3.383 \(\int \frac{x^{7/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx\)

Optimal. Leaf size=316 \[ -\frac{5 (A b-9 a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{3/4} b^{13/4}}+\frac{5 (A b-9 a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{3/4} b^{13/4}}-\frac{5 (A b-9 a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{3/4} b^{13/4}}+\frac{5 (A b-9 a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt{2} a^{3/4} b^{13/4}}-\frac{5 \sqrt{x} (A b-9 a B)}{16 a b^3}+\frac{x^{5/2} (A b-9 a B)}{16 a b^2 \left (a+b x^2\right )}+\frac{x^{9/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2} \]

[Out]

(-5*(A*b - 9*a*B)*Sqrt[x])/(16*a*b^3) + ((A*b - a*B)*x^(9/2))/(4*a*b*(a + b*x^2)
^2) + ((A*b - 9*a*B)*x^(5/2))/(16*a*b^2*(a + b*x^2)) - (5*(A*b - 9*a*B)*ArcTan[1
- (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*a^(3/4)*b^(13/4)) + (5*(A*b -
9*a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*a^(3/4)*b^(13
/4)) - (5*(A*b - 9*a*B)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*
x])/(64*Sqrt[2]*a^(3/4)*b^(13/4)) + (5*(A*b - 9*a*B)*Log[Sqrt[a] + Sqrt[2]*a^(1/
4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(64*Sqrt[2]*a^(3/4)*b^(13/4))

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Rubi [A]  time = 0.526481, antiderivative size = 316, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454 \[ -\frac{5 (A b-9 a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{3/4} b^{13/4}}+\frac{5 (A b-9 a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{3/4} b^{13/4}}-\frac{5 (A b-9 a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{3/4} b^{13/4}}+\frac{5 (A b-9 a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt{2} a^{3/4} b^{13/4}}-\frac{5 \sqrt{x} (A b-9 a B)}{16 a b^3}+\frac{x^{5/2} (A b-9 a B)}{16 a b^2 \left (a+b x^2\right )}+\frac{x^{9/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(x^(7/2)*(A + B*x^2))/(a + b*x^2)^3,x]

[Out]

(-5*(A*b - 9*a*B)*Sqrt[x])/(16*a*b^3) + ((A*b - a*B)*x^(9/2))/(4*a*b*(a + b*x^2)
^2) + ((A*b - 9*a*B)*x^(5/2))/(16*a*b^2*(a + b*x^2)) - (5*(A*b - 9*a*B)*ArcTan[1
- (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*a^(3/4)*b^(13/4)) + (5*(A*b -
9*a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*a^(3/4)*b^(13
/4)) - (5*(A*b - 9*a*B)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*
x])/(64*Sqrt[2]*a^(3/4)*b^(13/4)) + (5*(A*b - 9*a*B)*Log[Sqrt[a] + Sqrt[2]*a^(1/
4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(64*Sqrt[2]*a^(3/4)*b^(13/4))

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Rubi in Sympy [A]  time = 88.5812, size = 298, normalized size = 0.94 \[ \frac{x^{\frac{9}{2}} \left (A b - B a\right )}{4 a b \left (a + b x^{2}\right )^{2}} + \frac{x^{\frac{5}{2}} \left (A b - 9 B a\right )}{16 a b^{2} \left (a + b x^{2}\right )} - \frac{5 \sqrt{x} \left (A b - 9 B a\right )}{16 a b^{3}} - \frac{5 \sqrt{2} \left (A b - 9 B a\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{128 a^{\frac{3}{4}} b^{\frac{13}{4}}} + \frac{5 \sqrt{2} \left (A b - 9 B a\right ) \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{128 a^{\frac{3}{4}} b^{\frac{13}{4}}} - \frac{5 \sqrt{2} \left (A b - 9 B a\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{64 a^{\frac{3}{4}} b^{\frac{13}{4}}} + \frac{5 \sqrt{2} \left (A b - 9 B a\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{64 a^{\frac{3}{4}} b^{\frac{13}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(7/2)*(B*x**2+A)/(b*x**2+a)**3,x)

