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

Optimal. Leaf size=298 \[ \frac{(3 a B+5 A b) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{9/4} b^{7/4}}-\frac{(3 a B+5 A b) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{9/4} b^{7/4}}-\frac{(3 a B+5 A b) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{9/4} b^{7/4}}+\frac{(3 a B+5 A b) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt{2} a^{9/4} b^{7/4}}+\frac{x^{3/2} (3 a B+5 A b)}{16 a^2 b \left (a+b x^2\right )}+\frac{x^{3/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2} \]

[Out]

((A*b - a*B)*x^(3/2))/(4*a*b*(a + b*x^2)^2) + ((5*A*b + 3*a*B)*x^(3/2))/(16*a^2*
b*(a + b*x^2)) - ((5*A*b + 3*a*B)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])
/(32*Sqrt[2]*a^(9/4)*b^(7/4)) + ((5*A*b + 3*a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqr
t[x])/a^(1/4)])/(32*Sqrt[2]*a^(9/4)*b^(7/4)) + ((5*A*b + 3*a*B)*Log[Sqrt[a] - Sq
rt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(64*Sqrt[2]*a^(9/4)*b^(7/4)) - ((5*A
*b + 3*a*B)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(64*Sqrt
[2]*a^(9/4)*b^(7/4))

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Rubi [A]  time = 0.46374, antiderivative size = 298, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409 \[ \frac{(3 a B+5 A b) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{9/4} b^{7/4}}-\frac{(3 a B+5 A b) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{9/4} b^{7/4}}-\frac{(3 a B+5 A b) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{9/4} b^{7/4}}+\frac{(3 a B+5 A b) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt{2} a^{9/4} b^{7/4}}+\frac{x^{3/2} (3 a B+5 A b)}{16 a^2 b \left (a+b x^2\right )}+\frac{x^{3/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

((A*b - a*B)*x^(3/2))/(4*a*b*(a + b*x^2)^2) + ((5*A*b + 3*a*B)*x^(3/2))/(16*a^2*
b*(a + b*x^2)) - ((5*A*b + 3*a*B)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])
/(32*Sqrt[2]*a^(9/4)*b^(7/4)) + ((5*A*b + 3*a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqr
t[x])/a^(1/4)])/(32*Sqrt[2]*a^(9/4)*b^(7/4)) + ((5*A*b + 3*a*B)*Log[Sqrt[a] - Sq
rt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(64*Sqrt[2]*a^(9/4)*b^(7/4)) - ((5*A
*b + 3*a*B)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(64*Sqrt
[2]*a^(9/4)*b^(7/4))

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Rubi in Sympy [A]  time = 81.3056, size = 277, normalized size = 0.93 \[ \frac{x^{\frac{3}{2}} \left (A b - B a\right )}{4 a b \left (a + b x^{2}\right )^{2}} + \frac{x^{\frac{3}{2}} \left (5 A b + 3 B a\right )}{16 a^{2} b \left (a + b x^{2}\right )} + \frac{\sqrt{2} \left (5 A b + 3 B a\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{128 a^{\frac{9}{4}} b^{\frac{7}{4}}} - \frac{\sqrt{2} \left (5 A b + 3 B a\right ) \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{128 a^{\frac{9}{4}} b^{\frac{7}{4}}} - \frac{\sqrt{2} \left (5 A b + 3 B a\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{64 a^{\frac{9}{4}} b^{\frac{7}{4}}} + \frac{\sqrt{2} \left (5 A b + 3 B a\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{64 a^{\frac{9}{4}} b^{\frac{7}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)*x**(1/2)/(b*x**2+a)**3,x)

[Out]

x**(3/2)*(A*b - B*a)/(4*a*b*(a + b*x**2)**2) + x**(3/2)*(5*A*b + 3*B*a)/(16*a**2
*b*(a + b*x**2)) + sqrt(2)*(5*A*b + 3*B*a)*log(-sqrt(2)*a**(1/4)*b**(1/4)*sqrt(x
) + sqrt(a) + sqrt(b)*x)/(128*a**(9/4)*b**(7/4)) - sqrt(2)*(5*A*b + 3*B*a)*log(s
qrt(2)*a**(1/4)*b**(1/4)*sqrt(x) + sqrt(a) + sqrt(b)*x)/(128*a**(9/4)*b**(7/4))
- sqrt(2)*(5*A*b + 3*B*a)*atan(1 - sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4))/(64*a**(9/
4)*b**(7/4)) + sqrt(2)*(5*A*b + 3*B*a)*atan(1 + sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4
))/(64*a**(9/4)*b**(7/4))

