3.296 \(\int \frac{x^{7/2}}{\left (a+b x^2\right )^2} \, dx\)

Optimal. Leaf size=230 \[ \frac{5 \sqrt [4]{a} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} b^{9/4}}-\frac{5 \sqrt [4]{a} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} b^{9/4}}+\frac{5 \sqrt [4]{a} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} b^{9/4}}-\frac{5 \sqrt [4]{a} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} b^{9/4}}-\frac{x^{5/2}}{2 b \left (a+b x^2\right )}+\frac{5 \sqrt{x}}{2 b^2} \]

[Out]

(5*Sqrt[x])/(2*b^2) - x^(5/2)/(2*b*(a + b*x^2)) + (5*a^(1/4)*ArcTan[1 - (Sqrt[2]
*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*b^(9/4)) - (5*a^(1/4)*ArcTan[1 + (Sqrt[2]
*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*b^(9/4)) + (5*a^(1/4)*Log[Sqrt[a] - Sqrt[
2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*b^(9/4)) - (5*a^(1/4)*Log[Sq
rt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*b^(9/4))

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Rubi [A]  time = 0.425478, antiderivative size = 230, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6 \[ \frac{5 \sqrt [4]{a} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} b^{9/4}}-\frac{5 \sqrt [4]{a} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} b^{9/4}}+\frac{5 \sqrt [4]{a} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} b^{9/4}}-\frac{5 \sqrt [4]{a} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} b^{9/4}}-\frac{x^{5/2}}{2 b \left (a+b x^2\right )}+\frac{5 \sqrt{x}}{2 b^2} \]

Antiderivative was successfully verified.

[In]  Int[x^(7/2)/(a + b*x^2)^2,x]

[Out]

(5*Sqrt[x])/(2*b^2) - x^(5/2)/(2*b*(a + b*x^2)) + (5*a^(1/4)*ArcTan[1 - (Sqrt[2]
*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*b^(9/4)) - (5*a^(1/4)*ArcTan[1 + (Sqrt[2]
*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*b^(9/4)) + (5*a^(1/4)*Log[Sqrt[a] - Sqrt[
2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*b^(9/4)) - (5*a^(1/4)*Log[Sq
rt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*b^(9/4))

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Rubi in Sympy [A]  time = 70.1972, size = 216, normalized size = 0.94 \[ \frac{5 \sqrt{2} \sqrt [4]{a} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{16 b^{\frac{9}{4}}} - \frac{5 \sqrt{2} \sqrt [4]{a} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{16 b^{\frac{9}{4}}} + \frac{5 \sqrt{2} \sqrt [4]{a} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 b^{\frac{9}{4}}} - \frac{5 \sqrt{2} \sqrt [4]{a} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 b^{\frac{9}{4}}} - \frac{x^{\frac{5}{2}}}{2 b \left (a + b x^{2}\right )} + \frac{5 \sqrt{x}}{2 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(7/2)/(b*x**2+a)**2,x)

[Out]

5*sqrt(2)*a**(1/4)*log(-sqrt(2)*a**(1/4)*b**(1/4)*sqrt(x) + sqrt(a) + sqrt(b)*x)
/(16*b**(9/4)) - 5*sqrt(2)*a**(1/4)*log(sqrt(2)*a**(1/4)*b**(1/4)*sqrt(x) + sqrt
(a) + sqrt(b)*x)/(16*b**(9/4)) + 5*sqrt(2)*a**(1/4)*atan(1 - sqrt(2)*b**(1/4)*sq
rt(x)/a**(1/4))/(8*b**(9/4)) - 5*sqrt(2)*a**(1/4)*atan(1 + sqrt(2)*b**(1/4)*sqrt
(x)/a**(1/4))/(8*b**(9/4)) - x**(5/2)/(2*b*(a + b*x**2)) + 5*sqrt(x)/(2*b**2)

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Mathematica [A]  time = 0.238733, size = 212, normalized size = 0.92 \[ \frac{\frac{8 a \sqrt [4]{b} \sqrt{x}}{a+b x^2}+5 \sqrt{2} \sqrt [4]{a} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-5 \sqrt{2} \sqrt [4]{a} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+10 \sqrt{2} \sqrt [4]{a} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )-10 \sqrt{2} \sqrt [4]{a} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )+32 \sqrt [4]{b} \sqrt{x}}{16 b^{9/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(7/2)/(a + b*x^2)^2,x]

[Out]

