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

Optimal. Leaf size=83 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{b^{3/2} \sqrt{d}}+\frac{a \sqrt{c+d x^2}}{b \sqrt{a+b x^2} (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.231633, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{b^{3/2} \sqrt{d}}+\frac{a \sqrt{c+d x^2}}{b \sqrt{a+b x^2} (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 21.1407, size = 71, normalized size = 0.86 \[ - \frac{a \sqrt{c + d x^{2}}}{b \sqrt{a + b x^{2}} \left (a d - b c\right )} + \frac{\operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x^{2}}}{\sqrt{d} \sqrt{a + b x^{2}}} \right )}}{b^{\frac{3}{2}} \sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.121371, size = 101, normalized size = 1.22 \[ \frac{\log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x^2} \sqrt{c+d x^2}+a d+b c+2 b d x^2\right )}{2 b^{3/2} \sqrt{d}}+\frac{a \sqrt{c+d x^2}}{b \sqrt{a+b x^2} (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.031, size = 292, normalized size = 3.5 \[{\frac{1}{2\, \left ( ad-bc \right ) b} \left ( \ln \left ({\frac{1}{2} \left ( 2\,bd{x}^{2}+2\,\sqrt{ \left ( b{x}^{2}+a \right ) \left ( d{x}^{2}+c \right ) }\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ){x}^{2}abd-\ln \left ({\frac{1}{2} \left ( 2\,bd{x}^{2}+2\,\sqrt{ \left ( b{x}^{2}+a \right ) \left ( d{x}^{2}+c \right ) }\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ){x}^{2}{b}^{2}c+\ln \left ({\frac{1}{2} \left ( 2\,bd{x}^{2}+2\,\sqrt{ \left ( b{x}^{2}+a \right ) \left ( d{x}^{2}+c \right ) }\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ){a}^{2}d-\ln \left ({\frac{1}{2} \left ( 2\,bd{x}^{2}+2\,\sqrt{ \left ( b{x}^{2}+a \right ) \left ( d{x}^{2}+c \right ) }\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ) abc-2\,a\sqrt{ \left ( b{x}^{2}+a \right ) \left ( d{x}^{2}+c \right ) }\sqrt{bd} \right ) \sqrt{d{x}^{2}+c}{\frac{1}{\sqrt{b{x}^{2}+a}}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ \left ( b{x}^{2}+a \right ) \left ( d{x}^{2}+c \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(ln(1/2*(2*b*d*x^2+2*((b*x^2+a)*(d*x^2+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^
(1/2))*x^2*a*b*d-ln(1/2*(2*b*d*x^2+2*((b*x^2+a)*(d*x^2+c))^(1/2)*(b*d)^(1/2)+a*d
+b*c)/(b*d)^(1/2))*x^2*b^2*c+ln(1/2*(2*b*d*x^2+2*((b*x^2+a)*(d*x^2+c))^(1/2)*(b*
d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^2*d-ln(1/2*(2*b*d*x^2+2*((b*x^2+a)*(d*x^2+c))^(
1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a*b*c-2*a*((b*x^2+a)*(d*x^2+c))^(1/2)*(b*
d)^(1/2))/b*(d*x^2+c)^(1/2)/(b*x^2+a)^(1/2)/(b*d)^(1/2)/(a*d-b*c)/((b*x^2+a)*(d*
x^2+c))^(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(x^3/((b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.338405, size = 1, normalized size = 0.01 \[ \left [\frac{4 \, \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} \sqrt{b d} a +{\left (a b c - a^{2} d +{\left (b^{2} c - a b d\right )} x^{2}\right )} \log \left (4 \,{\left (2 \, b^{2} d^{2} x^{2} + b^{2} c d + a b d^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} +{\left (8 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x^{2}\right )} \sqrt{b d}\right )}{4 \,{\left (a b^{2} c - a^{2} b d +{\left (b^{3} c - a b^{2} d\right )} x^{2}\right )} \sqrt{b d}}, \frac{2 \, \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} \sqrt{-b d} a +{\left (a b c - a^{2} d +{\left (b^{2} c - a b d\right )} x^{2}\right )} \arctan \left (\frac{{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} b d}\right )}{2 \,{\left (a b^{2} c - a^{2} b d +{\left (b^{3} c - a b^{2} d\right )} x^{2}\right )} \sqrt{-b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*(4*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(b*d)*a + (a*b*c - a^2*d + (b^2*c -
a*b*d)*x^2)*log(4*(2*b^2*d^2*x^2 + b^2*c*d + a*b*d^2)*sqrt(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)))/((a*b^2*c - a^2*b*d + (b^3*c - a*b^2*d)*x^2)*sqrt(b*d)), 1/2*(2*
sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(-b*d)*a + (a*b*c - a^2*d + (b^2*c - a*b*d)*
x^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)))/((a*b^2*c - a^2*b*d + (b^3*c - a*b^2*d)*x^2)*sqrt(-b*d))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.592749, size = 4, normalized size = 0.05 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x