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

Optimal. Leaf size=289 \[ \frac{\left (3 \sqrt{a} B-A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{8 \sqrt{2} a^{3/4} c^{7/4}}-\frac{\left (3 \sqrt{a} B-A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{8 \sqrt{2} a^{3/4} c^{7/4}}-\frac{\left (3 \sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{3/4} c^{7/4}}+\frac{\left (3 \sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{3/4} c^{7/4}}-\frac{\sqrt{x} (A+B x)}{2 c \left (a+c x^2\right )} \]

[Out]

-(Sqrt[x]*(A + B*x))/(2*c*(a + c*x^2)) - ((3*Sqrt[a]*B + A*Sqrt[c])*ArcTan[1 - (
Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(3/4)*c^(7/4)) + ((3*Sqrt[a]*B +
A*Sqrt[c])*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(3/4)*c^
(7/4)) + ((3*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^(3/4)*c^(7/4)) - ((3*Sqrt[a]*B - A*Sqrt[c])*Log[Sqr
t[a] + Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(8*Sqrt[2]*a^(3/4)*c^(7/4))

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Rubi [A]  time = 0.499712, antiderivative size = 289, 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 \[ \frac{\left (3 \sqrt{a} B-A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{8 \sqrt{2} a^{3/4} c^{7/4}}-\frac{\left (3 \sqrt{a} B-A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{8 \sqrt{2} a^{3/4} c^{7/4}}-\frac{\left (3 \sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{3/4} c^{7/4}}+\frac{\left (3 \sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{3/4} c^{7/4}}-\frac{\sqrt{x} (A+B x)}{2 c \left (a+c x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[x]*(A + B*x))/(2*c*(a + c*x^2)) - ((3*Sqrt[a]*B + A*Sqrt[c])*ArcTan[1 - (
Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(3/4)*c^(7/4)) + ((3*Sqrt[a]*B +
A*Sqrt[c])*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(3/4)*c^
(7/4)) + ((3*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^(3/4)*c^(7/4)) - ((3*Sqrt[a]*B - A*Sqrt[c])*Log[Sqr
t[a] + Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(8*Sqrt[2]*a^(3/4)*c^(7/4))

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Rubi in Sympy [A]  time = 91.009, size = 270, normalized size = 0.93 \[ - \frac{\sqrt{x} \left (2 A + 2 B x\right )}{4 c \left (a + c x^{2}\right )} - \frac{\sqrt{2} \left (A \sqrt{c} - 3 B \sqrt{a}\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} \sqrt{x} + \sqrt{a} \sqrt{c} + c x \right )}}{16 a^{\frac{3}{4}} c^{\frac{7}{4}}} + \frac{\sqrt{2} \left (A \sqrt{c} - 3 B \sqrt{a}\right ) \log{\left (\sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} \sqrt{x} + \sqrt{a} \sqrt{c} + c x \right )}}{16 a^{\frac{3}{4}} c^{\frac{7}{4}}} - \frac{\sqrt{2} \left (A \sqrt{c} + 3 B \sqrt{a}\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{3}{4}} c^{\frac{7}{4}}} + \frac{\sqrt{2} \left (A \sqrt{c} + 3 B \sqrt{a}\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{3}{4}} c^{\frac{7}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(x)*(2*A + 2*B*x)/(4*c*(a + c*x**2)) - sqrt(2)*(A*sqrt(c) - 3*B*sqrt(a))*lo
g(-sqrt(2)*a**(1/4)*c**(3/4)*sqrt(x) + sqrt(a)*sqrt(c) + c*x)/(16*a**(3/4)*c**(7
/4)) + sqrt(2)*(A*sqrt(c) - 3*B*sqrt(a))*log(sqrt(2)*a**(1/4)*c**(3/4)*sqrt(x) +
sqrt(a)*sqrt(c) + c*x)/(16*a**(3/4)*c**(7/4)) - sqrt(2)*(A*sqrt(c) + 3*B*sqrt(a
))*atan(1 - sqrt(2)*c**(1/4)*sqrt(x)/a**(1/4))/(8*a**(3/4)*c**(7/4)) + sqrt(2)*(
A*sqrt(c) + 3*B*sqrt(a))*atan(1 + sqrt(2)*c**(1/4)*sqrt(x)/a**(1/4))/(8*a**(3/4)
*c**(7/4))