(32*b^(1/4)*Sqrt[x] + (8*a*b^(1/4)*Sqrt[x])/(a + b*x^2) + 10*Sqrt[2]*a^(1/4)*Arc
Tan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 10*Sqrt[2]*a^(1/4)*ArcTan[1 + (Sqrt
[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 5*Sqrt[2]*a^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*
b^(1/4)*Sqrt[x] + Sqrt[b]*x] - 5*Sqrt[2]*a^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b
^(1/4)*Sqrt[x] + Sqrt[b]*x])/(16*b^(9/4))

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Maple [A]  time = 0.018, size = 158, normalized size = 0.7 \[ 2\,{\frac{\sqrt{x}}{{b}^{2}}}+{\frac{a}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }\sqrt{x}}-{\frac{5\,\sqrt{2}}{16\,{b}^{2}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }-{\frac{5\,\sqrt{2}}{8\,{b}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }-{\frac{5\,\sqrt{2}}{8\,{b}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(7/2)/(b*x^2+a)^2,x)

[Out]

2*x^(1/2)/b^2+1/2*a/b^2*x^(1/2)/(b*x^2+a)-5/16/b^2*(a/b)^(1/4)*2^(1/2)*ln((x+(a/
b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2))/(x-(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2)
))-5/8/b^2*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)-5/8/b^2*(a/
b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)

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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(x^(7/2)/(b*x^2 + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.288531, size = 239, normalized size = 1.04 \[ \frac{20 \,{\left (b^{3} x^{2} + a b^{2}\right )} \left (-\frac{a}{b^{9}}\right )^{\frac{1}{4}} \arctan \left (\frac{b^{2} \left (-\frac{a}{b^{9}}\right )^{\frac{1}{4}}}{\sqrt{b^{4} \sqrt{-\frac{a}{b^{9}}} + x} + \sqrt{x}}\right ) - 5 \,{\left (b^{3} x^{2} + a b^{2}\right )} \left (-\frac{a}{b^{9}}\right )^{\frac{1}{4}} \log \left (5 \, b^{2} \left (-\frac{a}{b^{9}}\right )^{\frac{1}{4}} + 5 \, \sqrt{x}\right ) + 5 \,{\left (b^{3} x^{2} + a b^{2}\right )} \left (-\frac{a}{b^{9}}\right )^{\frac{1}{4}} \log \left (-5 \, b^{2} \left (-\frac{a}{b^{9}}\right )^{\frac{1}{4}} + 5 \, \sqrt{x}\right ) + 4 \,{\left (4 \, b x^{2} + 5 \, a\right )} \sqrt{x}}{8 \,{\left (b^{3} x^{2} + a b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(7/2)/(b*x^2 + a)^2,x, algorithm="fricas")

[Out]

1/8*(20*(b^3*x^2 + a*b^2)*(-a/b^9)^(1/4)*arctan(b^2*(-a/b^9)^(1/4)/(sqrt(b^4*sqr
t(-a/b^9) + x) + sqrt(x))) - 5*(b^3*x^2 + a*b^2)*(-a/b^9)^(1/4)*log(5*b^2*(-a/b^
9)^(1/4) + 5*sqrt(x)) + 5*(b^3*x^2 + a*b^2)*(-a/b^9)^(1/4)*log(-5*b^2*(-a/b^9)^(
1/4) + 5*sqrt(x)) + 4*(4*b*x^2 + 5*a)*sqrt(x))/(b^3*x^2 + a*b^2)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(7/2)/(b*x**2+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.218843, size = 265, normalized size = 1.15 \[ -\frac{5 \, \sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, b^{3}} - \frac{5 \, \sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, b^{3}} - \frac{5 \, \sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{16 \, b^{3}} + \frac{5 \, \sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{16 \, b^{3}} + \frac{a \sqrt{x}}{2 \,{\left (b x^{2} + a\right )} b^{2}} + \frac{2 \, \sqrt{x}}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(7/2)/(b*x^2 + a)^2,x, algorithm="giac")

[Out]

-5/8*sqrt(2)*(a*b^3)^(1/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/
(a/b)^(1/4))/b^3 - 5/8*sqrt(2)*(a*b^3)^(1/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^
(1/4) - 2*sqrt(x))/(a/b)^(1/4))/b^3 - 5/16*sqrt(2)*(a*b^3)^(1/4)*ln(sqrt(2)*sqrt
(x)*(a/b)^(1/4) + x + sqrt(a/b))/b^3 + 5/16*sqrt(2)*(a*b^3)^(1/4)*ln(-sqrt(2)*sq
rt(x)*(a/b)^(1/4) + x + sqrt(a/b))/b^3 + 1/2*a*sqrt(x)/((b*x^2 + a)*b^2) + 2*sqr
t(x)/b^2