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

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

[Out]

(-2*A)/(a*Sqrt[x]) - ((Sqrt[a]*B - A*Sqrt[c])*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x
])/a^(1/4)])/(Sqrt[2]*a^(5/4)*c^(1/4)) + ((Sqrt[a]*B - A*Sqrt[c])*ArcTan[1 + (Sq
rt[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(5/4)*c^(1/4)) - ((Sqrt[a]*B + A*Sqr
t[c])*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(2*Sqrt[2]*a^(
5/4)*c^(1/4)) + ((Sqrt[a]*B + A*Sqrt[c])*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*S
qrt[x] + Sqrt[c]*x])/(2*Sqrt[2]*a^(5/4)*c^(1/4))

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Rubi [A]  time = 0.451004, 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 \[ -\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{2 \sqrt{2} a^{5/4} \sqrt [4]{c}}+\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{2 \sqrt{2} a^{5/4} \sqrt [4]{c}}-\frac{\left (\sqrt{a} B-A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{5/4} \sqrt [4]{c}}+\frac{\left (\sqrt{a} B-A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{5/4} \sqrt [4]{c}}-\frac{2 A}{a \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*A)/(a*Sqrt[x]) - ((Sqrt[a]*B - A*Sqrt[c])*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x
])/a^(1/4)])/(Sqrt[2]*a^(5/4)*c^(1/4)) + ((Sqrt[a]*B - A*Sqrt[c])*ArcTan[1 + (Sq
rt[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(5/4)*c^(1/4)) - ((Sqrt[a]*B + A*Sqr
t[c])*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(2*Sqrt[2]*a^(
5/4)*c^(1/4)) + ((Sqrt[a]*B + A*Sqrt[c])*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*S
qrt[x] + Sqrt[c]*x])/(2*Sqrt[2]*a^(5/4)*c^(1/4))

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Rubi in Sympy [A]  time = 86.4184, size = 250, normalized size = 0.94 \[ - \frac{2 A}{a \sqrt{x}} + \frac{\sqrt{2} \left (A \sqrt{c} - B \sqrt{a}\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}}{2 a^{\frac{5}{4}} \sqrt [4]{c}} - \frac{\sqrt{2} \left (A \sqrt{c} - B \sqrt{a}\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}}{2 a^{\frac{5}{4}} \sqrt [4]{c}} - \frac{\sqrt{2} \left (A \sqrt{c} + B \sqrt{a}\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} \sqrt{x} + \sqrt{a} \sqrt{c} + c x \right )}}{4 a^{\frac{5}{4}} \sqrt [4]{c}} + \frac{\sqrt{2} \left (A \sqrt{c} + B \sqrt{a}\right ) \log{\left (\sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} \sqrt{x} + \sqrt{a} \sqrt{c} + c x \right )}}{4 a^{\frac{5}{4}} \sqrt [4]{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A/(a*sqrt(x)) + sqrt(2)*(A*sqrt(c) - B*sqrt(a))*atan(1 - sqrt(2)*c**(1/4)*sqr
t(x)/a**(1/4))/(2*a**(5/4)*c**(1/4)) - sqrt(2)*(A*sqrt(c) - B*sqrt(a))*atan(1 +
sqrt(2)*c**(1/4)*sqrt(x)/a**(1/4))/(2*a**(5/4)*c**(1/4)) - 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)/(4*a*
*(5/4)*c**(1/4)) + 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)/(4*a**(5/4)*c**(1/4))

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Mathematica [A]  time = 0.604513, size = 266, normalized size = 1. \[ \frac{-\frac{\sqrt{2} \left (a^{3/4} A \sqrt{c}+a^{5/4} B\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{\sqrt [4]{c}}+\frac{\sqrt{2} \left (a^{3/4} A \sqrt{c}+a^{5/4} B\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{\sqrt [4]{c}}-\frac{2 \sqrt{2} a^{3/4} \left (\sqrt{a} B-A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{c}}+\frac{2 \sqrt{2} a^{3/4} \left (\sqrt{a} B-A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{c}}-\frac{8 a A}{\sqrt{x}}}{4 a^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.014, size = 277, normalized size = 1.1 \[{\frac{B\sqrt{2}}{2\,a}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }+{\frac{B\sqrt{2}}{2\,a}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) }+{\frac{B\sqrt{2}}{4\,a}\sqrt [4]{{\frac{a}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ) }-{\frac{A\sqrt{2}}{4\,a}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-{\frac{A\sqrt{2}}{2\,a}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-{\frac{A\sqrt{2}}{2\,a}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-2\,{\frac{A}{\sqrt{x}a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.292647, size = 1042, normalized size = 3.93 \[ -\frac{a \sqrt{x} \sqrt{\frac{a^{2} \sqrt{-\frac{B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c}} + 2 \, A B}{a^{2}}} \log \left (-{\left (B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt{x} +{\left (A a^{4} c \sqrt{-\frac{B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c}} + B^{3} a^{3} - A^{2} B a^{2} c\right )} \sqrt{\frac{a^{2} \sqrt{-\frac{B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c}} + 2 \, A B}{a^{2}}}\right ) - a \sqrt{x} \sqrt{\frac{a^{2} \sqrt{-\frac{B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c}} + 2 \, A B}{a^{2}}} \log \left (-{\left (B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt{x} -{\left (A a^{4} c \sqrt{-\frac{B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c}} + B^{3} a^{3} - A^{2} B a^{2} c\right )} \sqrt{\frac{a^{2} \sqrt{-\frac{B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c}} + 2 \, A B}{a^{2}}}\right ) - a \sqrt{x} \sqrt{-\frac{a^{2} \sqrt{-\frac{B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c}} - 2 \, A B}{a^{2}}} \log \left (-{\left (B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt{x} +{\left (A a^{4} c \sqrt{-\frac{B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c}} - B^{3} a^{3} + A^{2} B a^{2} c\right )} \sqrt{-\frac{a^{2} \sqrt{-\frac{B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c}} - 2 \, A B}{a^{2}}}\right ) + a \sqrt{x} \sqrt{-\frac{a^{2} \sqrt{-\frac{B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c}} - 2 \, A B}{a^{2}}} \log \left (-{\left (B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt{x} -{\left (A a^{4} c \sqrt{-\frac{B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c}} - B^{3} a^{3} + A^{2} B a^{2} c\right )} \sqrt{-\frac{a^{2} \sqrt{-\frac{B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c}} - 2 \, A B}{a^{2}}}\right ) + 4 \, A}{2 \, a \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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+A)/x**(3/2)/(c*x**2+a),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.283726, size = 336, normalized size = 1.27 \[ -\frac{2 \, A}{a \sqrt{x}} + \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} B a c - \left (a c^{3}\right )^{\frac{3}{4}} A\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{2 \, a^{2} c^{2}} + \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} B a c - \left (a c^{3}\right )^{\frac{3}{4}} A\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{2 \, a^{2} c^{2}} + \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} B a c + \left (a c^{3}\right )^{\frac{3}{4}} A\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{c}}\right )}{4 \, a^{2} c^{2}} - \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} B a c + \left (a c^{3}\right )^{\frac{3}{4}} A\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{c}}\right )}{4 \, a^{2} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A/(a*sqrt(x)) + 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^2*c^2) + 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^2*c^2) + 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^2*c^2) - 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^2*c^2)