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

Optimal. Leaf size=136 \[ \frac{b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{\sqrt{d}}-\frac{\sqrt{a} (3 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 c^{3/2}}-\frac{a \sqrt{a+b x^2} \sqrt{c+d x^2}}{2 c x^2} \]

[Out]

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

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Rubi [A]  time = 0.440704, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{\sqrt{d}}-\frac{\sqrt{a} (3 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 c^{3/2}}-\frac{a \sqrt{a+b x^2} \sqrt{c+d x^2}}{2 c x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 42.9618, size = 122, normalized size = 0.9 \[ \frac{\sqrt{a} \left (a d - 3 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x^{2}}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{2 c^{\frac{3}{2}}} - \frac{a \sqrt{a + b x^{2}} \sqrt{c + d x^{2}}}{2 c x^{2}} + \frac{b^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x^{2}}}{\sqrt{b} \sqrt{c + d x^{2}}} \right )}}{\sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.512971, size = 327, normalized size = 2.4 \[ \frac{a \left (-\frac{4 b^2 c^2 x^4 F_1\left (1;\frac{1}{2},\frac{1}{2};2;-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{x^2 \left (a d F_1\left (2;\frac{1}{2},\frac{3}{2};3;-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+b c F_1\left (2;\frac{3}{2},\frac{1}{2};3;-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )-4 a c F_1\left (1;\frac{1}{2},\frac{1}{2};2;-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}+\frac{2 b d x^4 (3 b c-a d) F_1\left (1;\frac{1}{2},\frac{1}{2};2;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )}{-4 b d x^2 F_1\left (1;\frac{1}{2},\frac{1}{2};2;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )+b c F_1\left (2;\frac{1}{2},\frac{3}{2};3;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )+a d F_1\left (2;\frac{3}{2},\frac{1}{2};3;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )}-\left (a+b x^2\right ) \left (c+d x^2\right )\right )}{2 c x^2 \sqrt{a+b x^2} \sqrt{c+d x^2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(a*(-((a + b*x^2)*(c + d*x^2)) + (2*b*d*(3*b*c - a*d)*x^4*AppellF1[1, 1/2, 1/2,
2, -(a/(b*x^2)), -(c/(d*x^2))])/(-4*b*d*x^2*AppellF1[1, 1/2, 1/2, 2, -(a/(b*x^2)
), -(c/(d*x^2))] + b*c*AppellF1[2, 1/2, 3/2, 3, -(a/(b*x^2)), -(c/(d*x^2))] + a*
d*AppellF1[2, 3/2, 1/2, 3, -(a/(b*x^2)), -(c/(d*x^2))]) - (4*b^2*c^2*x^4*AppellF
1[1, 1/2, 1/2, 2, -((b*x^2)/a), -((d*x^2)/c)])/(-4*a*c*AppellF1[1, 1/2, 1/2, 2,
-((b*x^2)/a), -((d*x^2)/c)] + x^2*(a*d*AppellF1[2, 1/2, 3/2, 3, -((b*x^2)/a), -(
(d*x^2)/c)] + b*c*AppellF1[2, 3/2, 1/2, 3, -((b*x^2)/a), -((d*x^2)/c)]))))/(2*c*
x^2*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])

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Maple [B]  time = 0.023, size = 298, normalized size = 2.2 \[{\frac{1}{4\,c{x}^{2}}\sqrt{b{x}^{2}+a}\sqrt{d{x}^{2}+c} \left ( 2\,\ln \left ( 1/2\,{\frac{2\,bd{x}^{2}+2\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}{b}^{2}c\sqrt{ac}+\ln \left ({\frac{1}{{x}^{2}} \left ( ad{x}^{2}+c{x}^{2}b+2\,\sqrt{ac}\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}+2\,ac \right ) } \right ){x}^{2}{a}^{2}d\sqrt{bd}-3\,\ln \left ({\frac{ad{x}^{2}+c{x}^{2}b+2\,\sqrt{ac}\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}+2\,ac}{{x}^{2}}} \right ){x}^{2}abc\sqrt{bd}-2\,a\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}\sqrt{ac}\sqrt{bd} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}}}{\frac{1}{\sqrt{bd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/c*(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))*x^2*b^2*c*(a*c)^(1/2)+ln((a*d
*x^2+c*x^2*b+2*(a*c)^(1/2)*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+2*a*c)/x^2)*x^2*a
^2*d*(b*d)^(1/2)-3*ln((a*d*x^2+c*x^2*b+2*(a*c)^(1/2)*(b*d*x^4+a*d*x^2+b*c*x^2+a*
c)^(1/2)+2*a*c)/x^2)*x^2*a*b*c*(b*d)^(1/2)-2*a*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/
2)*(a*c)^(1/2)*(b*d)^(1/2))/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/x^2/(a*c)^(1/2)/
(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)/(sqrt(d*x^2 + c)*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.745405, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*(2*b*c*x^2*sqrt(b/d)*log(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 + 4*(2*b*d^2*x^2 + b*c*d + a*d^2)*sqrt(b*x^2 + a)*sqrt(d
*x^2 + c)*sqrt(b/d)) - (3*b*c - a*d)*x^2*sqrt(a/c)*log(((b^2*c^2 + 6*a*b*c*d + a
^2*d^2)*x^4 + 8*a^2*c^2 + 8*(a*b*c^2 + a^2*c*d)*x^2 + 4*(2*a*c^2 + (b*c^2 + a*c*
d)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(a/c))/x^4) - 4*sqrt(b*x^2 + a)*sqrt
(d*x^2 + c)*a)/(c*x^2), 1/8*(4*b*c*x^2*sqrt(-b/d)*arctan(1/2*(2*b*d*x^2 + b*c +
a*d)/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*d*sqrt(-b/d))) - (3*b*c - a*d)*x^2*sqrt(a/
c)*log(((b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^4 + 8*a^2*c^2 + 8*(a*b*c^2 + a^2*c*d)*
x^2 + 4*(2*a*c^2 + (b*c^2 + a*c*d)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(a/c
))/x^4) - 4*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*a)/(c*x^2), 1/4*(b*c*x^2*sqrt(b/d)*l
og(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 + 4
*(2*b*d^2*x^2 + b*c*d + a*d^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(b/d)) - (3*b
*c - a*d)*x^2*sqrt(-a/c)*arctan(1/2*((b*c + a*d)*x^2 + 2*a*c)/(sqrt(b*x^2 + a)*s
qrt(d*x^2 + c)*c*sqrt(-a/c))) - 2*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*a)/(c*x^2), 1/
4*(2*b*c*x^2*sqrt(-b/d)*arctan(1/2*(2*b*d*x^2 + b*c + a*d)/(sqrt(b*x^2 + a)*sqrt
(d*x^2 + c)*d*sqrt(-b/d))) - (3*b*c - a*d)*x^2*sqrt(-a/c)*arctan(1/2*((b*c + a*d
)*x^2 + 2*a*c)/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*c*sqrt(-a/c))) - 2*sqrt(b*x^2 +
a)*sqrt(d*x^2 + c)*a)/(c*x^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.602651, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x