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

Optimal. Leaf size=109 \[ -\frac{2 a^2 A}{\sqrt{x}}+\frac{2}{5} x^{5/2} \left (2 a B c+2 A b c+b^2 B\right )+\frac{2}{3} x^{3/2} \left (A \left (2 a c+b^2\right )+2 a b B\right )+2 a \sqrt{x} (a B+2 A b)+\frac{2}{7} c x^{7/2} (A c+2 b B)+\frac{2}{9} B c^2 x^{9/2} \]

[Out]

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

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Rubi [A]  time = 0.144485, antiderivative size = 109, 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 \[ -\frac{2 a^2 A}{\sqrt{x}}+\frac{2}{5} x^{5/2} \left (2 a B c+2 A b c+b^2 B\right )+\frac{2}{3} x^{3/2} \left (A \left (2 a c+b^2\right )+2 a b B\right )+2 a \sqrt{x} (a B+2 A b)+\frac{2}{7} c x^{7/2} (A c+2 b B)+\frac{2}{9} B c^2 x^{9/2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 19.7571, size = 121, normalized size = 1.11 \[ - \frac{2 A a^{2}}{\sqrt{x}} + \frac{2 B c^{2} x^{\frac{9}{2}}}{9} + 2 a \sqrt{x} \left (2 A b + B a\right ) + \frac{2 c x^{\frac{7}{2}} \left (A c + 2 B b\right )}{7} + x^{\frac{5}{2}} \left (\frac{4 A b c}{5} + \frac{4 B a c}{5} + \frac{2 B b^{2}}{5}\right ) + x^{\frac{3}{2}} \left (\frac{4 A a c}{3} + \frac{2 A b^{2}}{3} + \frac{4 B a b}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A*a**2/sqrt(x) + 2*B*c**2*x**(9/2)/9 + 2*a*sqrt(x)*(2*A*b + B*a) + 2*c*x**(7/
2)*(A*c + 2*B*b)/7 + x**(5/2)*(4*A*b*c/5 + 4*B*a*c/5 + 2*B*b**2/5) + x**(3/2)*(4
*A*a*c/3 + 2*A*b**2/3 + 4*B*a*b/3)

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Mathematica [A]  time = 0.105222, size = 93, normalized size = 0.85 \[ \frac{2 \left (-315 a^2 A+63 x^3 \left (2 a B c+2 A b c+b^2 B\right )+105 x^2 \left (A \left (2 a c+b^2\right )+2 a b B\right )+315 a x (a B+2 A b)+45 c x^4 (A c+2 b B)+35 B c^2 x^5\right )}{315 \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-315*a^2*A + 315*a*(2*A*b + a*B)*x + 105*(2*a*b*B + A*(b^2 + 2*a*c))*x^2 + 6
3*(b^2*B + 2*A*b*c + 2*a*B*c)*x^3 + 45*c*(2*b*B + A*c)*x^4 + 35*B*c^2*x^5))/(315
*Sqrt[x])

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Maple [A]  time = 0.009, size = 102, normalized size = 0.9 \[ -{\frac{-70\,B{c}^{2}{x}^{5}-90\,A{c}^{2}{x}^{4}-180\,B{x}^{4}bc-252\,A{x}^{3}bc-252\,aBc{x}^{3}-126\,B{b}^{2}{x}^{3}-420\,aAc{x}^{2}-210\,A{b}^{2}{x}^{2}-420\,B{x}^{2}ab-1260\,aAbx-630\,{a}^{2}Bx+630\,A{a}^{2}}{315}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/315*(-35*B*c^2*x^5-45*A*c^2*x^4-90*B*b*c*x^4-126*A*b*c*x^3-126*B*a*c*x^3-63*B
*b^2*x^3-210*A*a*c*x^2-105*A*b^2*x^2-210*B*a*b*x^2-630*A*a*b*x-315*B*a^2*x+315*A
*a^2)/x^(1/2)

