3.957 \(\int \frac{\left (a+b x^2\right )^{5/2}}{x \sqrt{c+d x^2}} \, dx\)

Optimal. Leaf size=187 \[ -\frac{a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{\sqrt{c}}+\frac{\sqrt{b} \left (15 a^2 d^2-10 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{8 d^{5/2}}-\frac{b \sqrt{a+b x^2} \sqrt{c+d x^2} (3 b c-7 a d)}{8 d^2}+\frac{b \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{4 d} \]

[Out]

-(b*(3*b*c - 7*a*d)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(8*d^2) + (b*(a + b*x^2)^(3
/2)*Sqrt[c + d*x^2])/(4*d) - (a^(5/2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x^2])/(Sqrt[a]
*Sqrt[c + d*x^2])])/Sqrt[c] + (Sqrt[b]*(3*b^2*c^2 - 10*a*b*c*d + 15*a^2*d^2)*Arc
Tanh[(Sqrt[d]*Sqrt[a + b*x^2])/(Sqrt[b]*Sqrt[c + d*x^2])])/(8*d^(5/2))

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Rubi [A]  time = 0.674128, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ -\frac{a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{\sqrt{c}}+\frac{\sqrt{b} \left (15 a^2 d^2-10 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{8 d^{5/2}}-\frac{b \sqrt{a+b x^2} \sqrt{c+d x^2} (3 b c-7 a d)}{8 d^2}+\frac{b \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{4 d} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(5/2)/(x*Sqrt[c + d*x^2]),x]

[Out]

-(b*(3*b*c - 7*a*d)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(8*d^2) + (b*(a + b*x^2)^(3
/2)*Sqrt[c + d*x^2])/(4*d) - (a^(5/2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x^2])/(Sqrt[a]
*Sqrt[c + d*x^2])])/Sqrt[c] + (Sqrt[b]*(3*b^2*c^2 - 10*a*b*c*d + 15*a^2*d^2)*Arc
Tanh[(Sqrt[d]*Sqrt[a + b*x^2])/(Sqrt[b]*Sqrt[c + d*x^2])])/(8*d^(5/2))

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Rubi in Sympy [A]  time = 65.934, size = 173, normalized size = 0.93 \[ - \frac{a^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x^{2}}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{\sqrt{c}} + \frac{\sqrt{b} \left (15 a^{2} d^{2} - 10 a b c d + 3 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x^{2}}}{\sqrt{d} \sqrt{a + b x^{2}}} \right )}}{8 d^{\frac{5}{2}}} + \frac{b \left (a + b x^{2}\right )^{\frac{3}{2}} \sqrt{c + d x^{2}}}{4 d} + \frac{b \sqrt{a + b x^{2}} \sqrt{c + d x^{2}} \left (7 a d - 3 b c\right )}{8 d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(5/2)/x/(d*x**2+c)**(1/2),x)

[Out]

-a**(5/2)*atanh(sqrt(c)*sqrt(a + b*x**2)/(sqrt(a)*sqrt(c + d*x**2)))/sqrt(c) + s
qrt(b)*(15*a**2*d**2 - 10*a*b*c*d + 3*b**2*c**2)*atanh(sqrt(b)*sqrt(c + d*x**2)/
(sqrt(d)*sqrt(a + b*x**2)))/(8*d**(5/2)) + b*(a + b*x**2)**(3/2)*sqrt(c + d*x**2
)/(4*d) + b*sqrt(a + b*x**2)*sqrt(c + d*x**2)*(7*a*d - 3*b*c)/(8*d**2)

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Mathematica [C]  time = 0.944195, size = 357, normalized size = 1.91 \[ \frac{\frac{8 a^3 b d x^2 F_1\left (1;\frac{1}{2},\frac{1}{2};2;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )}{-4 b d x^2 F_1\left (1;\frac{1}{2},\frac{1}{2};2;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )+b c F_1\left (2;\frac{1}{2},\frac{3}{2};3;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )+a d F_1\left (2;\frac{3}{2},\frac{1}{2};3;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )}+\frac{b \left (\left (a+b x^2\right ) \left (c+d x^2\right ) \left (9 a d-3 b c+2 b d x^2\right )-\frac{2 a c x^2 \left (15 a^2 d^2-10 a b c d+3 b^2 c^2\right ) F_1\left (1;\frac{1}{2},\frac{1}{2};2;-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{x^2 \left (a d F_1\left (2;\frac{1}{2},\frac{3}{2};3;-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+b c F_1\left (2;\frac{3}{2},\frac{1}{2};3;-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )-4 a c F_1\left (1;\frac{1}{2},\frac{1}{2};2;-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}\right )}{2 d^2}}{4 \sqrt{a+b x^2} \sqrt{c+d x^2}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(a + b*x^2)^(5/2)/(x*Sqrt[c + d*x^2]),x]

[Out]

((8*a^3*b*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))]) + (b*((a + b*x^2)*(c + d*x^2)*(-3*b*c + 9*a*d + 2*b*d*x^2) - (2*a*
c*(3*b^2*c^2 - 10*a*b*c*d + 15*a^2*d^2)*x^2*AppellF1[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*d^2))/(4*Sqrt[a + b*x^2]*Sqrt[c
+ d*x^2])

