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

Optimal. Leaf size=265 \[ \frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{2 \sqrt{2} \sqrt [4]{a} c^{5/4}}-\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{2 \sqrt{2} \sqrt [4]{a} c^{5/4}}+\frac{\left (\sqrt{a} B-A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} \sqrt [4]{a} c^{5/4}}-\frac{\left (\sqrt{a} B-A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} \sqrt [4]{a} c^{5/4}}+\frac{2 B \sqrt{x}}{c} \]

[Out]

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

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Rubi [A]  time = 0.470408, antiderivative size = 265, 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 \[ \frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{2 \sqrt{2} \sqrt [4]{a} c^{5/4}}-\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{2 \sqrt{2} \sqrt [4]{a} c^{5/4}}+\frac{\left (\sqrt{a} B-A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} \sqrt [4]{a} c^{5/4}}-\frac{\left (\sqrt{a} B-A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} \sqrt [4]{a} c^{5/4}}+\frac{2 B \sqrt{x}}{c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 86.7728, size = 250, normalized size = 0.94 \[ \frac{2 B \sqrt{x}}{c} - \frac{\sqrt{2} \left (A \sqrt{c} - B \sqrt{a}\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}}{2 \sqrt [4]{a} c^{\frac{5}{4}}} + \frac{\sqrt{2} \left (A \sqrt{c} - B \sqrt{a}\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}}{2 \sqrt [4]{a} c^{\frac{5}{4}}} + \frac{\sqrt{2} \left (A \sqrt{c} + B \sqrt{a}\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} \sqrt{x} + \sqrt{a} \sqrt{c} + c x \right )}}{4 \sqrt [4]{a} c^{\frac{5}{4}}} - \frac{\sqrt{2} \left (A \sqrt{c} + B \sqrt{a}\right ) \log{\left (\sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} \sqrt{x} + \sqrt{a} \sqrt{c} + c x \right )}}{4 \sqrt [4]{a} c^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*sqrt(x)/c - sqrt(2)*(A*sqrt(c) - B*sqrt(a))*atan(1 - sqrt(2)*c**(1/4)*sqrt(x
)/a**(1/4))/(2*a**(1/4)*c**(5/4)) + sqrt(2)*(A*sqrt(c) - B*sqrt(a))*atan(1 + sqr
t(2)*c**(1/4)*sqrt(x)/a**(1/4))/(2*a**(1/4)*c**(5/4)) + sqrt(2)*(A*sqrt(c) + B*s
qrt(a))*log(-sqrt(2)*a**(1/4)*c**(3/4)*sqrt(x) + sqrt(a)*sqrt(c) + c*x)/(4*a**(1
/4)*c**(5/4)) - sqrt(2)*(A*sqrt(c) + B*sqrt(a))*log(sqrt(2)*a**(1/4)*c**(3/4)*sq
rt(x) + sqrt(a)*sqrt(c) + c*x)/(4*a**(1/4)*c**(5/4))

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Mathematica [A]  time = 0.472319, size = 260, normalized size = 0.98 \[ \frac{\sqrt{2} \left (a^{3/4} A c+a^{5/4} B \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )-\sqrt{2} \left (a^{3/4} A c+a^{5/4} B \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )+2 \sqrt{2} \left (a^{5/4} B \sqrt{c}-a^{3/4} A c\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )-2 \sqrt{2} \left (a^{5/4} B \sqrt{c}-a^{3/4} A c\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )+8 a B c^{3/4} \sqrt{x}}{4 a c^{7/4}} \]

Antiderivative was successfully verified.

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

[Out]

(8*a*B*c^(3/4)*Sqrt[x] + 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)] + 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] - 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])/(4*a*c^(7/4))

