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

Optimal. Leaf size=276 \[ -\frac{a^{5/4} (A b-a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} b^{13/4}}+\frac{a^{5/4} (A b-a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} b^{13/4}}-\frac{a^{5/4} (A b-a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} b^{13/4}}+\frac{a^{5/4} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} b^{13/4}}-\frac{2 a \sqrt{x} (A b-a B)}{b^3}+\frac{2 x^{5/2} (A b-a B)}{5 b^2}+\frac{2 B x^{9/2}}{9 b} \]

[Out]

(-2*a*(A*b - a*B)*Sqrt[x])/b^3 + (2*(A*b - a*B)*x^(5/2))/(5*b^2) + (2*B*x^(9/2))
/(9*b) - (a^(5/4)*(A*b - a*B)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sq
rt[2]*b^(13/4)) + (a^(5/4)*(A*b - a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1
/4)])/(Sqrt[2]*b^(13/4)) - (a^(5/4)*(A*b - a*B)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^
(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*b^(13/4)) + (a^(5/4)*(A*b - a*B)*Log[Sqrt
[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*b^(13/4))

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Rubi [A]  time = 0.567414, antiderivative size = 276, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409 \[ -\frac{a^{5/4} (A b-a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} b^{13/4}}+\frac{a^{5/4} (A b-a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} b^{13/4}}-\frac{a^{5/4} (A b-a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} b^{13/4}}+\frac{a^{5/4} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} b^{13/4}}-\frac{2 a \sqrt{x} (A b-a B)}{b^3}+\frac{2 x^{5/2} (A b-a B)}{5 b^2}+\frac{2 B x^{9/2}}{9 b} \]

Antiderivative was successfully verified.

[In]  Int[(x^(7/2)*(A + B*x^2))/(a + b*x^2),x]

[Out]

(-2*a*(A*b - a*B)*Sqrt[x])/b^3 + (2*(A*b - a*B)*x^(5/2))/(5*b^2) + (2*B*x^(9/2))
/(9*b) - (a^(5/4)*(A*b - a*B)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sq
rt[2]*b^(13/4)) + (a^(5/4)*(A*b - a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1
/4)])/(Sqrt[2]*b^(13/4)) - (a^(5/4)*(A*b - a*B)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^
(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*b^(13/4)) + (a^(5/4)*(A*b - a*B)*Log[Sqrt
[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*b^(13/4))

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Rubi in Sympy [A]  time = 85.8017, size = 258, normalized size = 0.93 \[ \frac{2 B x^{\frac{9}{2}}}{9 b} - \frac{\sqrt{2} a^{\frac{5}{4}} \left (A b - B a\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{4 b^{\frac{13}{4}}} + \frac{\sqrt{2} a^{\frac{5}{4}} \left (A b - B a\right ) \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{4 b^{\frac{13}{4}}} - \frac{\sqrt{2} a^{\frac{5}{4}} \left (A b - B a\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{2 b^{\frac{13}{4}}} + \frac{\sqrt{2} a^{\frac{5}{4}} \left (A b - B a\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{2 b^{\frac{13}{4}}} - \frac{2 a \sqrt{x} \left (A b - B a\right )}{b^{3}} + \frac{2 x^{\frac{5}{2}} \left (A b - B a\right )}{5 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*x**(9/2)/(9*b) - sqrt(2)*a**(5/4)*(A*b - B*a)*log(-sqrt(2)*a**(1/4)*b**(1/4)
*sqrt(x) + sqrt(a) + sqrt(b)*x)/(4*b**(13/4)) + sqrt(2)*a**(5/4)*(A*b - B*a)*log
(sqrt(2)*a**(1/4)*b**(1/4)*sqrt(x) + sqrt(a) + sqrt(b)*x)/(4*b**(13/4)) - sqrt(2
)*a**(5/4)*(A*b - B*a)*atan(1 - sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4))/(2*b**(13/4))
+ sqrt(2)*a**(5/4)*(A*b - B*a)*atan(1 + sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4))/(2*b
**(13/4)) - 2*a*sqrt(x)*(A*b - B*a)/b**3 + 2*x**(5/2)*(A*b - B*a)/(5*b**2)

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Mathematica [A]  time = 0.307242, size = 264, normalized size = 0.96 \[ \frac{45 \sqrt{2} a^{5/4} (a B-A b) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-45 \sqrt{2} a^{5/4} (a B-A b) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+90 \sqrt{2} a^{5/4} (a B-A b) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )-90 \sqrt{2} a^{5/4} (a B-A b) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )+72 b^{5/4} x^{5/2} (A b-a B)+360 a \sqrt [4]{b} \sqrt{x} (a B-A b)+40 b^{9/4} B x^{9/2}}{180 b^{13/4}} \]

Antiderivative was successfully verified.

