3.121 \(\int \left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2} \, dx\)

Optimal. Leaf size=96 \[ \frac{2 b \left (b x^2+c x^4\right )^{5/2} (4 b B-9 A c)}{315 c^3 x^5}-\frac{\left (b x^2+c x^4\right )^{5/2} (4 b B-9 A c)}{63 c^2 x^3}+\frac{B \left (b x^2+c x^4\right )^{5/2}}{9 c x} \]

[Out]

(2*b*(4*b*B - 9*A*c)*(b*x^2 + c*x^4)^(5/2))/(315*c^3*x^5) - ((4*b*B - 9*A*c)*(b*
x^2 + c*x^4)^(5/2))/(63*c^2*x^3) + (B*(b*x^2 + c*x^4)^(5/2))/(9*c*x)

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Rubi [A]  time = 0.142594, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ \frac{2 b \left (b x^2+c x^4\right )^{5/2} (4 b B-9 A c)}{315 c^3 x^5}-\frac{\left (b x^2+c x^4\right )^{5/2} (4 b B-9 A c)}{63 c^2 x^3}+\frac{B \left (b x^2+c x^4\right )^{5/2}}{9 c x} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x^2)*(b*x^2 + c*x^4)^(3/2),x]

[Out]

(2*b*(4*b*B - 9*A*c)*(b*x^2 + c*x^4)^(5/2))/(315*c^3*x^5) - ((4*b*B - 9*A*c)*(b*
x^2 + c*x^4)^(5/2))/(63*c^2*x^3) + (B*(b*x^2 + c*x^4)^(5/2))/(9*c*x)

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Rubi in Sympy [A]  time = 14.6535, size = 87, normalized size = 0.91 \[ \frac{B \left (b x^{2} + c x^{4}\right )^{\frac{5}{2}}}{9 c x} - \frac{2 b \left (9 A c - 4 B b\right ) \left (b x^{2} + c x^{4}\right )^{\frac{5}{2}}}{315 c^{3} x^{5}} + \frac{\left (9 A c - 4 B b\right ) \left (b x^{2} + c x^{4}\right )^{\frac{5}{2}}}{63 c^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)*(c*x**4+b*x**2)**(3/2),x)

[Out]

B*(b*x**2 + c*x**4)**(5/2)/(9*c*x) - 2*b*(9*A*c - 4*B*b)*(b*x**2 + c*x**4)**(5/2
)/(315*c**3*x**5) + (9*A*c - 4*B*b)*(b*x**2 + c*x**4)**(5/2)/(63*c**2*x**3)

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Mathematica [A]  time = 0.0768993, size = 71, normalized size = 0.74 \[ \frac{x \left (b+c x^2\right )^3 \left (-2 b c \left (9 A+10 B x^2\right )+5 c^2 x^2 \left (9 A+7 B x^2\right )+8 b^2 B\right )}{315 c^3 \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x^2)*(b*x^2 + c*x^4)^(3/2),x]

[Out]

(x*(b + c*x^2)^3*(8*b^2*B + 5*c^2*x^2*(9*A + 7*B*x^2) - 2*b*c*(9*A + 10*B*x^2)))
/(315*c^3*Sqrt[x^2*(b + c*x^2)])

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Maple [A]  time = 0.008, size = 67, normalized size = 0.7 \[ -{\frac{ \left ( c{x}^{2}+b \right ) \left ( -35\,B{c}^{2}{x}^{4}-45\,A{x}^{2}{c}^{2}+20\,B{x}^{2}bc+18\,Abc-8\,{b}^{2}B \right ) }{315\,{c}^{3}{x}^{3}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)*(c*x^4+b*x^2)^(3/2),x)

[Out]

-1/315*(c*x^2+b)*(-35*B*c^2*x^4-45*A*c^2*x^2+20*B*b*c*x^2+18*A*b*c-8*B*b^2)*(c*x
^4+b*x^2)^(3/2)/c^3/x^3

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Maxima [A]  time = 1.39758, size = 142, normalized size = 1.48 \[ \frac{{\left (5 \, c^{3} x^{6} + 8 \, b c^{2} x^{4} + b^{2} c x^{2} - 2 \, b^{3}\right )} \sqrt{c x^{2} + b} A}{35 \, c^{2}} + \frac{{\left (35 \, c^{4} x^{8} + 50 \, b c^{3} x^{6} + 3 \, b^{2} c^{2} x^{4} - 4 \, b^{3} c x^{2} + 8 \, b^{4}\right )} \sqrt{c x^{2} + b} B}{315 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^(3/2)*(B*x^2 + A),x, algorithm="maxima")

