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

Optimal. Leaf size=125 \[ \frac{3 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{8 \sqrt{b} d^{5/2}}-\frac{3 \sqrt{a+b x^2} \sqrt{c+d x^2} (b c-a d)}{8 d^2}+\frac{\left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{4 d} \]

[Out]

(-3*(b*c - a*d)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(8*d^2) + ((a + b*x^2)^(3/2)*Sq
rt[c + d*x^2])/(4*d) + (3*(b*c - a*d)^2*ArcTanh[(Sqrt[d]*Sqrt[a + b*x^2])/(Sqrt[
b]*Sqrt[c + d*x^2])])/(8*Sqrt[b]*d^(5/2))

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Rubi [A]  time = 0.24393, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{3 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{8 \sqrt{b} d^{5/2}}-\frac{3 \sqrt{a+b x^2} \sqrt{c+d x^2} (b c-a d)}{8 d^2}+\frac{\left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{4 d} \]

Antiderivative was successfully verified.

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

[Out]

(-3*(b*c - a*d)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(8*d^2) + ((a + b*x^2)^(3/2)*Sq
rt[c + d*x^2])/(4*d) + (3*(b*c - a*d)^2*ArcTanh[(Sqrt[d]*Sqrt[a + b*x^2])/(Sqrt[
b]*Sqrt[c + d*x^2])])/(8*Sqrt[b]*d^(5/2))

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Rubi in Sympy [A]  time = 25.9904, size = 110, normalized size = 0.88 \[ \frac{\left (a + b x^{2}\right )^{\frac{3}{2}} \sqrt{c + d x^{2}}}{4 d} + \frac{3 \sqrt{a + b x^{2}} \sqrt{c + d x^{2}} \left (a d - b c\right )}{8 d^{2}} + \frac{3 \left (a d - b c\right )^{2} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x^{2}}}{\sqrt{d} \sqrt{a + b x^{2}}} \right )}}{8 \sqrt{b} d^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a + b*x**2)**(3/2)*sqrt(c + d*x**2)/(4*d) + 3*sqrt(a + b*x**2)*sqrt(c + d*x**2)
*(a*d - b*c)/(8*d**2) + 3*(a*d - b*c)**2*atanh(sqrt(b)*sqrt(c + d*x**2)/(sqrt(d)
*sqrt(a + b*x**2)))/(8*sqrt(b)*d**(5/2))

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Mathematica [A]  time = 0.0925797, size = 119, normalized size = 0.95 \[ \frac{3 (b c-a d)^2 \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x^2} \sqrt{c+d x^2}+a d+b c+2 b d x^2\right )}{16 \sqrt{b} d^{5/2}}+\frac{\sqrt{a+b x^2} \sqrt{c+d x^2} \left (5 a d-3 b c+2 b d x^2\right )}{8 d^2} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(-3*b*c + 5*a*d + 2*b*d*x^2))/(8*d^2) + (3*(b*c
- a*d)^2*Log[b*c + a*d + 2*b*d*x^2 + 2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x^2]*Sqrt[c +
d*x^2]])/(16*Sqrt[b]*d^(5/2))

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Maple [B]  time = 0.018, size = 337, normalized size = 2.7 \[{\frac{1}{16\,{d}^{2}}\sqrt{b{x}^{2}+a}\sqrt{d{x}^{2}+c} \left ( 4\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}{x}^{2}db\sqrt{bd}+3\,\ln \left ( 1/2\,{\frac{2\,bd{x}^{2}+2\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{2}{d}^{2}-6\,\ln \left ( 1/2\,{\frac{2\,bd{x}^{2}+2\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) cadb+3\,{b}^{2}\ln \left ( 1/2\,{\frac{2\,bd{x}^{2}+2\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){c}^{2}+10\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}ad\sqrt{bd}-6\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}cb\sqrt{bd} \right ){\frac{1}{\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}}}{\frac{1}{\sqrt{bd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(4*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*x^2*
d*b*(b*d)^(1/2)+3*ln(1/2*(2*b*d*x^2+2*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(b*d)^
(1/2)+a*d+b*c)/(b*d)^(1/2))*a^2*d^2-6*ln(1/2*(2*b*d*x^2+2*(b*d*x^4+a*d*x^2+b*c*x
^2+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*c*a*d*b+3*b^2*ln(1/2*(2*b*d*x^2+
2*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*c^2+10*(
b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*a*d*(b*d)^(1/2)-6*(b*d*x^4+a*d*x^2+b*c*x^2+a*
c)^(1/2)*c*b*(b*d)^(1/2))/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/d^2/(b*d)^(1/2)

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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)^(3/2)*x/sqrt(d*x^2 + c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.2834, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (2 \, b d x^{2} - 3 \, b c + 5 \, a d\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} \sqrt{b d} + 3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (4 \,{\left (2 \, b^{2} d^{2} x^{2} + b^{2} c d + a b d^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} +{\left (8 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x^{2}\right )} \sqrt{b d}\right )}{32 \, \sqrt{b d} d^{2}}, \frac{2 \,{\left (2 \, b d x^{2} - 3 \, b c + 5 \, a d\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} \sqrt{-b d} + 3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} b d}\right )}{16 \, \sqrt{-b d} d^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/32*(4*(2*b*d*x^2 - 3*b*c + 5*a*d)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(b*d) +
3*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*log(4*(2*b^2*d^2*x^2 + b^2*c*d + a*b*d^2)*sqr
t(b*x^2 + a)*sqrt(d*x^2 + c) + (8*b^2*d^2*x^4 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 +
8*(b^2*c*d + a*b*d^2)*x^2)*sqrt(b*d)))/(sqrt(b*d)*d^2), 1/16*(2*(2*b*d*x^2 - 3*b
*c + 5*a*d)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(-b*d) + 3*(b^2*c^2 - 2*a*b*c*d
+ a^2*d^2)*arctan(1/2*(2*b*d*x^2 + b*c + a*d)*sqrt(-b*d)/(sqrt(b*x^2 + a)*sqrt(d
*x^2 + c)*b*d)))/(sqrt(-b*d)*d^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x*(a + b*x**2)**(3/2)/sqrt(c + d*x**2), x)

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GIAC/XCAS [A]  time = 0.247332, size = 201, normalized size = 1.61 \[ \frac{{\left (\sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d} \sqrt{b x^{2} + a}{\left (\frac{2 \,{\left (b x^{2} + a\right )}}{b d} - \frac{3 \,{\left (b c d - a d^{2}\right )}}{b d^{3}}\right )} - \frac{3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}{\rm ln}\left ({\left | -\sqrt{b x^{2} + a} \sqrt{b d} + \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} d^{2}}\right )} b}{8 \,{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(sqrt(b^2*c + (b*x^2 + a)*b*d - a*b*d)*sqrt(b*x^2 + a)*(2*(b*x^2 + a)/(b*d)
- 3*(b*c*d - a*d^2)/(b*d^3)) - 3*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*ln(abs(-sqrt(b*
x^2 + a)*sqrt(b*d) + sqrt(b^2*c + (b*x^2 + a)*b*d - a*b*d)))/(sqrt(b*d)*d^2))*b/
abs(b)