[Out]

x**(9/2)*(A*b - B*a)/(4*a*b*(a + b*x**2)**2) + x**(5/2)*(A*b - 9*B*a)/(16*a*b**2
*(a + b*x**2)) - 5*sqrt(x)*(A*b - 9*B*a)/(16*a*b**3) - 5*sqrt(2)*(A*b - 9*B*a)*l
og(-sqrt(2)*a**(1/4)*b**(1/4)*sqrt(x) + sqrt(a) + sqrt(b)*x)/(128*a**(3/4)*b**(1
3/4)) + 5*sqrt(2)*(A*b - 9*B*a)*log(sqrt(2)*a**(1/4)*b**(1/4)*sqrt(x) + sqrt(a)
+ sqrt(b)*x)/(128*a**(3/4)*b**(13/4)) - 5*sqrt(2)*(A*b - 9*B*a)*atan(1 - sqrt(2)
*b**(1/4)*sqrt(x)/a**(1/4))/(64*a**(3/4)*b**(13/4)) + 5*sqrt(2)*(A*b - 9*B*a)*at
an(1 + sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4))/(64*a**(3/4)*b**(13/4))

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Mathematica [A]  time = 0.503512, size = 285, normalized size = 0.9 \[ \frac{\frac{5 \sqrt{2} (9 a B-A b) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{3/4}}+\frac{5 \sqrt{2} (A b-9 a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{3/4}}+\frac{10 \sqrt{2} (9 a B-A b) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{a^{3/4}}+\frac{10 \sqrt{2} (A b-9 a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{a^{3/4}}-\frac{8 \sqrt [4]{b} \sqrt{x} (9 A b-17 a B)}{a+b x^2}+\frac{32 a \sqrt [4]{b} \sqrt{x} (A b-a B)}{\left (a+b x^2\right )^2}+256 \sqrt [4]{b} B \sqrt{x}}{128 b^{13/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^(7/2)*(A + B*x^2))/(a + b*x^2)^3,x]

[Out]

(256*b^(1/4)*B*Sqrt[x] + (32*a*b^(1/4)*(A*b - a*B)*Sqrt[x])/(a + b*x^2)^2 - (8*b
^(1/4)*(9*A*b - 17*a*B)*Sqrt[x])/(a + b*x^2) + (10*Sqrt[2]*(-(A*b) + 9*a*B)*ArcT
an[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/a^(3/4) + (10*Sqrt[2]*(A*b - 9*a*B)*A
rcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/a^(3/4) + (5*Sqrt[2]*(-(A*b) + 9*a
*B)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(3/4) + (5*Sqr
t[2]*(A*b - 9*a*B)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a
^(3/4))/(128*b^(13/4))

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Maple [A]  time = 0.024, size = 363, normalized size = 1.2 \[ 2\,{\frac{B\sqrt{x}}{{b}^{3}}}-{\frac{9\,A}{16\,b \left ( b{x}^{2}+a \right ) ^{2}}{x}^{{\frac{5}{2}}}}+{\frac{17\,Ba}{16\,{b}^{2} \left ( b{x}^{2}+a \right ) ^{2}}{x}^{{\frac{5}{2}}}}-{\frac{5\,Aa}{16\,{b}^{2} \left ( b{x}^{2}+a \right ) ^{2}}\sqrt{x}}+{\frac{13\,{a}^{2}B}{16\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{2}}\sqrt{x}}+{\frac{5\,\sqrt{2}A}{64\,{b}^{2}a}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{5\,\sqrt{2}A}{64\,{b}^{2}a}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }+{\frac{5\,\sqrt{2}A}{128\,{b}^{2}a}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }-{\frac{45\,\sqrt{2}B}{64\,{b}^{3}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }-{\frac{45\,\sqrt{2}B}{64\,{b}^{3}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }-{\frac{45\,\sqrt{2}B}{128\,{b}^{3}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(7/2)*(B*x^2+A)/(b*x^2+a)^3,x)

[Out]

