3.1004 \(\int \frac{(A+B x) \left (a+b x+c x^2\right )^3}{x^{3/2}} \, dx\)

Optimal. Leaf size=176 \[ -\frac{2 a^3 A}{\sqrt{x}}+2 a^2 \sqrt{x} (a B+3 A b)+\frac{2}{3} c x^{9/2} \left (a B c+A b c+b^2 B\right )+2 a x^{3/2} \left (A \left (a c+b^2\right )+a b B\right )+\frac{2}{7} x^{7/2} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{2}{5} x^{5/2} \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{2}{11} c^2 x^{11/2} (A c+3 b B)+\frac{2}{13} B c^3 x^{13/2} \]

[Out]

(-2*a^3*A)/Sqrt[x] + 2*a^2*(3*A*b + a*B)*Sqrt[x] + 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))*x^(5/2))/5 + (2*(b^3*B + 3*A
*b^2*c + 6*a*b*B*c + 3*a*A*c^2)*x^(7/2))/7 + (2*c*(b^2*B + A*b*c + a*B*c)*x^(9/2
))/3 + (2*c^2*(3*b*B + A*c)*x^(11/2))/11 + (2*B*c^3*x^(13/2))/13

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Rubi [A]  time = 0.27205, antiderivative size = 176, 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 \[ -\frac{2 a^3 A}{\sqrt{x}}+2 a^2 \sqrt{x} (a B+3 A b)+\frac{2}{3} c x^{9/2} \left (a B c+A b c+b^2 B\right )+2 a x^{3/2} \left (A \left (a c+b^2\right )+a b B\right )+\frac{2}{7} x^{7/2} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{2}{5} x^{5/2} \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{2}{11} c^2 x^{11/2} (A c+3 b B)+\frac{2}{13} B c^3 x^{13/2} \]

Antiderivative was successfully verified.

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

[Out]

(-2*a^3*A)/Sqrt[x] + 2*a^2*(3*A*b + a*B)*Sqrt[x] + 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))*x^(5/2))/5 + (2*(b^3*B + 3*A
*b^2*c + 6*a*b*B*c + 3*a*A*c^2)*x^(7/2))/7 + (2*c*(b^2*B + A*b*c + a*B*c)*x^(9/2
))/3 + (2*c^2*(3*b*B + A*c)*x^(11/2))/11 + (2*B*c^3*x^(13/2))/13

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Rubi in Sympy [A]  time = 38.7972, size = 201, normalized size = 1.14 \[ - \frac{2 A a^{3}}{\sqrt{x}} + \frac{2 B c^{3} x^{\frac{13}{2}}}{13} + 2 a^{2} \sqrt{x} \left (3 A b + B a\right ) + 2 a x^{\frac{3}{2}} \left (A a c + A b^{2} + B a b\right ) + \frac{2 c^{2} x^{\frac{11}{2}} \left (A c + 3 B b\right )}{11} + \frac{2 c x^{\frac{9}{2}} \left (A b c + B a c + B b^{2}\right )}{3} + x^{\frac{7}{2}} \left (\frac{6 A a c^{2}}{7} + \frac{6 A b^{2} c}{7} + \frac{12 B a b c}{7} + \frac{2 B b^{3}}{7}\right ) + x^{\frac{5}{2}} \left (\frac{12 A a b c}{5} + \frac{2 A b^{3}}{5} + \frac{6 B a^{2} c}{5} + \frac{6 B a b^{2}}{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A*a**3/sqrt(x) + 2*B*c**3*x**(13/2)/13 + 2*a**2*sqrt(x)*(3*A*b + B*a) + 2*a*x
**(3/2)*(A*a*c + A*b**2 + B*a*b) + 2*c**2*x**(11/2)*(A*c + 3*B*b)/11 + 2*c*x**(9
/2)*(A*b*c + B*a*c + B*b**2)/3 + x**(7/2)*(6*A*a*c**2/7 + 6*A*b**2*c/7 + 12*B*a*
b*c/7 + 2*B*b**3/7) + x**(5/2)*(12*A*a*b*c/5 + 2*A*b**3/5 + 6*B*a**2*c/5 + 6*B*a
*b**2/5)

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Mathematica [A]  time = 0.244813, size = 176, normalized size = 1. \[ -\frac{2 a^3 A}{\sqrt{x}}+2 a^2 \sqrt{x} (a B+3 A b)+\frac{2}{3} c x^{9/2} \left (a B c+A b c+b^2 B\right )+2 a x^{3/2} \left (A \left (a c+b^2\right )+a b B\right )+\frac{2}{7} x^{7/2} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{2}{5} x^{5/2} \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{2}{11} c^2 x^{11/2} (A c+3 b B)+\frac{2}{13} B c^3 x^{13/2} \]

Antiderivative was successfully verified.

