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

Optimal. Leaf size=133 \[ -\frac{a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{\sqrt{c}}-\frac{\sqrt{b} (b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{2 d^{3/2}}+\frac{b \sqrt{a+b x^2} \sqrt{c+d x^2}}{2 d} \]

[Out]

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

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Rubi [A]  time = 0.439869, antiderivative size = 133, 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 \[ -\frac{a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{\sqrt{c}}-\frac{\sqrt{b} (b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{2 d^{3/2}}+\frac{b \sqrt{a+b x^2} \sqrt{c+d x^2}}{2 d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 42.3388, size = 119, normalized size = 0.89 \[ - \frac{a^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x^{2}}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{\sqrt{c}} + \frac{\sqrt{b} \left (3 a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x^{2}}}{\sqrt{d} \sqrt{a + b x^{2}}} \right )}}{2 d^{\frac{3}{2}}} + \frac{b \sqrt{a + b x^{2}} \sqrt{c + d x^{2}}}{2 d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.998953, size = 400, normalized size = 3.01 \[ \frac{b \left (\frac{4 a^2 d x^2 F_1\left (1;\frac{1}{2},\frac{1}{2};2;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )}{-4 b d x^2 F_1\left (1;\frac{1}{2},\frac{1}{2};2;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )+b c F_1\left (2;\frac{1}{2},\frac{3}{2};3;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )+a d F_1\left (2;\frac{3}{2},\frac{1}{2};3;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )}+\frac{x^2 \left (a+b x^2\right ) \left (c+d x^2\right ) \left (a d F_1\left (2;\frac{1}{2},\frac{3}{2};3;-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+b c F_1\left (2;\frac{3}{2},\frac{1}{2};3;-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )-2 a c \left (2 a c+5 a d x^2+b c x^2+2 b d x^4\right ) F_1\left (1;\frac{1}{2},\frac{1}{2};2;-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{d \left (x^2 \left (a d F_1\left (2;\frac{1}{2},\frac{3}{2};3;-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+b c F_1\left (2;\frac{3}{2},\frac{1}{2};3;-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )-4 a c F_1\left (1;\frac{1}{2},\frac{1}{2};2;-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )}\right )}{2 \sqrt{a+b x^2} \sqrt{c+d x^2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(b*((4*a^2*d*x^2*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))]) + (-2*a*c*(2*a*c + b*c*x^2 + 5*a*d*x^2 + 2*b*d*x^4)*AppellF1[1, 1
/2, 1/2, 2, -((b*x^2)/a), -((d*x^2)/c)] + x^2*(a + b*x^2)*(c + d*x^2)*(a*d*Appel
lF1[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)]))/(d*(-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*Sqrt[a + b*x^2]*
Sqrt[c + d*x^2])

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Maple [B]  time = 0.021, size = 287, normalized size = 2.2 \[{\frac{1}{4\,d}\sqrt{b{x}^{2}+a}\sqrt{d{x}^{2}+c} \left ( 3\,a\ln \left ( 1/2\,{\frac{2\,bd{x}^{2}+2\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) b\sqrt{ac}d-c{b}^{2}\ln \left ({\frac{1}{2} \left ( 2\,bd{x}^{2}+2\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ) \sqrt{ac}-2\,{a}^{2}\ln \left ({\frac{ad{x}^{2}+c{x}^{2}b+2\,\sqrt{ac}\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}+2\,ac}{{x}^{2}}} \right ) \sqrt{bd}d+2\,b\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}\sqrt{bd}\sqrt{ac} \right ){\frac{1}{\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.888444, 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),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.250013, size = 284, normalized size = 2.14 \[ -\frac{{\left (\frac{4 \, \sqrt{b d} a^{2} \arctan \left (-\frac{b^{2} c + a b d -{\left (\sqrt{b x^{2} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt{-a b c d} b}\right )}{\sqrt{-a b c d} b} - \frac{2 \, \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d} \sqrt{b x^{2} + a}}{b d} - \frac{{\left (\sqrt{b d} b c - 3 \, \sqrt{b d} a d\right )}{\rm ln}\left ({\left (\sqrt{b x^{2} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d}\right )}^{2}\right )}{b d^{2}}\right )} b^{2}}{4 \,{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(4*sqrt(b*d)*a^2*arctan(-1/2*(b^2*c + a*b*d - (sqrt(b*x^2 + a)*sqrt(b*d) -
sqrt(b^2*c + (b*x^2 + a)*b*d - a*b*d))^2)/(sqrt(-a*b*c*d)*b))/(sqrt(-a*b*c*d)*b)
- 2*sqrt(b^2*c + (b*x^2 + a)*b*d - a*b*d)*sqrt(b*x^2 + a)/(b*d) - (sqrt(b*d)*b*
c - 3*sqrt(b*d)*a*d)*ln((sqrt(b*x^2 + a)*sqrt(b*d) - sqrt(b^2*c + (b*x^2 + a)*b*
d - a*b*d))^2)/(b*d^2))*b^2/abs(b)