2*B*x^(1/2)/b^3-9/16/b/(b*x^2+a)^2*x^(5/2)*A+17/16/b^2/(b*x^2+a)^2*x^(5/2)*a*B-5
/16/b^2/(b*x^2+a)^2*A*x^(1/2)*a+13/16/b^3/(b*x^2+a)^2*B*x^(1/2)*a^2+5/64/b^2*(a/
b)^(1/4)/a*2^(1/2)*A*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+5/64/b^2*(a/b)^(1/4)/
a*2^(1/2)*A*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)+5/128/b^2*(a/b)^(1/4)/a*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)))-45/64/b^3*(a/b)^(1/4)*2^(1/2)*B*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/
2)+1)-45/64/b^3*(a/b)^(1/4)*2^(1/2)*B*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)-45/1
28/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)))

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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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.245301, size = 909, normalized size = 2.88 \[ -\frac{20 \,{\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )} \left (-\frac{6561 \, B^{4} a^{4} - 2916 \, A B^{3} a^{3} b + 486 \, A^{2} B^{2} a^{2} b^{2} - 36 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{13}}\right )^{\frac{1}{4}} \arctan \left (-\frac{a b^{3} \left (-\frac{6561 \, B^{4} a^{4} - 2916 \, A B^{3} a^{3} b + 486 \, A^{2} B^{2} a^{2} b^{2} - 36 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{13}}\right )^{\frac{1}{4}}}{{\left (9 \, B a - A b\right )} \sqrt{x} - \sqrt{a^{2} b^{6} \sqrt{-\frac{6561 \, B^{4} a^{4} - 2916 \, A B^{3} a^{3} b + 486 \, A^{2} B^{2} a^{2} b^{2} - 36 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{13}}} +{\left (81 \, B^{2} a^{2} - 18 \, A B a b + A^{2} b^{2}\right )} x}}\right ) - 5 \,{\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )} \left (-\frac{6561 \, B^{4} a^{4} - 2916 \, A B^{3} a^{3} b + 486 \, A^{2} B^{2} a^{2} b^{2} - 36 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{13}}\right )^{\frac{1}{4}} \log \left (5 \, a b^{3} \left (-\frac{6561 \, B^{4} a^{4} - 2916 \, A B^{3} a^{3} b + 486 \, A^{2} B^{2} a^{2} b^{2} - 36 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{13}}\right )^{\frac{1}{4}} - 5 \,{\left (9 \, B a - A b\right )} \sqrt{x}\right ) + 5 \,{\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )} \left (-\frac{6561 \, B^{4} a^{4} - 2916 \, A B^{3} a^{3} b + 486 \, A^{2} B^{2} a^{2} b^{2} - 36 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{13}}\right )^{\frac{1}{4}} \log \left (-5 \, a b^{3} \left (-\frac{6561 \, B^{4} a^{4} - 2916 \, A B^{3} a^{3} b + 486 \, A^{2} B^{2} a^{2} b^{2} - 36 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{13}}\right )^{\frac{1}{4}} - 5 \,{\left (9 \, B a - A b\right )} \sqrt{x}\right ) - 4 \,{\left (32 \, B b^{2} x^{4} + 45 \, B a^{2} - 5 \, A a b + 9 \,{\left (9 \, B a b - A b^{2}\right )} x^{2}\right )} \sqrt{x}}{64 \,{\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^(7/2)/(b*x^2 + a)^3,x, algorithm="fricas")

[Out]