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Mathematica [A]  time = 0.370157, size = 267, normalized size = 0.9 \[ \frac{-\frac{32 a^{5/4} b^{3/4} x^{3/2} (a B-A b)}{\left (a+b x^2\right )^2}+\frac{8 \sqrt [4]{a} b^{3/4} x^{3/2} (3 a B+5 A b)}{a+b x^2}+\sqrt{2} (3 a B+5 A b) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-\sqrt{2} (3 a B+5 A b) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-2 \sqrt{2} (3 a B+5 A b) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )+2 \sqrt{2} (3 a B+5 A b) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{128 a^{9/4} b^{7/4}} \]

Antiderivative was successfully verified.

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

[Out]

((-32*a^(5/4)*b^(3/4)*(-(A*b) + a*B)*x^(3/2))/(a + b*x^2)^2 + (8*a^(1/4)*b^(3/4)
*(5*A*b + 3*a*B)*x^(3/2))/(a + b*x^2) - 2*Sqrt[2]*(5*A*b + 3*a*B)*ArcTan[1 - (Sq
rt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 2*Sqrt[2]*(5*A*b + 3*a*B)*ArcTan[1 + (Sqrt[2]*
b^(1/4)*Sqrt[x])/a^(1/4)] + Sqrt[2]*(5*A*b + 3*a*B)*Log[Sqrt[a] - Sqrt[2]*a^(1/4
)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - Sqrt[2]*(5*A*b + 3*a*B)*Log[Sqrt[a] + Sqrt[2]*a
^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(128*a^(9/4)*b^(7/4))

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Maple [A]  time = 0.023, size = 335, normalized size = 1.1 \[ 2\,{\frac{1}{ \left ( b{x}^{2}+a \right ) ^{2}} \left ( 1/32\,{\frac{ \left ( 5\,Ab+3\,Ba \right ){x}^{7/2}}{{a}^{2}}}+1/32\,{\frac{ \left ( 9\,Ab-Ba \right ){x}^{3/2}}{ab}} \right ) }+{\frac{5\,\sqrt{2}A}{64\,{a}^{2}b}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{5\,\sqrt{2}A}{64\,{a}^{2}b}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{5\,\sqrt{2}A}{128\,{a}^{2}b}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{3\,\sqrt{2}B}{64\,a{b}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{3\,\sqrt{2}B}{64\,a{b}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{3\,\sqrt{2}B}{128\,a{b}^{2}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)*x^(1/2)/(b*x^2+a)^3,x)

[Out]