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Maple [B]  time = 0.026, size = 446, normalized size = 2.4 \[{\frac{1}{16\,{d}^{2}}\sqrt{b{x}^{2}+a}\sqrt{d{x}^{2}+c} \left ( 4\,{b}^{2}\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}{x}^{2}d\sqrt{bd}\sqrt{ac}+15\,b\ln \left ( 1/2\,{\frac{2\,bd{x}^{2}+2\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{2}{d}^{2}\sqrt{ac}-10\,{b}^{2}\ln \left ( 1/2\,{\frac{2\,bd{x}^{2}+2\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) cad\sqrt{ac}+3\,{b}^{3}\ln \left ( 1/2\,{\frac{2\,bd{x}^{2}+2\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){c}^{2}\sqrt{ac}-8\,{a}^{3}\ln \left ({\frac{ad{x}^{2}+c{x}^{2}b+2\,\sqrt{ac}\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}+2\,ac}{{x}^{2}}} \right ){d}^{2}\sqrt{bd}+18\,b\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}ad\sqrt{bd}\sqrt{ac}-6\,{b}^{2}\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}c\sqrt{bd}\sqrt{ac} \right ){\frac{1}{\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(5/2)/x/(d*x^2+c)^(1/2),x)

[Out]

1/16*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(4*b^2*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*
x^2*d*(b*d)^(1/2)*(a*c)^(1/2)+15*b*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*(a*c)^(1/2)-10*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*a*d*(a*c)^(1/2)+3*b^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))*c^2*(a*c)^(1/2)-8*a^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)*d^2*(b*d)^(1/2)+18*b
*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*a*d*(b*d)^(1/2)*(a*c)^(1/2)-6*b^2*(b*d*x^4+
a*d*x^2+b*c*x^2+a*c)^(1/2)*c*(b*d)^(1/2)*(a*c)^(1/2))/(b*d*x^4+a*d*x^2+b*c*x^2+a
*c)^(1/2)/d^2/(b*d)^(1/2)/(a*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((b*x^2 + a)^(5/2)/(sqrt(d*x^2 + c)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.73185, 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)^(5/2)/(sqrt(d*x^2 + c)*x),x, algorithm="fricas")

[Out]

[1/32*(8*a^2*d^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) + (3*b^2*c^2 - 10*a*b*c*d + 15*a^2*d^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)) + 4
*(2*b^2*d*x^2 - 3*b^2*c + 9*a*b*d)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/d^2, 1/16*(4
*a^2*d^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) + (3*b^2*c^2 - 10*a*b*c*d + 15*a^2*d^2)*sqrt(-b/d)*arcta
n(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*(2*b^2*d*x^2 - 3*b^2*c + 9*a*b*d)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/d^2, -1/32*
(16*a^2*d^2*sqrt(-a/c)*arctan(1/2*((b*c + a*d)*x^2 + 2*a*c)/(sqrt(b*x^2 + a)*sqr
t(d*x^2 + c)*c*sqrt(-a/c))) - (3*b^2*c^2 - 10*a*b*c*d + 15*a^2*d^2)*sqrt(b/d)*lo
g(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*(2*
b^2*d*x^2 - 3*b^2*c + 9*a*b*d)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/d^2, -1/16*(8*a^
2*d^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))) - (3*b^2*c^2 - 10*a*b*c*d + 15*a^2*d^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))) - 2*
(2*b^2*d*x^2 - 3*b^2*c + 9*a*b*d)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/d^2]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x^{2}\right )^{\frac{5}{2}}}{x \sqrt{c + d x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**(5/2)/x/(d*x**2+c)**(1/2),x)

[Out]

Integral((a + b*x**2)**(5/2)/(x*sqrt(c + d*x**2)), x)

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GIAC/XCAS [A]  time = 0.262097, size = 354, normalized size = 1.89 \[ -\frac{{\left (\frac{16 \, \sqrt{b d} a^{3} \arctan \left (-\frac{b^{2} c + a b d -{\left (\sqrt{b x^{2} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt{-a b c d} b}\right )}{\sqrt{-a b c d} b} - 2 \, \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d} \sqrt{b x^{2} + a}{\left (\frac{2 \,{\left (b x^{2} + a\right )}}{b d} - \frac{3 \, b^{2} c d - 7 \, a b d^{2}}{b^{2} d^{3}}\right )} + \frac{{\left (3 \, \sqrt{b d} b^{2} c^{2} - 10 \, \sqrt{b d} a b c d + 15 \, \sqrt{b d} a^{2} d^{2}\right )}{\rm ln}\left ({\left (\sqrt{b x^{2} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d}\right )}^{2}\right )}{b d^{3}}\right )} b^{2}}{16 \,{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(5/2)/(sqrt(d*x^2 + c)*x),x, algorithm="giac")

[Out]

-1/16*(16*sqrt(b*d)*a^3*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)*(2*(b*x^2 + a)/(b*d
) - (3*b^2*c*d - 7*a*b*d^2)/(b^2*d^3)) + (3*sqrt(b*d)*b^2*c^2 - 10*sqrt(b*d)*a*b
*c*d + 15*sqrt(b*d)*a^2*d^2)*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^3))*b^2/abs(b)