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

[Out]

(360*a*b^(1/4)*(-(A*b) + a*B)*Sqrt[x] + 72*b^(5/4)*(A*b - a*B)*x^(5/2) + 40*b^(9
/4)*B*x^(9/2) + 90*Sqrt[2]*a^(5/4)*(-(A*b) + a*B)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sq
rt[x])/a^(1/4)] - 90*Sqrt[2]*a^(5/4)*(-(A*b) + a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*
Sqrt[x])/a^(1/4)] + 45*Sqrt[2]*a^(5/4)*(-(A*b) + a*B)*Log[Sqrt[a] - Sqrt[2]*a^(1
/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - 45*Sqrt[2]*a^(5/4)*(-(A*b) + a*B)*Log[Sqrt[a]
+ Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(180*b^(13/4))

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Maple [A]  time = 0.017, size = 330, normalized size = 1.2 \[{\frac{2\,B}{9\,b}{x}^{{\frac{9}{2}}}}+{\frac{2\,A}{5\,b}{x}^{{\frac{5}{2}}}}-{\frac{2\,Ba}{5\,{b}^{2}}{x}^{{\frac{5}{2}}}}-2\,{\frac{aA\sqrt{x}}{{b}^{2}}}+2\,{\frac{B\sqrt{x}{a}^{2}}{{b}^{3}}}+{\frac{a\sqrt{2}A}{2\,{b}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }+{\frac{a\sqrt{2}A}{4\,{b}^{2}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{a\sqrt{2}A}{2\,{b}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }-{\frac{{a}^{2}\sqrt{2}B}{2\,{b}^{3}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }-{\frac{{a}^{2}\sqrt{2}B}{4\,{b}^{3}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }-{\frac{{a}^{2}\sqrt{2}B}{2\,{b}^{3}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(7/2)*(B*x^2+A)/(b*x^2+a),x)

[Out]

2/9*B*x^(9/2)/b+2/5/b*A*x^(5/2)-2/5/b^2*B*x^(5/2)*a-2/b^2*A*x^(1/2)*a+2/b^3*B*x^
(1/2)*a^2+1/2*a/b^2*(a/b)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)+
1/4*a/b^2*(a/b)^(1/4)*2^(1/2)*A*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)))+1/2*a/b^2*(a/b)^(1/4)*2^(1/2)*A*arct
an(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)-1/2*a^2/b^3*(a/b)^(1/4)*2^(1/2)*B*arctan(2^(1/
2)/(a/b)^(1/4)*x^(1/2)-1)-1/4*a^2/b^3*(a/b)^(1/4)*2^(1/2)*B*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)))-1/2*a^2/
b^3*(a/b)^(1/4)*2^(1/2)*B*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((B*x^2 + A)*x^(7/2)/(b*x^2 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.249058, size = 805, normalized size = 2.92 \[ -\frac{180 \, b^{3} \left (-\frac{B^{4} a^{9} - 4 \, A B^{3} a^{8} b + 6 \, A^{2} B^{2} a^{7} b^{2} - 4 \, A^{3} B a^{6} b^{3} + A^{4} a^{5} b^{4}}{b^{13}}\right )^{\frac{1}{4}} \arctan \left (-\frac{b^{3} \left (-\frac{B^{4} a^{9} - 4 \, A B^{3} a^{8} b + 6 \, A^{2} B^{2} a^{7} b^{2} - 4 \, A^{3} B a^{6} b^{3} + A^{4} a^{5} b^{4}}{b^{13}}\right )^{\frac{1}{4}}}{{\left (B a^{2} - A a b\right )} \sqrt{x} - \sqrt{b^{6} \sqrt{-\frac{B^{4} a^{9} - 4 \, A B^{3} a^{8} b + 6 \, A^{2} B^{2} a^{7} b^{2} - 4 \, A^{3} B a^{6} b^{3} + A^{4} a^{5} b^{4}}{b^{13}}} +{\left (B^{2} a^{4} - 2 \, A B a^{3} b + A^{2} a^{2} b^{2}\right )} x}}\right ) - 45 \, b^{3} \left (-\frac{B^{4} a^{9} - 4 \, A B^{3} a^{8} b + 6 \, A^{2} B^{2} a^{7} b^{2} - 4 \, A^{3} B a^{6} b^{3} + A^{4} a^{5} b^{4}}{b^{13}}\right )^{\frac{1}{4}} \log \left (b^{3} \left (-\frac{B^{4} a^{9} - 4 \, A B^{3} a^{8} b + 6 \, A^{2} B^{2} a^{7} b^{2} - 4 \, A^{3} B a^{6} b^{3} + A^{4} a^{5} b^{4}}{b^{13}}\right )^{\frac{1}{4}} -{\left (B a^{2} - A a b\right )} \sqrt{x}\right ) + 45 \, b^{3} \left (-\frac{B^{4} a^{9} - 4 \, A B^{3} a^{8} b + 6 \, A^{2} B^{2} a^{7} b^{2} - 4 \, A^{3} B a^{6} b^{3} + A^{4} a^{5} b^{4}}{b^{13}}\right )^{\frac{1}{4}} \log \left (-b^{3} \left (-\frac{B^{4} a^{9} - 4 \, A B^{3} a^{8} b + 6 \, A^{2} B^{2} a^{7} b^{2} - 4 \, A^{3} B a^{6} b^{3} + A^{4} a^{5} b^{4}}{b^{13}}\right )^{\frac{1}{4}} -{\left (B a^{2} - A a b\right )} \sqrt{x}\right ) - 4 \,{\left (5 \, B b^{2} x^{4} + 45 \, B a^{2} - 45 \, A a b - 9 \,{\left (B a b - A b^{2}\right )} x^{2}\right )} \sqrt{x}}{90 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/90*(180*b^3*(-(B^4*a^9 - 4*A*B^3*a^8*b + 6*A^2*B^2*a^7*b^2 - 4*A^3*B*a^6*b^3
+ A^4*a^5*b^4)/b^13)^(1/4)*arctan(-b^3*(-(B^4*a^9 - 4*A*B^3*a^8*b + 6*A^2*B^2*a^
7*b^2 - 4*A^3*B*a^6*b^3 + A^4*a^5*b^4)/b^13)^(1/4)/((B*a^2 - A*a*b)*sqrt(x) - sq
rt(b^6*sqrt(-(B^4*a^9 - 4*A*B^3*a^8*b + 6*A^2*B^2*a^7*b^2 - 4*A^3*B*a^6*b^3 + A^
4*a^5*b^4)/b^13) + (B^2*a^4 - 2*A*B*a^3*b + A^2*a^2*b^2)*x))) - 45*b^3*(-(B^4*a^
9 - 4*A*B^3*a^8*b + 6*A^2*B^2*a^7*b^2 - 4*A^3*B*a^6*b^3 + A^4*a^5*b^4)/b^13)^(1/
4)*log(b^3*(-(B^4*a^9 - 4*A*B^3*a^8*b + 6*A^2*B^2*a^7*b^2 - 4*A^3*B*a^6*b^3 + A^
4*a^5*b^4)/b^13)^(1/4) - (B*a^2 - A*a*b)*sqrt(x)) + 45*b^3*(-(B^4*a^9 - 4*A*B^3*
a^8*b + 6*A^2*B^2*a^7*b^2 - 4*A^3*B*a^6*b^3 + A^4*a^5*b^4)/b^13)^(1/4)*log(-b^3*
(-(B^4*a^9 - 4*A*B^3*a^8*b + 6*A^2*B^2*a^7*b^2 - 4*A^3*B*a^6*b^3 + A^4*a^5*b^4)/
b^13)^(1/4) - (B*a^2 - A*a*b)*sqrt(x)) - 4*(5*B*b^2*x^4 + 45*B*a^2 - 45*A*a*b -
9*(B*a*b - A*b^2)*x^2)*sqrt(x))/b^3