2*(1/32*(5*A*b+3*B*a)/a^2*x^(7/2)+1/32*(9*A*b-B*a)/a/b*x^(3/2))/(b*x^2+a)^2+5/64
/a^2/b/(a/b)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+5/64/a^2/b/(a
/b)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)+5/128/a^2/b/(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)))+3/64/a/b^2/(a/b)^(1/4)*2^(1/2)*B*arctan(2^(1/2)/(a/b)^(1
/4)*x^(1/2)+1)+3/64/a/b^2/(a/b)^(1/4)*2^(1/2)*B*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/
2)-1)+3/128/a/b^2/(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)*sqrt(x)/(b*x^2 + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.259989, size = 1189, normalized size = 3.99 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/64*(4*(a^2*b^3*x^4 + 2*a^3*b^2*x^2 + a^4*b)*(-(81*B^4*a^4 + 540*A*B^3*a^3*b +
1350*A^2*B^2*a^2*b^2 + 1500*A^3*B*a*b^3 + 625*A^4*b^4)/(a^9*b^7))^(1/4)*arctan(a
^7*b^5*(-(81*B^4*a^4 + 540*A*B^3*a^3*b + 1350*A^2*B^2*a^2*b^2 + 1500*A^3*B*a*b^3
+ 625*A^4*b^4)/(a^9*b^7))^(3/4)/((27*B^3*a^3 + 135*A*B^2*a^2*b + 225*A^2*B*a*b^
2 + 125*A^3*b^3)*sqrt(x) + sqrt((729*B^6*a^6 + 7290*A*B^5*a^5*b + 30375*A^2*B^4*
a^4*b^2 + 67500*A^3*B^3*a^3*b^3 + 84375*A^4*B^2*a^2*b^4 + 56250*A^5*B*a*b^5 + 15
625*A^6*b^6)*x - (81*B^4*a^9*b^3 + 540*A*B^3*a^8*b^4 + 1350*A^2*B^2*a^7*b^5 + 15
00*A^3*B*a^6*b^6 + 625*A^4*a^5*b^7)*sqrt(-(81*B^4*a^4 + 540*A*B^3*a^3*b + 1350*A
^2*B^2*a^2*b^2 + 1500*A^3*B*a*b^3 + 625*A^4*b^4)/(a^9*b^7))))) + (a^2*b^3*x^4 +
2*a^3*b^2*x^2 + a^4*b)*(-(81*B^4*a^4 + 540*A*B^3*a^3*b + 1350*A^2*B^2*a^2*b^2 +
1500*A^3*B*a*b^3 + 625*A^4*b^4)/(a^9*b^7))^(1/4)*log(a^7*b^5*(-(81*B^4*a^4 + 540
*A*B^3*a^3*b + 1350*A^2*B^2*a^2*b^2 + 1500*A^3*B*a*b^3 + 625*A^4*b^4)/(a^9*b^7))
^(3/4) + (27*B^3*a^3 + 135*A*B^2*a^2*b + 225*A^2*B*a*b^2 + 125*A^3*b^3)*sqrt(x))
- (a^2*b^3*x^4 + 2*a^3*b^2*x^2 + a^4*b)*(-(81*B^4*a^4 + 540*A*B^3*a^3*b + 1350*
A^2*B^2*a^2*b^2 + 1500*A^3*B*a*b^3 + 625*A^4*b^4)/(a^9*b^7))^(1/4)*log(-a^7*b^5*
(-(81*B^4*a^4 + 540*A*B^3*a^3*b + 1350*A^2*B^2*a^2*b^2 + 1500*A^3*B*a*b^3 + 625*
A^4*b^4)/(a^9*b^7))^(3/4) + (27*B^3*a^3 + 135*A*B^2*a^2*b + 225*A^2*B*a*b^2 + 12
5*A^3*b^3)*sqrt(x)) + 4*((3*B*a*b + 5*A*b^2)*x^3 - (B*a^2 - 9*A*a*b)*x)*sqrt(x))
/(a^2*b^3*x^4 + 2*a^3*b^2*x^2 + a^4*b)

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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((B*x**2+A)*x**(1/2)/(b*x**2+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.245794, size = 402, normalized size = 1.35 \[ \frac{3 \, B a b x^{\frac{7}{2}} + 5 \, A b^{2} x^{\frac{7}{2}} - B a^{2} x^{\frac{3}{2}} + 9 \, A a b x^{\frac{3}{2}}}{16 \,{\left (b x^{2} + a\right )}^{2} a^{2} b} + \frac{\sqrt{2}{\left (3 \, \left (a b^{3}\right )^{\frac{3}{4}} B a + 5 \, \left (a b^{3}\right )^{\frac{3}{4}} A b\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{64 \, a^{3} b^{4}} + \frac{\sqrt{2}{\left (3 \, \left (a b^{3}\right )^{\frac{3}{4}} B a + 5 \, \left (a b^{3}\right )^{\frac{3}{4}} A b\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{64 \, a^{3} b^{4}} - \frac{\sqrt{2}{\left (3 \, \left (a b^{3}\right )^{\frac{3}{4}} B a + 5 \, \left (a b^{3}\right )^{\frac{3}{4}} A b\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{128 \, a^{3} b^{4}} + \frac{\sqrt{2}{\left (3 \, \left (a b^{3}\right )^{\frac{3}{4}} B a + 5 \, \left (a b^{3}\right )^{\frac{3}{4}} A b\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{128 \, a^{3} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*(3*B*a*b*x^(7/2) + 5*A*b^2*x^(7/2) - B*a^2*x^(3/2) + 9*A*a*b*x^(3/2))/((b*x
^2 + a)^2*a^2*b) + 1/64*sqrt(2)*(3*(a*b^3)^(3/4)*B*a + 5*(a*b^3)^(3/4)*A*b)*arct
an(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(a^3*b^4) + 1/64*s
qrt(2)*(3*(a*b^3)^(3/4)*B*a + 5*(a*b^3)^(3/4)*A*b)*arctan(-1/2*sqrt(2)*(sqrt(2)*
(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(a^3*b^4) - 1/128*sqrt(2)*(3*(a*b^3)^(3/4)
*B*a + 5*(a*b^3)^(3/4)*A*b)*ln(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^3
*b^4) + 1/128*sqrt(2)*(3*(a*b^3)^(3/4)*B*a + 5*(a*b^3)^(3/4)*A*b)*ln(-sqrt(2)*sq
rt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^3*b^4)