3.422 \(\int \frac{\sqrt{x} (A+B x)}{\left (a+c x^2\right )^2} \, dx\)

Optimal. Leaf size=292 \[ -\frac{\left (\sqrt{a} B-A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{8 \sqrt{2} a^{5/4} c^{5/4}}+\frac{\left (\sqrt{a} B-A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{8 \sqrt{2} a^{5/4} c^{5/4}}-\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{5/4} c^{5/4}}+\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{5/4} c^{5/4}}-\frac{\sqrt{x} (a B-A c x)}{2 a c \left (a+c x^2\right )} \]

[Out]

-(Sqrt[x]*(a*B - A*c*x))/(2*a*c*(a + c*x^2)) - ((Sqrt[a]*B + A*Sqrt[c])*ArcTan[1
- (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(5/4)*c^(5/4)) + ((Sqrt[a]*B
+ A*Sqrt[c])*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(5/4)*
c^(5/4)) - ((Sqrt[a]*B - A*Sqrt[c])*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x
] + Sqrt[c]*x])/(8*Sqrt[2]*a^(5/4)*c^(5/4)) + ((Sqrt[a]*B - A*Sqrt[c])*Log[Sqrt[
a] + Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(8*Sqrt[2]*a^(5/4)*c^(5/4))

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Rubi [A]  time = 0.472119, antiderivative size = 292, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{\left (\sqrt{a} B-A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{8 \sqrt{2} a^{5/4} c^{5/4}}+\frac{\left (\sqrt{a} B-A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{8 \sqrt{2} a^{5/4} c^{5/4}}-\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{5/4} c^{5/4}}+\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{5/4} c^{5/4}}-\frac{\sqrt{x} (a B-A c x)}{2 a c \left (a+c x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[x]*(a*B - A*c*x))/(2*a*c*(a + c*x^2)) - ((Sqrt[a]*B + A*Sqrt[c])*ArcTan[1
- (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(5/4)*c^(5/4)) + ((Sqrt[a]*B
+ A*Sqrt[c])*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(5/4)*
c^(5/4)) - ((Sqrt[a]*B - A*Sqrt[c])*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x
] + Sqrt[c]*x])/(8*Sqrt[2]*a^(5/4)*c^(5/4)) + ((Sqrt[a]*B - A*Sqrt[c])*Log[Sqrt[
a] + Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(8*Sqrt[2]*a^(5/4)*c^(5/4))

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Rubi in Sympy [A]  time = 90.3069, size = 265, normalized size = 0.91 \[ - \frac{\sqrt{x} \left (- A c x + B a\right )}{2 a c \left (a + c x^{2}\right )} + \frac{\sqrt{2} \left (A \sqrt{c} - B \sqrt{a}\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} \sqrt{x} + \sqrt{a} \sqrt{c} + c x \right )}}{16 a^{\frac{5}{4}} c^{\frac{5}{4}}} - \frac{\sqrt{2} \left (A \sqrt{c} - B \sqrt{a}\right ) \log{\left (\sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} \sqrt{x} + \sqrt{a} \sqrt{c} + c x \right )}}{16 a^{\frac{5}{4}} c^{\frac{5}{4}}} - \frac{\sqrt{2} \left (A \sqrt{c} + B \sqrt{a}\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{5}{4}} c^{\frac{5}{4}}} + \frac{\sqrt{2} \left (A \sqrt{c} + B \sqrt{a}\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{5}{4}} c^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(x)*(-A*c*x + B*a)/(2*a*c*(a + c*x**2)) + sqrt(2)*(A*sqrt(c) - B*sqrt(a))*l
og(-sqrt(2)*a**(1/4)*c**(3/4)*sqrt(x) + sqrt(a)*sqrt(c) + c*x)/(16*a**(5/4)*c**(
5/4)) - sqrt(2)*(A*sqrt(c) - B*sqrt(a))*log(sqrt(2)*a**(1/4)*c**(3/4)*sqrt(x) +
sqrt(a)*sqrt(c) + c*x)/(16*a**(5/4)*c**(5/4)) - sqrt(2)*(A*sqrt(c) + B*sqrt(a))*
atan(1 - sqrt(2)*c**(1/4)*sqrt(x)/a**(1/4))/(8*a**(5/4)*c**(5/4)) + sqrt(2)*(A*s
qrt(c) + B*sqrt(a))*atan(1 + sqrt(2)*c**(1/4)*sqrt(x)/a**(1/4))/(8*a**(5/4)*c**(
5/4))