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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)/(b*x**2+a),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.229697, size = 402, normalized size = 1.46 \[ -\frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} B a^{2} - \left (a b^{3}\right )^{\frac{1}{4}} A a b\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{2 \, b^{4}} - \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} B a^{2} - \left (a b^{3}\right )^{\frac{1}{4}} A a b\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{2 \, b^{4}} - \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} B a^{2} - \left (a b^{3}\right )^{\frac{1}{4}} A a b\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{4 \, b^{4}} + \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} B a^{2} - \left (a b^{3}\right )^{\frac{1}{4}} A a b\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{4 \, b^{4}} + \frac{2 \,{\left (5 \, B b^{8} x^{\frac{9}{2}} - 9 \, B a b^{7} x^{\frac{5}{2}} + 9 \, A b^{8} x^{\frac{5}{2}} + 45 \, B a^{2} b^{6} \sqrt{x} - 45 \, A a b^{7} \sqrt{x}\right )}}{45 \, b^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*sqrt(2)*((a*b^3)^(1/4)*B*a^2 - (a*b^3)^(1/4)*A*a*b)*arctan(1/2*sqrt(2)*(sqr
t(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/b^4 - 1/2*sqrt(2)*((a*b^3)^(1/4)*B*a^
2 - (a*b^3)^(1/4)*A*a*b)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(
a/b)^(1/4))/b^4 - 1/4*sqrt(2)*((a*b^3)^(1/4)*B*a^2 - (a*b^3)^(1/4)*A*a*b)*ln(sqr
t(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/b^4 + 1/4*sqrt(2)*((a*b^3)^(1/4)*B*a^2
- (a*b^3)^(1/4)*A*a*b)*ln(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/b^4 + 2
/45*(5*B*b^8*x^(9/2) - 9*B*a*b^7*x^(5/2) + 9*A*b^8*x^(5/2) + 45*B*a^2*b^6*sqrt(x
) - 45*A*a*b^7*sqrt(x))/b^9