3.958 \(\int \frac{\left (a+b x^2\right )^{5/2}}{x^3 \sqrt{c+d x^2}} \, dx\)

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

[Out]

(b*(b*c + a*d)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(2*c*d) - (a*(a + b*x^2)^(3/2)*S
qrt[c + d*x^2])/(2*c*x^2) - (a^(3/2)*(5*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)*(b*c - 5*a*d)*ArcTanh[(S
qrt[d]*Sqrt[a + b*x^2])/(Sqrt[b]*Sqrt[c + d*x^2])])/(2*d^(3/2))

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Rubi [A]  time = 0.683332, 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 \[ -\frac{a^{3/2} (5 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 c^{3/2}}-\frac{b^{3/2} (b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{2 d^{3/2}}-\frac{a \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{2 c x^2}+\frac{b \sqrt{a+b x^2} \sqrt{c+d x^2} (a d+b c)}{2 c d} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(5/2)/(x^3*Sqrt[c + d*x^2]),x]

[Out]

(b*(b*c + a*d)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(2*c*d) - (a*(a + b*x^2)^(3/2)*S
qrt[c + d*x^2])/(2*c*x^2) - (a^(3/2)*(5*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)*(b*c - 5*a*d)*ArcTanh[(S
qrt[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 = 71.3697, size = 167, normalized size = 0.89 \[ \frac{a^{\frac{3}{2}} \left (a d - 5 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x^{2}}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{2 c^{\frac{3}{2}}} - \frac{a \left (a + b x^{2}\right )^{\frac{3}{2}} \sqrt{c + d x^{2}}}{2 c x^{2}} + \frac{b^{\frac{3}{2}} \left (5 a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x^{2}}}{\sqrt{b} \sqrt{c + d x^{2}}} \right )}}{2 d^{\frac{3}{2}}} + \frac{b \sqrt{a + b x^{2}} \sqrt{c + d x^{2}} \left (a d + b c\right )}{2 c d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(5/2)/x**3/(d*x**2+c)**(1/2),x)

[Out]

a**(3/2)*(a*d - 5*b*c)*atanh(sqrt(c)*sqrt(a + b*x**2)/(sqrt(a)*sqrt(c + d*x**2))
)/(2*c**(3/2)) - a*(a + b*x**2)**(3/2)*sqrt(c + d*x**2)/(2*c*x**2) + b**(3/2)*(5
*a*d - b*c)*atanh(sqrt(d)*sqrt(a + b*x**2)/(sqrt(b)*sqrt(c + d*x**2)))/(2*d**(3/
2)) + b*sqrt(a + b*x**2)*sqrt(c + d*x**2)*(a*d + b*c)/(2*c*d)

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Mathematica [C]  time = 0.836098, size = 358, normalized size = 1.91 \[ \frac{\frac{2 a^2 b d^2 x^4 (5 b c-a d) F_1\left (1;\frac{1}{2},\frac{1}{2};2;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )}{-4 b d x^2 F_1\left (1;\frac{1}{2},\frac{1}{2};2;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )+b c F_1\left (2;\frac{1}{2},\frac{3}{2};3;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )+a d F_1\left (2;\frac{3}{2},\frac{1}{2};3;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )}+\left (a+b x^2\right ) \left (c+d x^2\right ) \left (b^2 c x^2-a^2 d\right )-\frac{2 a b^2 c^2 x^4 (5 a d-b c) F_1\left (1;\frac{1}{2},\frac{1}{2};2;-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{x^2 \left (a d F_1\left (2;\frac{1}{2},\frac{3}{2};3;-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+b c F_1\left (2;\frac{3}{2},\frac{1}{2};3;-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )-4 a c F_1\left (1;\frac{1}{2},\frac{1}{2};2;-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}}{2 c d x^2 \sqrt{a+b x^2} \sqrt{c+d x^2}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(a + b*x^2)^(5/2)/(x^3*Sqrt[c + d*x^2]),x]

[Out]

((a + b*x^2)*(-(a^2*d) + b^2*c*x^2)*(c + d*x^2) + (2*a^2*b*d^2*(5*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))]) -
(2*a*b^2*c^2*(-(b*c) + 5*a*d)*x^4*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*c*d*x^2*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])

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Maple [B]  time = 0.025, size = 423, normalized size = 2.3 \[{\frac{1}{4\,c{x}^{2}d}\sqrt{b{x}^{2}+a}\sqrt{d{x}^{2}+c} \left ( \ln \left ({\frac{1}{{x}^{2}} \left ( ad{x}^{2}+c{x}^{2}b+2\,\sqrt{ac}\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}+2\,ac \right ) } \right ){x}^{2}{a}^{3}{d}^{2}\sqrt{bd}-5\,\ln \left ({\frac{ad{x}^{2}+c{x}^{2}b+2\,\sqrt{ac}\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}+2\,ac}{{x}^{2}}} \right ){x}^{2}{a}^{2}bcd\sqrt{bd}+5\,\ln \left ( 1/2\,{\frac{2\,bd{x}^{2}+2\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}a{b}^{2}cd\sqrt{ac}-\ln \left ({\frac{1}{2} \left ( 2\,bd{x}^{2}+2\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ){x}^{2}{b}^{3}{c}^{2}\sqrt{ac}+2\,{x}^{2}{b}^{2}c\sqrt{ac}\sqrt{bd}\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}-2\,{a}^{2}d\sqrt{ac}\sqrt{bd}\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}}}{\frac{1}{\sqrt{bd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(5/2)/x^3/(d*x^2+c)^(1/2),x)

[Out]

1/4*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/c*(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^3*d^2*(b*d)^(1/2)-5*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*b*c
*d*(b*d)^(1/2)+5*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*a*b^2*c*d*(a*c)^(1/2)-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^3*c^2*(a*c)
^(1/2)+2*x^2*b^2*c*(a*c)^(1/2)*(b*d)^(1/2)*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)-2
*a^2*d*(a*c)^(1/2)*(b*d)^(1/2)*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(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)/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)^(5/2)/(sqrt(d*x^2 + c)*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.12796, 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^3),x, algorithm="fricas")

[Out]

[-1/8*((b^2*c^2 - 5*a*b*c*d)*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)) + (5*a*b*c*d - a^2*d^2)*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*(b^2*c*x^2 - a^2*d)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(c*d*x^2), -1/8*(2*(b^
2*c^2 - 5*a*b*c*d)*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))) + (5*a*b*c*d - a^2*d^2)*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*(b^2*c*x^2 - a^2*d)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(c*d*x^2), -1/8*(2*(5*a
*b*c*d - a^2*d^2)*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))) + (b^2*c^2 - 5*a*b*c*d)*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)) - 4*(b^2*
c*x^2 - a^2*d)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(c*d*x^2), -1/4*((5*a*b*c*d - a^
2*d^2)*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))) + (b^2*c^2 - 5*a*b*c*d)*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))) - 2*(b^2*c*
x^2 - a^2*d)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(c*d*x^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x^{2}\right )^{\frac{5}{2}}}{x^{3} \sqrt{c + d x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**(5/2)/x**3/(d*x**2+c)**(1/2),x)

[Out]

Integral((a + b*x**2)**(5/2)/(x**3*sqrt(c + d*x**2)), x)

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GIAC/XCAS [A]  time = 0.619278, 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^3),x, algorithm="giac")

[Out]

sage0*x