3.637 \(\int \frac{x^3 \left (a+b x^2\right )^2}{\sqrt{c+d x^2}} \, dx\)

Optimal. Leaf size=112 \[ -\frac{b \left (c+d x^2\right )^{5/2} (3 b c-2 a d)}{5 d^4}+\frac{\left (c+d x^2\right )^{3/2} (b c-a d) (3 b c-a d)}{3 d^4}-\frac{c \sqrt{c+d x^2} (b c-a d)^2}{d^4}+\frac{b^2 \left (c+d x^2\right )^{7/2}}{7 d^4} \]

[Out]

-((c*(b*c - a*d)^2*Sqrt[c + d*x^2])/d^4) + ((b*c - a*d)*(3*b*c - a*d)*(c + d*x^2
)^(3/2))/(3*d^4) - (b*(3*b*c - 2*a*d)*(c + d*x^2)^(5/2))/(5*d^4) + (b^2*(c + d*x
^2)^(7/2))/(7*d^4)

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Rubi [A]  time = 0.267724, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{b \left (c+d x^2\right )^{5/2} (3 b c-2 a d)}{5 d^4}+\frac{\left (c+d x^2\right )^{3/2} (b c-a d) (3 b c-a d)}{3 d^4}-\frac{c \sqrt{c+d x^2} (b c-a d)^2}{d^4}+\frac{b^2 \left (c+d x^2\right )^{7/2}}{7 d^4} \]

Antiderivative was successfully verified.

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

[Out]

-((c*(b*c - a*d)^2*Sqrt[c + d*x^2])/d^4) + ((b*c - a*d)*(3*b*c - a*d)*(c + d*x^2
)^(3/2))/(3*d^4) - (b*(3*b*c - 2*a*d)*(c + d*x^2)^(5/2))/(5*d^4) + (b^2*(c + d*x
^2)^(7/2))/(7*d^4)

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Rubi in Sympy [A]  time = 31.9884, size = 99, normalized size = 0.88 \[ \frac{b^{2} \left (c + d x^{2}\right )^{\frac{7}{2}}}{7 d^{4}} + \frac{b \left (c + d x^{2}\right )^{\frac{5}{2}} \left (2 a d - 3 b c\right )}{5 d^{4}} - \frac{c \sqrt{c + d x^{2}} \left (a d - b c\right )^{2}}{d^{4}} + \frac{\left (c + d x^{2}\right )^{\frac{3}{2}} \left (a d - 3 b c\right ) \left (a d - b c\right )}{3 d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0953668, size = 99, normalized size = 0.88 \[ \frac{\sqrt{c+d x^2} \left (35 a^2 d^2 \left (d x^2-2 c\right )+14 a b d \left (8 c^2-4 c d x^2+3 d^2 x^4\right )-3 b^2 \left (16 c^3-8 c^2 d x^2+6 c d^2 x^4-5 d^3 x^6\right )\right )}{105 d^4} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[c + d*x^2]*(35*a^2*d^2*(-2*c + d*x^2) + 14*a*b*d*(8*c^2 - 4*c*d*x^2 + 3*d^
2*x^4) - 3*b^2*(16*c^3 - 8*c^2*d*x^2 + 6*c*d^2*x^4 - 5*d^3*x^6)))/(105*d^4)

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Maple [A]  time = 0.01, size = 108, normalized size = 1. \[ -{\frac{-15\,{b}^{2}{x}^{6}{d}^{3}-42\,ab{d}^{3}{x}^{4}+18\,{b}^{2}c{d}^{2}{x}^{4}-35\,{a}^{2}{d}^{3}{x}^{2}+56\,abc{d}^{2}{x}^{2}-24\,{b}^{2}{c}^{2}d{x}^{2}+70\,{a}^{2}c{d}^{2}-112\,ab{c}^{2}d+48\,{b}^{2}{c}^{3}}{105\,{d}^{4}}\sqrt{d{x}^{2}+c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/105*(d*x^2+c)^(1/2)*(-15*b^2*d^3*x^6-42*a*b*d^3*x^4+18*b^2*c*d^2*x^4-35*a^2*d
^3*x^2+56*a*b*c*d^2*x^2-24*b^2*c^2*d*x^2+70*a^2*c*d^2-112*a*b*c^2*d+48*b^2*c^3)/
d^4

