3.260 \(\int x \sqrt{c+d x^2} \sqrt{a^2+2 a b x^2+b^2 x^4} \, dx\)

Optimal. Leaf size=108 \[ \frac{b \sqrt{a^2+2 a b x^2+b^2 x^4} \left (c+d x^2\right )^{5/2}}{5 d^2 \left (a+b x^2\right )}-\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \left (c+d x^2\right )^{3/2} (b c-a d)}{3 d^2 \left (a+b x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.280404, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086 \[ \frac{b \sqrt{a^2+2 a b x^2+b^2 x^4} \left (c+d x^2\right )^{5/2}}{5 d^2 \left (a+b x^2\right )}-\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \left (c+d x^2\right )^{3/2} (b c-a d)}{3 d^2 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[x*Sqrt[c + d*x^2]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]

[Out]

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

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Rubi in Sympy [A]  time = 18.1331, size = 75, normalized size = 0.69 \[ \frac{b \left (c + d x^{2}\right )^{\frac{5}{2}} \sqrt{\left (a + b x^{2}\right )^{2}}}{5 d^{2} \left (a + b x^{2}\right )} + \frac{\left (c + d x^{2}\right )^{\frac{3}{2}} \left (a d - b c\right ) \sqrt{\left (a + b x^{2}\right )^{2}}}{3 d^{2} \left (a + b x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0578292, size = 56, normalized size = 0.52 \[ \frac{\sqrt{\left (a+b x^2\right )^2} \left (c+d x^2\right )^{3/2} \left (5 a d-2 b c+3 b d x^2\right )}{15 d^2 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[x*Sqrt[c + d*x^2]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]

[Out]

(Sqrt[(a + b*x^2)^2]*(c + d*x^2)^(3/2)*(-2*b*c + 5*a*d + 3*b*d*x^2))/(15*d^2*(a
+ b*x^2))

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Maple [A]  time = 0.006, size = 51, normalized size = 0.5 \[{\frac{3\,b{x}^{2}d+5\,ad-2\,bc}{15\,{d}^{2} \left ( b{x}^{2}+a \right ) } \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(d*x^2+c)^(3/2)*(3*b*d*x^2+5*a*d-2*b*c)*((b*x^2+a)^2)^(1/2)/d^2/(b*x^2+a)

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Maxima [A]  time = 0.708693, size = 68, normalized size = 0.63 \[ \frac{{\left (3 \, b d^{2} x^{4} - 2 \, b c^{2} + 5 \, a c d +{\left (b c d + 5 \, a d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{15 \, d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(3*b*d^2*x^4 - 2*b*c^2 + 5*a*c*d + (b*c*d + 5*a*d^2)*x^2)*sqrt(d*x^2 + c)/d
^2

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Fricas [A]  time = 0.269568, size = 68, normalized size = 0.63 \[ \frac{{\left (3 \, b d^{2} x^{4} - 2 \, b c^{2} + 5 \, a c d +{\left (b c d + 5 \, a d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{15 \, d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(3*b*d^2*x^4 - 2*b*c^2 + 5*a*c*d + (b*c*d + 5*a*d^2)*x^2)*sqrt(d*x^2 + c)/d
^2

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.271715, size = 85, normalized size = 0.79 \[ \frac{5 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a{\rm sign}\left (b x^{2} + a\right ) + \frac{{\left (3 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} - 5 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} c\right )} b{\rm sign}\left (b x^{2} + a\right )}{d}}{15 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(5*(d*x^2 + c)^(3/2)*a*sign(b*x^2 + a) + (3*(d*x^2 + c)^(5/2) - 5*(d*x^2 +
c)^(3/2)*c)*b*sign(b*x^2 + a)/d)/d