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Mathematica [A]  time = 0.569117, size = 276, normalized size = 0.95 \[ \frac{\sqrt{2} \left (a^{3/4} A c-a^{5/4} B \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )+\sqrt{2} \left (a^{5/4} B \sqrt{c}-a^{3/4} A c\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )-2 \sqrt{2} \left (a^{3/4} A c+a^{5/4} B \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )+2 \sqrt{2} \left (a^{3/4} A c+a^{5/4} B \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )+\frac{8 a c^{3/4} \sqrt{x} (A c x-a B)}{a+c x^2}}{16 a^2 c^{7/4}} \]

Antiderivative was successfully verified.

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

[Out]

((8*a*c^(3/4)*Sqrt[x]*(-(a*B) + A*c*x))/(a + c*x^2) - 2*Sqrt[2]*(a^(5/4)*B*Sqrt[
c] + a^(3/4)*A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)] + 2*Sqrt[2]*(a^(
5/4)*B*Sqrt[c] + a^(3/4)*A*c)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)] + Sq
rt[2]*(-(a^(5/4)*B*Sqrt[c]) + a^(3/4)*A*c)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)
*Sqrt[x] + Sqrt[c]*x] + Sqrt[2]*(a^(5/4)*B*Sqrt[c] - a^(3/4)*A*c)*Log[Sqrt[a] +
Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(16*a^2*c^(7/4))

