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

Optimal. Leaf size=83 \[ 2 a^3 A \sqrt{x}+\frac{2}{5} a^2 x^{5/2} (a B+3 A b)+\frac{2}{13} b^2 x^{13/2} (3 a B+A b)+\frac{2}{3} a b x^{9/2} (a B+A b)+\frac{2}{17} b^3 B x^{17/2} \]

[Out]

2*a^3*A*Sqrt[x] + (2*a^2*(3*A*b + a*B)*x^(5/2))/5 + (2*a*b*(A*b + a*B)*x^(9/2))/
3 + (2*b^2*(A*b + 3*a*B)*x^(13/2))/13 + (2*b^3*B*x^(17/2))/17

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Rubi [A]  time = 0.111743, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ 2 a^3 A \sqrt{x}+\frac{2}{5} a^2 x^{5/2} (a B+3 A b)+\frac{2}{13} b^2 x^{13/2} (3 a B+A b)+\frac{2}{3} a b x^{9/2} (a B+A b)+\frac{2}{17} b^3 B x^{17/2} \]

Antiderivative was successfully verified.

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

[Out]

2*a^3*A*Sqrt[x] + (2*a^2*(3*A*b + a*B)*x^(5/2))/5 + (2*a*b*(A*b + a*B)*x^(9/2))/
3 + (2*b^2*(A*b + 3*a*B)*x^(13/2))/13 + (2*b^3*B*x^(17/2))/17

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Rubi in Sympy [A]  time = 16.635, size = 83, normalized size = 1. \[ 2 A a^{3} \sqrt{x} + \frac{2 B b^{3} x^{\frac{17}{2}}}{17} + \frac{2 a^{2} x^{\frac{5}{2}} \left (3 A b + B a\right )}{5} + \frac{2 a b x^{\frac{9}{2}} \left (A b + B a\right )}{3} + \frac{2 b^{2} x^{\frac{13}{2}} \left (A b + 3 B a\right )}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**3*(B*x**2+A)/x**(1/2),x)

[Out]

2*A*a**3*sqrt(x) + 2*B*b**3*x**(17/2)/17 + 2*a**2*x**(5/2)*(3*A*b + B*a)/5 + 2*a
*b*x**(9/2)*(A*b + B*a)/3 + 2*b**2*x**(13/2)*(A*b + 3*B*a)/13

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Mathematica [A]  time = 0.0420963, size = 71, normalized size = 0.86 \[ \frac{2 \sqrt{x} \left (3315 a^3 A+663 a^2 x^2 (a B+3 A b)+255 b^2 x^6 (3 a B+A b)+1105 a b x^4 (a B+A b)+195 b^3 B x^8\right )}{3315} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[x]*(3315*a^3*A + 663*a^2*(3*A*b + a*B)*x^2 + 1105*a*b*(A*b + a*B)*x^4 +
255*b^2*(A*b + 3*a*B)*x^6 + 195*b^3*B*x^8))/3315

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Maple [A]  time = 0.009, size = 80, normalized size = 1. \[{\frac{390\,{b}^{3}B{x}^{8}+510\,{x}^{6}{b}^{3}A+1530\,{x}^{6}a{b}^{2}B+2210\,{x}^{4}a{b}^{2}A+2210\,{x}^{4}{a}^{2}bB+3978\,{x}^{2}A{a}^{2}b+1326\,{x}^{2}B{a}^{3}+6630\,{a}^{3}A}{3315}\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^3*(B*x^2+A)/x^(1/2),x)

[Out]

2/3315*x^(1/2)*(195*B*b^3*x^8+255*A*b^3*x^6+765*B*a*b^2*x^6+1105*A*a*b^2*x^4+110
5*B*a^2*b*x^4+1989*A*a^2*b*x^2+663*B*a^3*x^2+3315*A*a^3)

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Maxima [A]  time = 1.34168, size = 99, normalized size = 1.19 \[ \frac{2}{17} \, B b^{3} x^{\frac{17}{2}} + \frac{2}{13} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac{13}{2}} + \frac{2}{3} \,{\left (B a^{2} b + A a b^{2}\right )} x^{\frac{9}{2}} + 2 \, A a^{3} \sqrt{x} + \frac{2}{5} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac{5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/17*B*b^3*x^(17/2) + 2/13*(3*B*a*b^2 + A*b^3)*x^(13/2) + 2/3*(B*a^2*b + A*a*b^2
)*x^(9/2) + 2*A*a^3*sqrt(x) + 2/5*(B*a^3 + 3*A*a^2*b)*x^(5/2)

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Fricas [A]  time = 0.225598, size = 101, normalized size = 1.22 \[ \frac{2}{3315} \,{\left (195 \, B b^{3} x^{8} + 255 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{6} + 1105 \,{\left (B a^{2} b + A a b^{2}\right )} x^{4} + 3315 \, A a^{3} + 663 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{2}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3315*(195*B*b^3*x^8 + 255*(3*B*a*b^2 + A*b^3)*x^6 + 1105*(B*a^2*b + A*a*b^2)*x
^4 + 3315*A*a^3 + 663*(B*a^3 + 3*A*a^2*b)*x^2)*sqrt(x)

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Sympy [A]  time = 16.9509, size = 112, normalized size = 1.35 \[ 2 A a^{3} \sqrt{x} + \frac{6 A a^{2} b x^{\frac{5}{2}}}{5} + \frac{2 A a b^{2} x^{\frac{9}{2}}}{3} + \frac{2 A b^{3} x^{\frac{13}{2}}}{13} + \frac{2 B a^{3} x^{\frac{5}{2}}}{5} + \frac{2 B a^{2} b x^{\frac{9}{2}}}{3} + \frac{6 B a b^{2} x^{\frac{13}{2}}}{13} + \frac{2 B b^{3} x^{\frac{17}{2}}}{17} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**3*(B*x**2+A)/x**(1/2),x)

[Out]

2*A*a**3*sqrt(x) + 6*A*a**2*b*x**(5/2)/5 + 2*A*a*b**2*x**(9/2)/3 + 2*A*b**3*x**(
13/2)/13 + 2*B*a**3*x**(5/2)/5 + 2*B*a**2*b*x**(9/2)/3 + 6*B*a*b**2*x**(13/2)/13
+ 2*B*b**3*x**(17/2)/17

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GIAC/XCAS [A]  time = 0.210854, size = 104, normalized size = 1.25 \[ \frac{2}{17} \, B b^{3} x^{\frac{17}{2}} + \frac{6}{13} \, B a b^{2} x^{\frac{13}{2}} + \frac{2}{13} \, A b^{3} x^{\frac{13}{2}} + \frac{2}{3} \, B a^{2} b x^{\frac{9}{2}} + \frac{2}{3} \, A a b^{2} x^{\frac{9}{2}} + \frac{2}{5} \, B a^{3} x^{\frac{5}{2}} + \frac{6}{5} \, A a^{2} b x^{\frac{5}{2}} + 2 \, A a^{3} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/17*B*b^3*x^(17/2) + 6/13*B*a*b^2*x^(13/2) + 2/13*A*b^3*x^(13/2) + 2/3*B*a^2*b*
x^(9/2) + 2/3*A*a*b^2*x^(9/2) + 2/5*B*a^3*x^(5/2) + 6/5*A*a^2*b*x^(5/2) + 2*A*a^
3*sqrt(x)