-1/64*(20*(b^5*x^4 + 2*a*b^4*x^2 + a^2*b^3)*(-(6561*B^4*a^4 - 2916*A*B^3*a^3*b +
486*A^2*B^2*a^2*b^2 - 36*A^3*B*a*b^3 + A^4*b^4)/(a^3*b^13))^(1/4)*arctan(-a*b^3
*(-(6561*B^4*a^4 - 2916*A*B^3*a^3*b + 486*A^2*B^2*a^2*b^2 - 36*A^3*B*a*b^3 + A^4
*b^4)/(a^3*b^13))^(1/4)/((9*B*a - A*b)*sqrt(x) - sqrt(a^2*b^6*sqrt(-(6561*B^4*a^
4 - 2916*A*B^3*a^3*b + 486*A^2*B^2*a^2*b^2 - 36*A^3*B*a*b^3 + A^4*b^4)/(a^3*b^13
)) + (81*B^2*a^2 - 18*A*B*a*b + A^2*b^2)*x))) - 5*(b^5*x^4 + 2*a*b^4*x^2 + a^2*b
^3)*(-(6561*B^4*a^4 - 2916*A*B^3*a^3*b + 486*A^2*B^2*a^2*b^2 - 36*A^3*B*a*b^3 +
A^4*b^4)/(a^3*b^13))^(1/4)*log(5*a*b^3*(-(6561*B^4*a^4 - 2916*A*B^3*a^3*b + 486*
A^2*B^2*a^2*b^2 - 36*A^3*B*a*b^3 + A^4*b^4)/(a^3*b^13))^(1/4) - 5*(9*B*a - A*b)*
sqrt(x)) + 5*(b^5*x^4 + 2*a*b^4*x^2 + a^2*b^3)*(-(6561*B^4*a^4 - 2916*A*B^3*a^3*
b + 486*A^2*B^2*a^2*b^2 - 36*A^3*B*a*b^3 + A^4*b^4)/(a^3*b^13))^(1/4)*log(-5*a*b
^3*(-(6561*B^4*a^4 - 2916*A*B^3*a^3*b + 486*A^2*B^2*a^2*b^2 - 36*A^3*B*a*b^3 + A
^4*b^4)/(a^3*b^13))^(1/4) - 5*(9*B*a - A*b)*sqrt(x)) - 4*(32*B*b^2*x^4 + 45*B*a^
2 - 5*A*a*b + 9*(9*B*a*b - A*b^2)*x^2)*sqrt(x))/(b^5*x^4 + 2*a*b^4*x^2 + a^2*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)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.260635, size = 410, normalized size = 1.3 \[ \frac{2 \, B \sqrt{x}}{b^{3}} - \frac{5 \, \sqrt{2}{\left (9 \, \left (a b^{3}\right )^{\frac{1}{4}} B a - \left (a b^{3}\right )^{\frac{1}{4}} A b\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{64 \, a b^{4}} - \frac{5 \, \sqrt{2}{\left (9 \, \left (a b^{3}\right )^{\frac{1}{4}} B a - \left (a b^{3}\right )^{\frac{1}{4}} A b\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{64 \, a b^{4}} - \frac{5 \, \sqrt{2}{\left (9 \, \left (a b^{3}\right )^{\frac{1}{4}} B a - \left (a b^{3}\right )^{\frac{1}{4}} A b\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{128 \, a b^{4}} + \frac{5 \, \sqrt{2}{\left (9 \, \left (a b^{3}\right )^{\frac{1}{4}} B a - \left (a b^{3}\right )^{\frac{1}{4}} A b\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{128 \, a b^{4}} + \frac{17 \, B a b x^{\frac{5}{2}} - 9 \, A b^{2} x^{\frac{5}{2}} + 13 \, B a^{2} \sqrt{x} - 5 \, A a b \sqrt{x}}{16 \,{\left (b x^{2} + a\right )}^{2} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^(7/2)/(b*x^2 + a)^3,x, algorithm="giac")

[Out]

2*B*sqrt(x)/b^3 - 5/64*sqrt(2)*(9*(a*b^3)^(1/4)*B*a - (a*b^3)^(1/4)*A*b)*arctan(
1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(a*b^4) - 5/64*sqrt(2
)*(9*(a*b^3)^(1/4)*B*a - (a*b^3)^(1/4)*A*b)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(
1/4) - 2*sqrt(x))/(a/b)^(1/4))/(a*b^4) - 5/128*sqrt(2)*(9*(a*b^3)^(1/4)*B*a - (a
*b^3)^(1/4)*A*b)*ln(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a*b^4) + 5/128
*sqrt(2)*(9*(a*b^3)^(1/4)*B*a - (a*b^3)^(1/4)*A*b)*ln(-sqrt(2)*sqrt(x)*(a/b)^(1/
4) + x + sqrt(a/b))/(a*b^4) + 1/16*(17*B*a*b*x^(5/2) - 9*A*b^2*x^(5/2) + 13*B*a^
2*sqrt(x) - 5*A*a*b*sqrt(x))/((b*x^2 + a)^2*b^3)