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Maple [A]  time = 0.012, size = 277, normalized size = 1.1 \[ 2\,{\frac{B\sqrt{x}}{c}}-{\frac{B\sqrt{2}}{2\,c}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) }-{\frac{B\sqrt{2}}{4\,c}\sqrt [4]{{\frac{a}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ) }-{\frac{B\sqrt{2}}{2\,c}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }+{\frac{A\sqrt{2}}{4\,c}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{A\sqrt{2}}{2\,c}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{A\sqrt{2}}{2\,c}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.290471, size = 1031, normalized size = 3.89 \[ \frac{c \sqrt{\frac{c^{2} \sqrt{-\frac{B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a c^{5}}} + 2 \, A B}{c^{2}}} \log \left (-{\left (B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt{x} +{\left (A a c^{4} \sqrt{-\frac{B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a c^{5}}} + B^{3} a^{2} c - A^{2} B a c^{2}\right )} \sqrt{\frac{c^{2} \sqrt{-\frac{B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a c^{5}}} + 2 \, A B}{c^{2}}}\right ) - c \sqrt{\frac{c^{2} \sqrt{-\frac{B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a c^{5}}} + 2 \, A B}{c^{2}}} \log \left (-{\left (B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt{x} -{\left (A a c^{4} \sqrt{-\frac{B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a c^{5}}} + B^{3} a^{2} c - A^{2} B a c^{2}\right )} \sqrt{\frac{c^{2} \sqrt{-\frac{B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a c^{5}}} + 2 \, A B}{c^{2}}}\right ) - c \sqrt{-\frac{c^{2} \sqrt{-\frac{B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a c^{5}}} - 2 \, A B}{c^{2}}} \log \left (-{\left (B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt{x} +{\left (A a c^{4} \sqrt{-\frac{B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a c^{5}}} - B^{3} a^{2} c + A^{2} B a c^{2}\right )} \sqrt{-\frac{c^{2} \sqrt{-\frac{B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a c^{5}}} - 2 \, A B}{c^{2}}}\right ) + c \sqrt{-\frac{c^{2} \sqrt{-\frac{B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a c^{5}}} - 2 \, A B}{c^{2}}} \log \left (-{\left (B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt{x} -{\left (A a c^{4} \sqrt{-\frac{B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a c^{5}}} - B^{3} a^{2} c + A^{2} B a c^{2}\right )} \sqrt{-\frac{c^{2} \sqrt{-\frac{B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a c^{5}}} - 2 \, A B}{c^{2}}}\right ) + 4 \, B \sqrt{x}}{2 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(c*sqrt((c^2*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a*c^5)) + 2*A*B)/c^2
)*log(-(B^4*a^2 - A^4*c^2)*sqrt(x) + (A*a*c^4*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A
^4*c^2)/(a*c^5)) + B^3*a^2*c - A^2*B*a*c^2)*sqrt((c^2*sqrt(-(B^4*a^2 - 2*A^2*B^2
*a*c + A^4*c^2)/(a*c^5)) + 2*A*B)/c^2)) - c*sqrt((c^2*sqrt(-(B^4*a^2 - 2*A^2*B^2
*a*c + A^4*c^2)/(a*c^5)) + 2*A*B)/c^2)*log(-(B^4*a^2 - A^4*c^2)*sqrt(x) - (A*a*c
^4*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a*c^5)) + B^3*a^2*c - A^2*B*a*c^2)
*sqrt((c^2*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a*c^5)) + 2*A*B)/c^2)) - c
*sqrt(-(c^2*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a*c^5)) - 2*A*B)/c^2)*log
(-(B^4*a^2 - A^4*c^2)*sqrt(x) + (A*a*c^4*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^
2)/(a*c^5)) - B^3*a^2*c + A^2*B*a*c^2)*sqrt(-(c^2*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c
+ A^4*c^2)/(a*c^5)) - 2*A*B)/c^2)) + c*sqrt(-(c^2*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*
c + A^4*c^2)/(a*c^5)) - 2*A*B)/c^2)*log(-(B^4*a^2 - A^4*c^2)*sqrt(x) - (A*a*c^4*
sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a*c^5)) - B^3*a^2*c + A^2*B*a*c^2)*sq
rt(-(c^2*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a*c^5)) - 2*A*B)/c^2)) + 4*B
*sqrt(x))/c

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Sympy [A]  time = 13.3477, size = 75, normalized size = 0.28 \[ 2 A \operatorname{RootSum}{\left (256 t^{4} a c^{3} + 1, \left ( t \mapsto t \log{\left (64 t^{3} a c^{2} + \sqrt{x} \right )} \right )\right )} - \frac{2 B a \operatorname{RootSum}{\left (256 t^{4} a^{3} c + 1, \left ( t \mapsto t \log{\left (4 t a + \sqrt{x} \right )} \right )\right )}}{c} + \frac{2 B \sqrt{x}}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*RootSum(256*_t**4*a*c**3 + 1, Lambda(_t, _t*log(64*_t**3*a*c**2 + sqrt(x))))
- 2*B*a*RootSum(256*_t**4*a**3*c + 1, Lambda(_t, _t*log(4*_t*a + sqrt(x))))/c +
2*B*sqrt(x)/c

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GIAC/XCAS [A]  time = 0.280558, size = 350, normalized size = 1.32 \[ \frac{2 \, B \sqrt{x}}{c} - \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} B a c - \left (a c^{3}\right )^{\frac{3}{4}} A\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{2 \, a c^{3}} + \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} B a c + \left (a c^{3}\right )^{\frac{3}{4}} A\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{c}}\right )}{4 \, a c^{3}} - \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} B a c^{3} - \left (a c^{3}\right )^{\frac{3}{4}} A c^{2}\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{2 \, a c^{5}} - \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} B a c^{3} + \left (a c^{3}\right )^{\frac{3}{4}} A c^{2}\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{c}}\right )}{4 \, a c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*sqrt(x)/c - 1/2*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*c^3) + 1/4*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*c^3) - 1/2*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*sqrt(x))/(a/c)^(1/4))/(a*c^5) - 1/4*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*c^5)