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.24873, size = 139, normalized size = 1.24 \[ \frac{{\left (15 \, b^{2} d^{3} x^{6} - 48 \, b^{2} c^{3} + 112 \, a b c^{2} d - 70 \, a^{2} c d^{2} - 6 \,{\left (3 \, b^{2} c d^{2} - 7 \, a b d^{3}\right )} x^{4} +{\left (24 \, b^{2} c^{2} d - 56 \, a b c d^{2} + 35 \, a^{2} d^{3}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{105 \, d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/105*(15*b^2*d^3*x^6 - 48*b^2*c^3 + 112*a*b*c^2*d - 70*a^2*c*d^2 - 6*(3*b^2*c*d
^2 - 7*a*b*d^3)*x^4 + (24*b^2*c^2*d - 56*a*b*c*d^2 + 35*a^2*d^3)*x^2)*sqrt(d*x^2
+ c)/d^4

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Sympy [A]  time = 5.10787, size = 240, normalized size = 2.14 \[ \begin{cases} - \frac{2 a^{2} c \sqrt{c + d x^{2}}}{3 d^{2}} + \frac{a^{2} x^{2} \sqrt{c + d x^{2}}}{3 d} + \frac{16 a b c^{2} \sqrt{c + d x^{2}}}{15 d^{3}} - \frac{8 a b c x^{2} \sqrt{c + d x^{2}}}{15 d^{2}} + \frac{2 a b x^{4} \sqrt{c + d x^{2}}}{5 d} - \frac{16 b^{2} c^{3} \sqrt{c + d x^{2}}}{35 d^{4}} + \frac{8 b^{2} c^{2} x^{2} \sqrt{c + d x^{2}}}{35 d^{3}} - \frac{6 b^{2} c x^{4} \sqrt{c + d x^{2}}}{35 d^{2}} + \frac{b^{2} x^{6} \sqrt{c + d x^{2}}}{7 d} & \text{for}\: d \neq 0 \\\frac{\frac{a^{2} x^{4}}{4} + \frac{a b x^{6}}{3} + \frac{b^{2} x^{8}}{8}}{\sqrt{c}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-2*a**2*c*sqrt(c + d*x**2)/(3*d**2) + a**2*x**2*sqrt(c + d*x**2)/(3*d
) + 16*a*b*c**2*sqrt(c + d*x**2)/(15*d**3) - 8*a*b*c*x**2*sqrt(c + d*x**2)/(15*d
**2) + 2*a*b*x**4*sqrt(c + d*x**2)/(5*d) - 16*b**2*c**3*sqrt(c + d*x**2)/(35*d**
4) + 8*b**2*c**2*x**2*sqrt(c + d*x**2)/(35*d**3) - 6*b**2*c*x**4*sqrt(c + d*x**2
)/(35*d**2) + b**2*x**6*sqrt(c + d*x**2)/(7*d), Ne(d, 0)), ((a**2*x**4/4 + a*b*x
**6/3 + b**2*x**8/8)/sqrt(c), True))

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GIAC/XCAS [A]  time = 0.229019, size = 203, normalized size = 1.81 \[ \frac{15 \,{\left (d x^{2} + c\right )}^{\frac{7}{2}} b^{2} - 63 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} b^{2} c + 105 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} b^{2} c^{2} - 105 \, \sqrt{d x^{2} + c} b^{2} c^{3} + 42 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} a b d - 140 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a b c d + 210 \, \sqrt{d x^{2} + c} a b c^{2} d + 35 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a^{2} d^{2} - 105 \, \sqrt{d x^{2} + c} a^{2} c d^{2}}{105 \, d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/105*(15*(d*x^2 + c)^(7/2)*b^2 - 63*(d*x^2 + c)^(5/2)*b^2*c + 105*(d*x^2 + c)^(
3/2)*b^2*c^2 - 105*sqrt(d*x^2 + c)*b^2*c^3 + 42*(d*x^2 + c)^(5/2)*a*b*d - 140*(d
*x^2 + c)^(3/2)*a*b*c*d + 210*sqrt(d*x^2 + c)*a*b*c^2*d + 35*(d*x^2 + c)^(3/2)*a
^2*d^2 - 105*sqrt(d*x^2 + c)*a^2*c*d^2)/d^4