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Maple [A]  time = 0.019, size = 316, normalized size = 1.1 \[ 2\,{\frac{1}{c{x}^{2}+a} \left ( 1/4\,{\frac{A{x}^{3/2}}{a}}-1/4\,{\frac{B\sqrt{x}}{c}} \right ) }+{\frac{B\sqrt{2}}{16\,ac}\sqrt [4]{{\frac{a}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ) }+{\frac{B\sqrt{2}}{8\,ac}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }+{\frac{B\sqrt{2}}{8\,ac}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) }+{\frac{A\sqrt{2}}{16\,ac}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{A\sqrt{2}}{8\,ac}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{A\sqrt{2}}{8\,ac}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(1/4*A/a*x^(3/2)-1/4*B*x^(1/2)/c)/(c*x^2+a)+1/16*B/a/c*(a/c)^(1/4)*2^(1/2)*ln(
(x+(a/c)^(1/4)*x^(1/2)*2^(1/2)+(a/c)^(1/2))/(x-(a/c)^(1/4)*x^(1/2)*2^(1/2)+(a/c)
^(1/2)))+1/8*B/a/c*(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x^(1/2)+1)+1/8
*B/a/c*(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x^(1/2)-1)+1/16*A/c/a/(a/c
)^(1/4)*2^(1/2)*ln((x-(a/c)^(1/4)*x^(1/2)*2^(1/2)+(a/c)^(1/2))/(x+(a/c)^(1/4)*x^
(1/2)*2^(1/2)+(a/c)^(1/2)))+1/8*A/c/a/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(
1/4)*x^(1/2)+1)+1/8*A/c/a/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(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 + A)*sqrt(x)/(c*x^2 + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.300313, size = 1216, normalized size = 4.16 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*((a*c^2*x^2 + a^2*c)*sqrt(-(a^2*c^2*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^
2)/(a^5*c^5)) + 2*A*B)/(a^2*c^2))*log(-(B^4*a^2 - A^4*c^2)*sqrt(x) + (A*a^4*c^4*
sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^5*c^5)) + B^3*a^3*c - A^2*B*a^2*c^2
)*sqrt(-(a^2*c^2*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^5*c^5)) + 2*A*B)/(
a^2*c^2))) - (a*c^2*x^2 + a^2*c)*sqrt(-(a^2*c^2*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c +
A^4*c^2)/(a^5*c^5)) + 2*A*B)/(a^2*c^2))*log(-(B^4*a^2 - A^4*c^2)*sqrt(x) - (A*a
^4*c^4*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^5*c^5)) + B^3*a^3*c - A^2*B*
a^2*c^2)*sqrt(-(a^2*c^2*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^5*c^5)) + 2
*A*B)/(a^2*c^2))) - (a*c^2*x^2 + a^2*c)*sqrt((a^2*c^2*sqrt(-(B^4*a^2 - 2*A^2*B^2
*a*c + A^4*c^2)/(a^5*c^5)) - 2*A*B)/(a^2*c^2))*log(-(B^4*a^2 - A^4*c^2)*sqrt(x)
+ (A*a^4*c^4*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^5*c^5)) - B^3*a^3*c +
A^2*B*a^2*c^2)*sqrt((a^2*c^2*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^5*c^5)
) - 2*A*B)/(a^2*c^2))) + (a*c^2*x^2 + a^2*c)*sqrt((a^2*c^2*sqrt(-(B^4*a^2 - 2*A^
2*B^2*a*c + A^4*c^2)/(a^5*c^5)) - 2*A*B)/(a^2*c^2))*log(-(B^4*a^2 - A^4*c^2)*sqr
t(x) - (A*a^4*c^4*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^5*c^5)) - B^3*a^3
*c + A^2*B*a^2*c^2)*sqrt((a^2*c^2*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^5
*c^5)) - 2*A*B)/(a^2*c^2))) - 4*(A*c*x - B*a)*sqrt(x))/(a*c^2*x^2 + a^2*c)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.284653, size = 379, normalized size = 1.3 \[ \frac{A c x^{\frac{3}{2}} - B a \sqrt{x}}{2 \,{\left (c x^{2} + a\right )} a c} + \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} B a c + \left (a c^{3}\right )^{\frac{3}{4}} A\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{8 \, a^{2} c^{3}} - \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} B a c - \left (a c^{3}\right )^{\frac{3}{4}} A\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{c}}\right )}{16 \, a^{2} c^{3}} + \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} B a c^{3} + \left (a c^{3}\right )^{\frac{3}{4}} A c^{2}\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{8 \, a^{2} c^{5}} + \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} B a c^{3} - \left (a c^{3}\right )^{\frac{3}{4}} A c^{2}\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{c}}\right )}{16 \, a^{2} c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(A*c*x^(3/2) - B*a*sqrt(x))/((c*x^2 + a)*a*c) + 1/8*sqrt(2)*((a*c^3)^(1/4)*B
*a*c + (a*c^3)^(3/4)*A)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/c)^(1/4) - 2*sqrt(x))/(a
/c)^(1/4))/(a^2*c^3) - 1/16*sqrt(2)*((a*c^3)^(1/4)*B*a*c - (a*c^3)^(3/4)*A)*ln(-
sqrt(2)*sqrt(x)*(a/c)^(1/4) + x + sqrt(a/c))/(a^2*c^3) + 1/8*sqrt(2)*((a*c^3)^(1
/4)*B*a*c^3 + (a*c^3)^(3/4)*A*c^2)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/c)^(1/4) + 2*s
qrt(x))/(a/c)^(1/4))/(a^2*c^5) + 1/16*sqrt(2)*((a*c^3)^(1/4)*B*a*c^3 - (a*c^3)^(
3/4)*A*c^2)*ln(sqrt(2)*sqrt(x)*(a/c)^(1/4) + x + sqrt(a/c))/(a^2*c^5)