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Mathematica [A]  time = 0.495209, size = 269, normalized size = 0.93 \[ \frac{\frac{\sqrt{2} \left (3 \sqrt{a} B-A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{a^{3/4}}-\frac{\sqrt{2} \left (3 \sqrt{a} B-A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{a^{3/4}}-\frac{2 \sqrt{2} \left (3 \sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{a^{3/4}}+\frac{2 \sqrt{2} \left (3 \sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{a^{3/4}}-\frac{8 c^{3/4} \sqrt{x} (A+B x)}{a+c x^2}}{16 c^{7/4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.021, size = 307, normalized size = 1.1 \[ 2\,{\frac{1}{c{x}^{2}+a} \left ( -1/4\,{\frac{B{x}^{3/2}}{c}}-1/4\,{\frac{A\sqrt{x}}{c}} \right ) }+{\frac{A\sqrt{2}}{16\,ac}\sqrt [4]{{\frac{a}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ) }+{\frac{A\sqrt{2}}{8\,ac}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }+{\frac{A\sqrt{2}}{8\,ac}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) }+{\frac{3\,B\sqrt{2}}{16\,{c}^{2}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{3\,B\sqrt{2}}{8\,{c}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{3\,B\sqrt{2}}{8\,{c}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.339997, size = 1199, normalized size = 4.15 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*((c^2*x^2 + a*c)*sqrt(-(a*c^3*sqrt(-(81*B^4*a^2 - 18*A^2*B^2*a*c + A^4*c^2)/
(a^3*c^7)) + 6*A*B)/(a*c^3))*log(-(81*B^4*a^2 - A^4*c^2)*sqrt(x) + (3*B*a^3*c^5*
sqrt(-(81*B^4*a^2 - 18*A^2*B^2*a*c + A^4*c^2)/(a^3*c^7)) - 9*A*B^2*a^2*c^2 + A^3
*a*c^3)*sqrt(-(a*c^3*sqrt(-(81*B^4*a^2 - 18*A^2*B^2*a*c + A^4*c^2)/(a^3*c^7)) +
6*A*B)/(a*c^3))) - (c^2*x^2 + a*c)*sqrt(-(a*c^3*sqrt(-(81*B^4*a^2 - 18*A^2*B^2*a
*c + A^4*c^2)/(a^3*c^7)) + 6*A*B)/(a*c^3))*log(-(81*B^4*a^2 - A^4*c^2)*sqrt(x) -
(3*B*a^3*c^5*sqrt(-(81*B^4*a^2 - 18*A^2*B^2*a*c + A^4*c^2)/(a^3*c^7)) - 9*A*B^2
*a^2*c^2 + A^3*a*c^3)*sqrt(-(a*c^3*sqrt(-(81*B^4*a^2 - 18*A^2*B^2*a*c + A^4*c^2)
/(a^3*c^7)) + 6*A*B)/(a*c^3))) - (c^2*x^2 + a*c)*sqrt((a*c^3*sqrt(-(81*B^4*a^2 -
18*A^2*B^2*a*c + A^4*c^2)/(a^3*c^7)) - 6*A*B)/(a*c^3))*log(-(81*B^4*a^2 - A^4*c
^2)*sqrt(x) + (3*B*a^3*c^5*sqrt(-(81*B^4*a^2 - 18*A^2*B^2*a*c + A^4*c^2)/(a^3*c^
7)) + 9*A*B^2*a^2*c^2 - A^3*a*c^3)*sqrt((a*c^3*sqrt(-(81*B^4*a^2 - 18*A^2*B^2*a*
c + A^4*c^2)/(a^3*c^7)) - 6*A*B)/(a*c^3))) + (c^2*x^2 + a*c)*sqrt((a*c^3*sqrt(-(
81*B^4*a^2 - 18*A^2*B^2*a*c + A^4*c^2)/(a^3*c^7)) - 6*A*B)/(a*c^3))*log(-(81*B^4
*a^2 - A^4*c^2)*sqrt(x) - (3*B*a^3*c^5*sqrt(-(81*B^4*a^2 - 18*A^2*B^2*a*c + A^4*
c^2)/(a^3*c^7)) + 9*A*B^2*a^2*c^2 - A^3*a*c^3)*sqrt((a*c^3*sqrt(-(81*B^4*a^2 - 1
8*A^2*B^2*a*c + A^4*c^2)/(a^3*c^7)) - 6*A*B)/(a*c^3))) - 4*(B*x + A)*sqrt(x))/(c
^2*x^2 + a*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**(3/2)*(B*x+A)/(c*x**2+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.284411, size = 366, normalized size = 1.27 \[ -\frac{B x^{\frac{3}{2}} + A \sqrt{x}}{2 \,{\left (c x^{2} + a\right )} c} + \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} A c^{2} + 3 \, \left (a c^{3}\right )^{\frac{3}{4}} B\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{8 \, a c^{4}} + \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} A c^{2} + 3 \, \left (a c^{3}\right )^{\frac{3}{4}} B\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{8 \, a c^{4}} + \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} A c^{2} - 3 \, \left (a c^{3}\right )^{\frac{3}{4}} B\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{c}}\right )}{16 \, a c^{4}} - \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} A c^{2} - 3 \, \left (a c^{3}\right )^{\frac{3}{4}} B\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{c}}\right )}{16 \, a c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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