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

Optimal. Leaf size=103 \[ -\frac{2 a^3 A}{\sqrt{x}}+2 a^3 B \sqrt{x}+2 a^2 A c x^{3/2}+\frac{6}{5} a^2 B c x^{5/2}+\frac{6}{7} a A c^2 x^{7/2}+\frac{2}{3} a B c^2 x^{9/2}+\frac{2}{11} A c^3 x^{11/2}+\frac{2}{13} B c^3 x^{13/2} \]

[Out]

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

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Rubi [A]  time = 0.0985656, antiderivative size = 103, 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 \[ -\frac{2 a^3 A}{\sqrt{x}}+2 a^3 B \sqrt{x}+2 a^2 A c x^{3/2}+\frac{6}{5} a^2 B c x^{5/2}+\frac{6}{7} a A c^2 x^{7/2}+\frac{2}{3} a B c^2 x^{9/2}+\frac{2}{11} A c^3 x^{11/2}+\frac{2}{13} B c^3 x^{13/2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.4506, size = 109, normalized size = 1.06 \[ - \frac{2 A a^{3}}{\sqrt{x}} + 2 A a^{2} c x^{\frac{3}{2}} + \frac{6 A a c^{2} x^{\frac{7}{2}}}{7} + \frac{2 A c^{3} x^{\frac{11}{2}}}{11} + 2 B a^{3} \sqrt{x} + \frac{6 B a^{2} c x^{\frac{5}{2}}}{5} + \frac{2 B a c^{2} x^{\frac{9}{2}}}{3} + \frac{2 B c^{3} x^{\frac{13}{2}}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A*a**3/sqrt(x) + 2*A*a**2*c*x**(3/2) + 6*A*a*c**2*x**(7/2)/7 + 2*A*c**3*x**(1
1/2)/11 + 2*B*a**3*sqrt(x) + 6*B*a**2*c*x**(5/2)/5 + 2*B*a*c**2*x**(9/2)/3 + 2*B
*c**3*x**(13/2)/13

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Mathematica [A]  time = 0.0641646, size = 80, normalized size = 0.78 \[ \frac{2 a^3 (B x-A)}{\sqrt{x}}+\frac{2}{5} a^2 c x^{3/2} (5 A+3 B x)+\frac{2}{21} a c^2 x^{7/2} (9 A+7 B x)+\frac{2}{143} c^3 x^{11/2} (13 A+11 B x) \]

Antiderivative was successfully verified.

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

[Out]

(2*a^3*(-A + B*x))/Sqrt[x] + (2*a^2*c*x^(3/2)*(5*A + 3*B*x))/5 + (2*a*c^2*x^(7/2
)*(9*A + 7*B*x))/21 + (2*c^3*x^(11/2)*(13*A + 11*B*x))/143

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Maple [A]  time = 0.009, size = 78, normalized size = 0.8 \[ -{\frac{-2310\,B{c}^{3}{x}^{7}-2730\,A{c}^{3}{x}^{6}-10010\,aB{c}^{2}{x}^{5}-12870\,aA{c}^{2}{x}^{4}-18018\,{a}^{2}Bc{x}^{3}-30030\,{a}^{2}Ac{x}^{2}-30030\,{a}^{3}Bx+30030\,A{a}^{3}}{15015}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15015*(-1155*B*c^3*x^7-1365*A*c^3*x^6-5005*B*a*c^2*x^5-6435*A*a*c^2*x^4-9009*
B*a^2*c*x^3-15015*A*a^2*c*x^2-15015*B*a^3*x+15015*A*a^3)/x^(1/2)

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Maxima [A]  time = 0.685267, size = 104, normalized size = 1.01 \[ \frac{2}{13} \, B c^{3} x^{\frac{13}{2}} + \frac{2}{11} \, A c^{3} x^{\frac{11}{2}} + \frac{2}{3} \, B a c^{2} x^{\frac{9}{2}} + \frac{6}{7} \, A a c^{2} x^{\frac{7}{2}} + \frac{6}{5} \, B a^{2} c x^{\frac{5}{2}} + 2 \, A a^{2} c x^{\frac{3}{2}} + 2 \, B a^{3} \sqrt{x} - \frac{2 \, A a^{3}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.26339, size = 104, normalized size = 1.01 \[ \frac{2 \,{\left (1155 \, B c^{3} x^{7} + 1365 \, A c^{3} x^{6} + 5005 \, B a c^{2} x^{5} + 6435 \, A a c^{2} x^{4} + 9009 \, B a^{2} c x^{3} + 15015 \, A a^{2} c x^{2} + 15015 \, B a^{3} x - 15015 \, A a^{3}\right )}}{15015 \, \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15015*(1155*B*c^3*x^7 + 1365*A*c^3*x^6 + 5005*B*a*c^2*x^5 + 6435*A*a*c^2*x^4 +
9009*B*a^2*c*x^3 + 15015*A*a^2*c*x^2 + 15015*B*a^3*x - 15015*A*a^3)/sqrt(x)

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Sympy [A]  time = 10.201, size = 109, normalized size = 1.06 \[ - \frac{2 A a^{3}}{\sqrt{x}} + 2 A a^{2} c x^{\frac{3}{2}} + \frac{6 A a c^{2} x^{\frac{7}{2}}}{7} + \frac{2 A c^{3} x^{\frac{11}{2}}}{11} + 2 B a^{3} \sqrt{x} + \frac{6 B a^{2} c x^{\frac{5}{2}}}{5} + \frac{2 B a c^{2} x^{\frac{9}{2}}}{3} + \frac{2 B c^{3} x^{\frac{13}{2}}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A*a**3/sqrt(x) + 2*A*a**2*c*x**(3/2) + 6*A*a*c**2*x**(7/2)/7 + 2*A*c**3*x**(1
1/2)/11 + 2*B*a**3*sqrt(x) + 6*B*a**2*c*x**(5/2)/5 + 2*B*a*c**2*x**(9/2)/3 + 2*B
*c**3*x**(13/2)/13

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GIAC/XCAS [A]  time = 0.268976, size = 104, normalized size = 1.01 \[ \frac{2}{13} \, B c^{3} x^{\frac{13}{2}} + \frac{2}{11} \, A c^{3} x^{\frac{11}{2}} + \frac{2}{3} \, B a c^{2} x^{\frac{9}{2}} + \frac{6}{7} \, A a c^{2} x^{\frac{7}{2}} + \frac{6}{5} \, B a^{2} c x^{\frac{5}{2}} + 2 \, A a^{2} c x^{\frac{3}{2}} + 2 \, B a^{3} \sqrt{x} - \frac{2 \, A a^{3}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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