[Out]

1/35*(5*c^3*x^6 + 8*b*c^2*x^4 + b^2*c*x^2 - 2*b^3)*sqrt(c*x^2 + b)*A/c^2 + 1/315
*(35*c^4*x^8 + 50*b*c^3*x^6 + 3*b^2*c^2*x^4 - 4*b^3*c*x^2 + 8*b^4)*sqrt(c*x^2 +
b)*B/c^3

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Fricas [A]  time = 0.222136, size = 143, normalized size = 1.49 \[ \frac{{\left (35 \, B c^{4} x^{8} + 5 \,{\left (10 \, B b c^{3} + 9 \, A c^{4}\right )} x^{6} + 8 \, B b^{4} - 18 \, A b^{3} c + 3 \,{\left (B b^{2} c^{2} + 24 \, A b c^{3}\right )} x^{4} -{\left (4 \, B b^{3} c - 9 \, A b^{2} c^{2}\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{315 \, c^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^(3/2)*(B*x^2 + A),x, algorithm="fricas")

[Out]

1/315*(35*B*c^4*x^8 + 5*(10*B*b*c^3 + 9*A*c^4)*x^6 + 8*B*b^4 - 18*A*b^3*c + 3*(B
*b^2*c^2 + 24*A*b*c^3)*x^4 - (4*B*b^3*c - 9*A*b^2*c^2)*x^2)*sqrt(c*x^4 + b*x^2)/
(c^3*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}} \left (A + B x^{2}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**2+A)*(c*x**4+b*x**2)**(3/2),x)

[Out]

Integral((x**2*(b + c*x**2))**(3/2)*(A + B*x**2), x)

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GIAC/XCAS [A]  time = 0.217142, size = 288, normalized size = 3. \[ \frac{\frac{21 \,{\left (3 \,{\left (c x^{2} + b\right )}^{\frac{5}{2}} - 5 \,{\left (c x^{2} + b\right )}^{\frac{3}{2}} b\right )} A b{\rm sign}\left (x\right )}{c} + \frac{3 \,{\left (15 \,{\left (c x^{2} + b\right )}^{\frac{7}{2}} - 42 \,{\left (c x^{2} + b\right )}^{\frac{5}{2}} b + 35 \,{\left (c x^{2} + b\right )}^{\frac{3}{2}} b^{2}\right )} B b{\rm sign}\left (x\right )}{c^{2}} + \frac{3 \,{\left (15 \,{\left (c x^{2} + b\right )}^{\frac{7}{2}} - 42 \,{\left (c x^{2} + b\right )}^{\frac{5}{2}} b + 35 \,{\left (c x^{2} + b\right )}^{\frac{3}{2}} b^{2}\right )} A{\rm sign}\left (x\right )}{c} + \frac{{\left (35 \,{\left (c x^{2} + b\right )}^{\frac{9}{2}} - 135 \,{\left (c x^{2} + b\right )}^{\frac{7}{2}} b + 189 \,{\left (c x^{2} + b\right )}^{\frac{5}{2}} b^{2} - 105 \,{\left (c x^{2} + b\right )}^{\frac{3}{2}} b^{3}\right )} B{\rm sign}\left (x\right )}{c^{2}}}{315 \, c} - \frac{2 \,{\left (4 \, B b^{\frac{9}{2}} - 9 \, A b^{\frac{7}{2}} c\right )}{\rm sign}\left (x\right )}{315 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^(3/2)*(B*x^2 + A),x, algorithm="giac")

[Out]

1/315*(21*(3*(c*x^2 + b)^(5/2) - 5*(c*x^2 + b)^(3/2)*b)*A*b*sign(x)/c + 3*(15*(c
*x^2 + b)^(7/2) - 42*(c*x^2 + b)^(5/2)*b + 35*(c*x^2 + b)^(3/2)*b^2)*B*b*sign(x)
/c^2 + 3*(15*(c*x^2 + b)^(7/2) - 42*(c*x^2 + b)^(5/2)*b + 35*(c*x^2 + b)^(3/2)*b
^2)*A*sign(x)/c + (35*(c*x^2 + b)^(9/2) - 135*(c*x^2 + b)^(7/2)*b + 189*(c*x^2 +
b)^(5/2)*b^2 - 105*(c*x^2 + b)^(3/2)*b^3)*B*sign(x)/c^2)/c - 2/315*(4*B*b^(9/2)
- 9*A*b^(7/2)*c)*sign(x)/c^3