3.393 \(\int x^{7/2} (A+B x) \left (a+c x^2\right )^2 \, dx\)

Optimal. Leaf size=77 \[ \frac{2}{9} a^2 A x^{9/2}+\frac{2}{11} a^2 B x^{11/2}+\frac{4}{13} a A c x^{13/2}+\frac{4}{15} a B c x^{15/2}+\frac{2}{17} A c^2 x^{17/2}+\frac{2}{19} B c^2 x^{19/2} \]

[Out]

(2*a^2*A*x^(9/2))/9 + (2*a^2*B*x^(11/2))/11 + (4*a*A*c*x^(13/2))/13 + (4*a*B*c*x
^(15/2))/15 + (2*A*c^2*x^(17/2))/17 + (2*B*c^2*x^(19/2))/19

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Rubi [A]  time = 0.0730284, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ \frac{2}{9} a^2 A x^{9/2}+\frac{2}{11} a^2 B x^{11/2}+\frac{4}{13} a A c x^{13/2}+\frac{4}{15} a B c x^{15/2}+\frac{2}{17} A c^2 x^{17/2}+\frac{2}{19} B c^2 x^{19/2} \]

Antiderivative was successfully verified.

[In]  Int[x^(7/2)*(A + B*x)*(a + c*x^2)^2,x]

[Out]

(2*a^2*A*x^(9/2))/9 + (2*a^2*B*x^(11/2))/11 + (4*a*A*c*x^(13/2))/13 + (4*a*B*c*x
^(15/2))/15 + (2*A*c^2*x^(17/2))/17 + (2*B*c^2*x^(19/2))/19

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Rubi in Sympy [A]  time = 8.77924, size = 80, normalized size = 1.04 \[ \frac{2 A a^{2} x^{\frac{9}{2}}}{9} + \frac{4 A a c x^{\frac{13}{2}}}{13} + \frac{2 A c^{2} x^{\frac{17}{2}}}{17} + \frac{2 B a^{2} x^{\frac{11}{2}}}{11} + \frac{4 B a c x^{\frac{15}{2}}}{15} + \frac{2 B c^{2} x^{\frac{19}{2}}}{19} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(7/2)*(B*x+A)*(c*x**2+a)**2,x)

[Out]

2*A*a**2*x**(9/2)/9 + 4*A*a*c*x**(13/2)/13 + 2*A*c**2*x**(17/2)/17 + 2*B*a**2*x*
*(11/2)/11 + 4*B*a*c*x**(15/2)/15 + 2*B*c**2*x**(19/2)/19

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Mathematica [A]  time = 0.0411556, size = 60, normalized size = 0.78 \[ \frac{2}{99} a^2 x^{9/2} (11 A+9 B x)+\frac{4}{195} a c x^{13/2} (15 A+13 B x)+\frac{2}{323} c^2 x^{17/2} (19 A+17 B x) \]

Antiderivative was successfully verified.

[In]  Integrate[x^(7/2)*(A + B*x)*(a + c*x^2)^2,x]

[Out]

(2*a^2*x^(9/2)*(11*A + 9*B*x))/99 + (4*a*c*x^(13/2)*(15*A + 13*B*x))/195 + (2*c^
2*x^(17/2)*(19*A + 17*B*x))/323

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Maple [A]  time = 0.009, size = 54, normalized size = 0.7 \[{\frac{218790\,B{c}^{2}{x}^{5}+244530\,A{c}^{2}{x}^{4}+554268\,aBc{x}^{3}+639540\,aAc{x}^{2}+377910\,{a}^{2}Bx+461890\,A{a}^{2}}{2078505}{x}^{{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(7/2)*(B*x+A)*(c*x^2+a)^2,x)

[Out]

2/2078505*x^(9/2)*(109395*B*c^2*x^5+122265*A*c^2*x^4+277134*B*a*c*x^3+319770*A*a
*c*x^2+188955*B*a^2*x+230945*A*a^2)

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Maxima [A]  time = 0.684646, size = 72, normalized size = 0.94 \[ \frac{2}{19} \, B c^{2} x^{\frac{19}{2}} + \frac{2}{17} \, A c^{2} x^{\frac{17}{2}} + \frac{4}{15} \, B a c x^{\frac{15}{2}} + \frac{4}{13} \, A a c x^{\frac{13}{2}} + \frac{2}{11} \, B a^{2} x^{\frac{11}{2}} + \frac{2}{9} \, A a^{2} x^{\frac{9}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^2*(B*x + A)*x^(7/2),x, algorithm="maxima")

[Out]

2/19*B*c^2*x^(19/2) + 2/17*A*c^2*x^(17/2) + 4/15*B*a*c*x^(15/2) + 4/13*A*a*c*x^(
13/2) + 2/11*B*a^2*x^(11/2) + 2/9*A*a^2*x^(9/2)

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Fricas [A]  time = 0.273964, size = 78, normalized size = 1.01 \[ \frac{2}{2078505} \,{\left (109395 \, B c^{2} x^{9} + 122265 \, A c^{2} x^{8} + 277134 \, B a c x^{7} + 319770 \, A a c x^{6} + 188955 \, B a^{2} x^{5} + 230945 \, A a^{2} x^{4}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^2*(B*x + A)*x^(7/2),x, algorithm="fricas")

[Out]

2/2078505*(109395*B*c^2*x^9 + 122265*A*c^2*x^8 + 277134*B*a*c*x^7 + 319770*A*a*c
*x^6 + 188955*B*a^2*x^5 + 230945*A*a^2*x^4)*sqrt(x)

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Sympy [A]  time = 41.4049, size = 80, normalized size = 1.04 \[ \frac{2 A a^{2} x^{\frac{9}{2}}}{9} + \frac{4 A a c x^{\frac{13}{2}}}{13} + \frac{2 A c^{2} x^{\frac{17}{2}}}{17} + \frac{2 B a^{2} x^{\frac{11}{2}}}{11} + \frac{4 B a c x^{\frac{15}{2}}}{15} + \frac{2 B c^{2} x^{\frac{19}{2}}}{19} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(7/2)*(B*x+A)*(c*x**2+a)**2,x)

[Out]

2*A*a**2*x**(9/2)/9 + 4*A*a*c*x**(13/2)/13 + 2*A*c**2*x**(17/2)/17 + 2*B*a**2*x*
*(11/2)/11 + 4*B*a*c*x**(15/2)/15 + 2*B*c**2*x**(19/2)/19

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GIAC/XCAS [A]  time = 0.268402, size = 72, normalized size = 0.94 \[ \frac{2}{19} \, B c^{2} x^{\frac{19}{2}} + \frac{2}{17} \, A c^{2} x^{\frac{17}{2}} + \frac{4}{15} \, B a c x^{\frac{15}{2}} + \frac{4}{13} \, A a c x^{\frac{13}{2}} + \frac{2}{11} \, B a^{2} x^{\frac{11}{2}} + \frac{2}{9} \, A a^{2} x^{\frac{9}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^2*(B*x + A)*x^(7/2),x, algorithm="giac")

[Out]

2/19*B*c^2*x^(19/2) + 2/17*A*c^2*x^(17/2) + 4/15*B*a*c*x^(15/2) + 4/13*A*a*c*x^(
13/2) + 2/11*B*a^2*x^(11/2) + 2/9*A*a^2*x^(9/2)