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Maxima [A]  time = 0.71622, size = 126, normalized size = 1.16 \[ \frac{2}{9} \, B c^{2} x^{\frac{9}{2}} + \frac{2}{7} \,{\left (2 \, B b c + A c^{2}\right )} x^{\frac{7}{2}} + \frac{2}{5} \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} x^{\frac{5}{2}} - \frac{2 \, A a^{2}}{\sqrt{x}} + \frac{2}{3} \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{\frac{3}{2}} + 2 \,{\left (B a^{2} + 2 \, A a b\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.272561, size = 126, normalized size = 1.16 \[ \frac{2 \,{\left (35 \, B c^{2} x^{5} + 45 \,{\left (2 \, B b c + A c^{2}\right )} x^{4} + 63 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} x^{3} - 315 \, A a^{2} + 105 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} + 315 \,{\left (B a^{2} + 2 \, A a b\right )} x\right )}}{315 \, \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(35*B*c^2*x^5 + 45*(2*B*b*c + A*c^2)*x^4 + 63*(B*b^2 + 2*(B*a + A*b)*c)*x^
3 - 315*A*a^2 + 105*(2*B*a*b + A*b^2 + 2*A*a*c)*x^2 + 315*(B*a^2 + 2*A*a*b)*x)/s
qrt(x)

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Sympy [A]  time = 12.7775, size = 156, normalized size = 1.43 \[ - \frac{2 A a^{2}}{\sqrt{x}} + 4 A a b \sqrt{x} + \frac{4 A a c x^{\frac{3}{2}}}{3} + \frac{2 A b^{2} x^{\frac{3}{2}}}{3} + \frac{4 A b c x^{\frac{5}{2}}}{5} + \frac{2 A c^{2} x^{\frac{7}{2}}}{7} + 2 B a^{2} \sqrt{x} + \frac{4 B a b x^{\frac{3}{2}}}{3} + \frac{4 B a c x^{\frac{5}{2}}}{5} + \frac{2 B b^{2} x^{\frac{5}{2}}}{5} + \frac{4 B b c x^{\frac{7}{2}}}{7} + \frac{2 B c^{2} x^{\frac{9}{2}}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A*a**2/sqrt(x) + 4*A*a*b*sqrt(x) + 4*A*a*c*x**(3/2)/3 + 2*A*b**2*x**(3/2)/3 +
4*A*b*c*x**(5/2)/5 + 2*A*c**2*x**(7/2)/7 + 2*B*a**2*sqrt(x) + 4*B*a*b*x**(3/2)/
3 + 4*B*a*c*x**(5/2)/5 + 2*B*b**2*x**(5/2)/5 + 4*B*b*c*x**(7/2)/7 + 2*B*c**2*x**
(9/2)/9

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GIAC/XCAS [A]  time = 0.274627, size = 139, normalized size = 1.28 \[ \frac{2}{9} \, B c^{2} x^{\frac{9}{2}} + \frac{4}{7} \, B b c x^{\frac{7}{2}} + \frac{2}{7} \, A c^{2} x^{\frac{7}{2}} + \frac{2}{5} \, B b^{2} x^{\frac{5}{2}} + \frac{4}{5} \, B a c x^{\frac{5}{2}} + \frac{4}{5} \, A b c x^{\frac{5}{2}} + \frac{4}{3} \, B a b x^{\frac{3}{2}} + \frac{2}{3} \, A b^{2} x^{\frac{3}{2}} + \frac{4}{3} \, A a c x^{\frac{3}{2}} + 2 \, B a^{2} \sqrt{x} + 4 \, A a b \sqrt{x} - \frac{2 \, A a^{2}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9*B*c^2*x^(9/2) + 4/7*B*b*c*x^(7/2) + 2/7*A*c^2*x^(7/2) + 2/5*B*b^2*x^(5/2) +
4/5*B*a*c*x^(5/2) + 4/5*A*b*c*x^(5/2) + 4/3*B*a*b*x^(3/2) + 2/3*A*b^2*x^(3/2) +
4/3*A*a*c*x^(3/2) + 2*B*a^2*sqrt(x) + 4*A*a*b*sqrt(x) - 2*A*a^2/sqrt(x)