{\frac{5}{2}}} - \frac{2 B a^{2} b}{x^{\frac{3}{2}}} - \frac{6 B a^{2} c}{\sqrt{x}} - \frac{6 B a b^{2}}{\sqrt{x}} + 12 B a b c \sqrt{x} + 2 B a c^{2} x^{\frac{3}{2}} + 2 B b^{3} \sqrt{x} + 2 B b^{2} c x^{\frac{3}{2}} + \frac{6 B b c^{2} x^{\frac{5}{2}}}{5} + \frac{2 B c^{3} x^{\frac{7}{2}}}{7} \]

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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GIAC/XCAS [A]  time = 0.269929, size = 259, normalized size = 1.49 \[ \frac{2}{7} \, B c^{3} x^{\frac{7}{2}} + \frac{6}{5} \, B b c^{2} x^{\frac{5}{2}} + \frac{2}{5} \, A c^{3} x^{\frac{5}{2}} + 2 \, B b^{2} c x^{\frac{3}{2}} + 2 \, B a c^{2} x^{\frac{3}{2}} + 2 \, A b c^{2} x^{\frac{3}{2}} + 2 \, B b^{3} \sqrt{x} + 12 \, B a b c \sqrt{x} + 6 \, A b^{2} c \sqrt{x} + 6 \, A a c^{2} \sqrt{x} - \frac{2 \,{\left (105 \, B a b^{2} x^{3} + 35 \, A b^{3} x^{3} + 105 \, B a^{2} c x^{3} + 210 \, A a b c x^{3} + 35 \, B a^{2} b x^{2} + 35 \, A a b^{2} x^{2} + 35 \, A a^{2} c x^{2} + 7 \, B a^{3} x + 21 \, A a^{2} b x + 5 \, A a^{3}\right )}}{35 \, x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^3*(B*x + A)/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)