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

[Out]

(-2*a^3*A)/Sqrt[x] + 2*a^2*(3*A*b + a*B)*Sqrt[x] + 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))*x^(5/2))/5 + (2*(b^3*B + 3*A
*b^2*c + 6*a*b*B*c + 3*a*A*c^2)*x^(7/2))/7 + (2*c*(b^2*B + A*b*c + a*B*c)*x^(9/2
))/3 + (2*c^2*(3*b*B + A*c)*x^(11/2))/11 + (2*B*c^3*x^(13/2))/13

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Maple [A]  time = 0.01, size = 192, normalized size = 1.1 \[ -{\frac{-2310\,B{c}^{3}{x}^{7}-2730\,A{c}^{3}{x}^{6}-8190\,B{x}^{6}b{c}^{2}-10010\,A{x}^{5}b{c}^{2}-10010\,aB{c}^{2}{x}^{5}-10010\,B{x}^{5}{b}^{2}c-12870\,aA{c}^{2}{x}^{4}-12870\,A{x}^{4}{b}^{2}c-25740\,B{x}^{4}abc-4290\,B{x}^{4}{b}^{3}-36036\,A{x}^{3}abc-6006\,A{b}^{3}{x}^{3}-18018\,{a}^{2}Bc{x}^{3}-18018\,B{x}^{3}a{b}^{2}-30030\,{a}^{2}Ac{x}^{2}-30030\,A{x}^{2}a{b}^{2}-30030\,B{x}^{2}{a}^{2}b-90090\,A{a}^{2}bx-30030\,{a}^{3}Bx+30030\,A{a}^{3}}{15015}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15015*(-1155*B*c^3*x^7-1365*A*c^3*x^6-4095*B*b*c^2*x^6-5005*A*b*c^2*x^5-5005*
B*a*c^2*x^5-5005*B*b^2*c*x^5-6435*A*a*c^2*x^4-6435*A*b^2*c*x^4-12870*B*a*b*c*x^4
-2145*B*b^3*x^4-18018*A*a*b*c*x^3-3003*A*b^3*x^3-9009*B*a^2*c*x^3-9009*B*a*b^2*x
^3-15015*A*a^2*c*x^2-15015*A*a*b^2*x^2-15015*B*a^2*b*x^2-45045*A*a^2*b*x-15015*B
*a^3*x+15015*A*a^3)/x^(1/2)

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Maxima [A]  time = 0.720326, size = 224, normalized size = 1.27 \[ \frac{2}{13} \, B c^{3} x^{\frac{13}{2}} + \frac{2}{11} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac{11}{2}} + \frac{2}{3} \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{\frac{9}{2}} + \frac{2}{7} \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{\frac{7}{2}} - \frac{2 \, A a^{3}}{\sqrt{x}} + \frac{2}{5} \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{\frac{5}{2}} + 2 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{\frac{3}{2}} + 2 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/13*B*c^3*x^(13/2) + 2/11*(3*B*b*c^2 + A*c^3)*x^(11/2) + 2/3*(B*b^2*c + (B*a +
A*b)*c^2)*x^(9/2) + 2/7*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^(7/2) - 2*
A*a^3/sqrt(x) + 2/5*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^(5/2) + 2*(B*a
^2*b + A*a*b^2 + A*a^2*c)*x^(3/2) + 2*(B*a^3 + 3*A*a^2*b)*sqrt(x)

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Fricas [A]  time = 0.327639, size = 224, normalized size = 1.27 \[ \frac{2 \,{\left (1155 \, B c^{3} x^{7} + 1365 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 5005 \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{5} + 2145 \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} - 15015 \, A a^{3} + 3003 \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} + 15015 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 15015 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )}}{15015 \, \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15015*(1155*B*c^3*x^7 + 1365*(3*B*b*c^2 + A*c^3)*x^6 + 5005*(B*b^2*c + (B*a +
A*b)*c^2)*x^5 + 2145*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^4 - 15015*A*a
^3 + 3003*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^3 + 15015*(B*a^2*b + A*a
*b^2 + A*a^2*c)*x^2 + 15015*(B*a^3 + 3*A*a^2*b)*x)/sqrt(x)

