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

Optimal. Leaf size=192 \[ -\frac{\sqrt{a} \left (3 a^2 d^2-10 a b c d+15 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{8 c^{5/2}}+\frac{b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{\sqrt{d}}-\frac{a \sqrt{a+b x^2} \sqrt{c+d x^2} (7 b c-3 a d)}{8 c^2 x^2}-\frac{a \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{4 c x^4} \]

[Out]

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

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Rubi [A]  time = 0.635704, antiderivative size = 192, 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 \[ -\frac{\sqrt{a} \left (3 a^2 d^2-10 a b c d+15 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{8 c^{5/2}}+\frac{b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{\sqrt{d}}-\frac{a \sqrt{a+b x^2} \sqrt{c+d x^2} (7 b c-3 a d)}{8 c^2 x^2}-\frac{a \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{4 c x^4} \]

Antiderivative was successfully verified.

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

[Out]

-(a*(7*b*c - 3*a*d)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(8*c^2*x^2) - (a*(a + b*x^2
)^(3/2)*Sqrt[c + d*x^2])/(4*c*x^4) - (Sqrt[a]*(15*b^2*c^2 - 10*a*b*c*d + 3*a^2*d
^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x^2])/(Sqrt[a]*Sqrt[c + d*x^2])])/(8*c^(5/2)) +
(b^(5/2)*ArcTanh[(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 = 64.1813, size = 180, normalized size = 0.94 \[ - \frac{\sqrt{a} \left (3 a^{2} d^{2} - 10 a b c d + 15 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x^{2}}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{8 c^{\frac{5}{2}}} - \frac{a \left (a + b x^{2}\right )^{\frac{3}{2}} \sqrt{c + d x^{2}}}{4 c x^{4}} + \frac{a \sqrt{a + b x^{2}} \sqrt{c + d x^{2}} \left (3 a d - 7 b c\right )}{8 c^{2} x^{2}} + \frac{b^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x^{2}}}{\sqrt{b} \sqrt{c + d x^{2}}} \right )}}{\sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.929212, size = 359, normalized size = 1.87 \[ \frac{a \left (\frac{2 b d x^6 \left (3 a^2 d^2-10 a b c d+15 b^2 c^2\right ) F_1\left (1;\frac{1}{2},\frac{1}{2};2;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )}{-4 b d x^2 F_1\left (1;\frac{1}{2},\frac{1}{2};2;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )+b c F_1\left (2;\frac{1}{2},\frac{3}{2};3;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )+a d F_1\left (2;\frac{3}{2},\frac{1}{2};3;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )}-\frac{16 b^3 c^3 x^6 F_1\left (1;\frac{1}{2},\frac{1}{2};2;-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{x^2 \left (a d F_1\left (2;\frac{1}{2},\frac{3}{2};3;-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+b c F_1\left (2;\frac{3}{2},\frac{1}{2};3;-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )-4 a c F_1\left (1;\frac{1}{2},\frac{1}{2};2;-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}+\left (a+b x^2\right ) \left (c+d x^2\right ) \left (-2 a c+3 a d x^2-9 b c x^2\right )\right )}{8 c^2 x^4 \sqrt{a+b x^2} \sqrt{c+d x^2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(a*((a + b*x^2)*(c + d*x^2)*(-2*a*c - 9*b*c*x^2 + 3*a*d*x^2) + (2*b*d*(15*b^2*c^
2 - 10*a*b*c*d + 3*a^2*d^2)*x^6*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*App
ellF1[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))]) - (16*b^3*c^3*x^6*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*AppellF
1[2, 3/2, 1/2, 3, -((b*x^2)/a), -((d*x^2)/c)]))))/(8*c^2*x^4*Sqrt[a + b*x^2]*Sqr
t[c + d*x^2])

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Maple [B]  time = 0.024, size = 464, normalized size = 2.4 \[ -{\frac{1}{16\,{c}^{2}{x}^{4}}\sqrt{b{x}^{2}+a}\sqrt{d{x}^{2}+c} \left ( 3\,\ln \left ({\frac{ad{x}^{2}+c{x}^{2}b+2\,\sqrt{ac}\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}+2\,ac}{{x}^{2}}} \right ){x}^{4}{a}^{3}{d}^{2}\sqrt{bd}-10\,\ln \left ({\frac{ad{x}^{2}+c{x}^{2}b+2\,\sqrt{ac}\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}+2\,ac}{{x}^{2}}} \right ){x}^{4}{a}^{2}bcd\sqrt{bd}+15\,\ln \left ({\frac{ad{x}^{2}+c{x}^{2}b+2\,\sqrt{ac}\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}+2\,ac}{{x}^{2}}} \right ){x}^{4}a{b}^{2}{c}^{2}\sqrt{bd}-8\,\ln \left ( 1/2\,{\frac{2\,bd{x}^{2}+2\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{4}{b}^{3}{c}^{2}\sqrt{ac}-6\,{x}^{2}{a}^{2}d\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}\sqrt{ac}\sqrt{bd}+18\,{x}^{2}abc\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}\sqrt{ac}\sqrt{bd}+4\,{a}^{2}c\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}\sqrt{ac}\sqrt{bd} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}}}{\frac{1}{\sqrt{bd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.78177, 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^5),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 1.88474, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x