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Sympy [A]  time = 36.0011, size = 284, normalized size = 1.61 \[ - \frac{2 A a^{3}}{\sqrt{x}} + 6 A a^{2} b \sqrt{x} + 2 A a^{2} c x^{\frac{3}{2}} + 2 A a b^{2} x^{\frac{3}{2}} + \frac{12 A a b c x^{\frac{5}{2}}}{5} + \frac{6 A a c^{2} x^{\frac{7}{2}}}{7} + \frac{2 A b^{3} x^{\frac{5}{2}}}{5} + \frac{6 A b^{2} c x^{\frac{7}{2}}}{7} + \frac{2 A b c^{2} x^{\frac{9}{2}}}{3} + \frac{2 A c^{3} x^{\frac{11}{2}}}{11} + 2 B a^{3} \sqrt{x} + 2 B a^{2} b x^{\frac{3}{2}} + \frac{6 B a^{2} c x^{\frac{5}{2}}}{5} + \frac{6 B a b^{2} x^{\frac{5}{2}}}{5} + \frac{12 B a b c x^{\frac{7}{2}}}{7} + \frac{2 B a c^{2} x^{\frac{9}{2}}}{3} + \frac{2 B b^{3} x^{\frac{7}{2}}}{7} + \frac{2 B b^{2} c x^{\frac{9}{2}}}{3} + \frac{6 B b c^{2} x^{\frac{11}{2}}}{11} + \frac{2 B c^{3} x^{\frac{13}{2}}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A*a**3/sqrt(x) + 6*A*a**2*b*sqrt(x) + 2*A*a**2*c*x**(3/2) + 2*A*a*b**2*x**(3/
2) + 12*A*a*b*c*x**(5/2)/5 + 6*A*a*c**2*x**(7/2)/7 + 2*A*b**3*x**(5/2)/5 + 6*A*b
**2*c*x**(7/2)/7 + 2*A*b*c**2*x**(9/2)/3 + 2*A*c**3*x**(11/2)/11 + 2*B*a**3*sqrt
(x) + 2*B*a**2*b*x**(3/2) + 6*B*a**2*c*x**(5/2)/5 + 6*B*a*b**2*x**(5/2)/5 + 12*B
*a*b*c*x**(7/2)/7 + 2*B*a*c**2*x**(9/2)/3 + 2*B*b**3*x**(7/2)/7 + 2*B*b**2*c*x**
(9/2)/3 + 6*B*b*c**2*x**(11/2)/11 + 2*B*c**3*x**(13/2)/13

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GIAC/XCAS [A]  time = 0.281505, size = 261, normalized size = 1.48 \[ \frac{2}{13} \, B c^{3} x^{\frac{13}{2}} + \frac{6}{11} \, B b c^{2} x^{\frac{11}{2}} + \frac{2}{11} \, A c^{3} x^{\frac{11}{2}} + \frac{2}{3} \, B b^{2} c x^{\frac{9}{2}} + \frac{2}{3} \, B a c^{2} x^{\frac{9}{2}} + \frac{2}{3} \, A b c^{2} x^{\frac{9}{2}} + \frac{2}{7} \, B b^{3} x^{\frac{7}{2}} + \frac{12}{7} \, B a b c x^{\frac{7}{2}} + \frac{6}{7} \, A b^{2} c x^{\frac{7}{2}} + \frac{6}{7} \, A a c^{2} x^{\frac{7}{2}} + \frac{6}{5} \, B a b^{2} x^{\frac{5}{2}} + \frac{2}{5} \, A b^{3} x^{\frac{5}{2}} + \frac{6}{5} \, B a^{2} c x^{\frac{5}{2}} + \frac{12}{5} \, A a b c x^{\frac{5}{2}} + 2 \, B a^{2} b x^{\frac{3}{2}} + 2 \, A a b^{2} x^{\frac{3}{2}} + 2 \, A a^{2} c x^{\frac{3}{2}} + 2 \, B a^{3} \sqrt{x} + 6 \, A a^{2} b \sqrt{x} - \frac{2 \, A a^{3}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/13*B*c^3*x^(13/2) + 6/11*B*b*c^2*x^(11/2) + 2/11*A*c^3*x^(11/2) + 2/3*B*b^2*c*
x^(9/2) + 2/3*B*a*c^2*x^(9/2) + 2/3*A*b*c^2*x^(9/2) + 2/7*B*b^3*x^(7/2) + 12/7*B
*a*b*c*x^(7/2) + 6/7*A*b^2*c*x^(7/2) + 6/7*A*a*c^2*x^(7/2) + 6/5*B*a*b^2*x^(5/2)
+ 2/5*A*b^3*x^(5/2) + 6/5*B*a^2*c*x^(5/2) + 12/5*A*a*b*c*x^(5/2) + 2*B*a^2*b*x^
(3/2) + 2*A*a*b^2*x^(3/2) + 2*A*a^2*c*x^(3/2) + 2*B*a^3*sqrt(x) + 6*A*a^2*b*sqrt
(x) - 2*A*a^3/sqrt(x)