3.412 \(\int \frac{x^{5/2} (A+B x)}{a+c x^2} \, dx\)

Optimal. Leaf size=292 \[ -\frac{a^{3/4} \left (\sqrt{a} B+A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{2 \sqrt{2} c^{9/4}}+\frac{a^{3/4} \left (\sqrt{a} B+A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{2 \sqrt{2} c^{9/4}}-\frac{a^{3/4} \left (\sqrt{a} B-A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} c^{9/4}}+\frac{a^{3/4} \left (\sqrt{a} B-A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} c^{9/4}}-\frac{2 a B \sqrt{x}}{c^2}+\frac{2 A x^{3/2}}{3 c}+\frac{2 B x^{5/2}}{5 c} \]

[Out]

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

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Rubi [A]  time = 0.692205, antiderivative size = 292, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{a^{3/4} \left (\sqrt{a} B+A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{2 \sqrt{2} c^{9/4}}+\frac{a^{3/4} \left (\sqrt{a} B+A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{2 \sqrt{2} c^{9/4}}-\frac{a^{3/4} \left (\sqrt{a} B-A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} c^{9/4}}+\frac{a^{3/4} \left (\sqrt{a} B-A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} c^{9/4}}-\frac{2 a B \sqrt{x}}{c^2}+\frac{2 A x^{3/2}}{3 c}+\frac{2 B x^{5/2}}{5 c} \]

Antiderivative was successfully verified.

[In]  Int[(x^(5/2)*(A + B*x))/(a + c*x^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 119.828, size = 277, normalized size = 0.95 \[ \frac{2 A x^{\frac{3}{2}}}{3 c} - \frac{2 B a \sqrt{x}}{c^{2}} + \frac{2 B x^{\frac{5}{2}}}{5 c} + \frac{\sqrt{2} a^{\frac{3}{4}} \left (A \sqrt{c} - B \sqrt{a}\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}}{2 c^{\frac{9}{4}}} - \frac{\sqrt{2} a^{\frac{3}{4}} \left (A \sqrt{c} - B \sqrt{a}\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}}{2 c^{\frac{9}{4}}} - \frac{\sqrt{2} a^{\frac{3}{4}} \left (A \sqrt{c} + B \sqrt{a}\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} \sqrt{x} + \sqrt{a} \sqrt{c} + c x \right )}}{4 c^{\frac{9}{4}}} + \frac{\sqrt{2} a^{\frac{3}{4}} \left (A \sqrt{c} + B \sqrt{a}\right ) \log{\left (\sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} \sqrt{x} + \sqrt{a} \sqrt{c} + c x \right )}}{4 c^{\frac{9}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(5/2)*(B*x+A)/(c*x**2+a),x)

[Out]

2*A*x**(3/2)/(3*c) - 2*B*a*sqrt(x)/c**2 + 2*B*x**(5/2)/(5*c) + sqrt(2)*a**(3/4)*
(A*sqrt(c) - B*sqrt(a))*atan(1 - sqrt(2)*c**(1/4)*sqrt(x)/a**(1/4))/(2*c**(9/4))
- sqrt(2)*a**(3/4)*(A*sqrt(c) - B*sqrt(a))*atan(1 + sqrt(2)*c**(1/4)*sqrt(x)/a*
*(1/4))/(2*c**(9/4)) - sqrt(2)*a**(3/4)*(A*sqrt(c) + B*sqrt(a))*log(-sqrt(2)*a**
(1/4)*c**(3/4)*sqrt(x) + sqrt(a)*sqrt(c) + c*x)/(4*c**(9/4)) + sqrt(2)*a**(3/4)*
(A*sqrt(c) + B*sqrt(a))*log(sqrt(2)*a**(1/4)*c**(3/4)*sqrt(x) + sqrt(a)*sqrt(c)
+ c*x)/(4*c**(9/4))

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Mathematica [A]  time = 0.494264, size = 284, normalized size = 0.97 \[ \frac{-15 \sqrt{2} \left (a^{3/4} A c+a^{5/4} B \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )+15 \sqrt{2} \left (a^{3/4} A c+a^{5/4} B \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )-30 \sqrt{2} \left (a^{5/4} B \sqrt{c}-a^{3/4} A c\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )+30 \sqrt{2} \left (a^{5/4} B \sqrt{c}-a^{3/4} A c\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )-120 a B c^{3/4} \sqrt{x}+40 A c^{7/4} x^{3/2}+24 B c^{7/4} x^{5/2}}{60 c^{11/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^(5/2)*(A + B*x))/(a + c*x^2),x]

[Out]

(-120*a*B*c^(3/4)*Sqrt[x] + 40*A*c^(7/4)*x^(3/2) + 24*B*c^(7/4)*x^(5/2) - 30*Sqr
t[2]*(a^(5/4)*B*Sqrt[c] - a^(3/4)*A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1
/4)] + 30*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)] - 15*Sqrt[2]*(a^(5/4)*B*Sqrt[c] + a^(3/4)*A*c)*Log[Sqrt[a] - S
qrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] + 15*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])/(60*c^(11
/4))

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Maple [A]  time = 0.017, size = 302, normalized size = 1. \[{\frac{2\,B}{5\,c}{x}^{{\frac{5}{2}}}}+{\frac{2\,A}{3\,c}{x}^{{\frac{3}{2}}}}-2\,{\frac{aB\sqrt{x}}{{c}^{2}}}+{\frac{Ba\sqrt{2}}{2\,{c}^{2}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }+{\frac{Ba\sqrt{2}}{2\,{c}^{2}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) }+{\frac{Ba\sqrt{2}}{4\,{c}^{2}}\sqrt [4]{{\frac{a}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ) }-{\frac{aA\sqrt{2}}{4\,{c}^{2}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-{\frac{aA\sqrt{2}}{2\,{c}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-{\frac{aA\sqrt{2}}{2\,{c}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)*(B*x+A)/(c*x^2+a),x)

[Out]

2/5*B*x^(5/2)/c+2/3*A*x^(3/2)/c-2*a*B*x^(1/2)/c^2+1/2*a/c^2*B*(a/c)^(1/4)*2^(1/2
)*arctan(2^(1/2)/(a/c)^(1/4)*x^(1/2)+1)+1/2*a/c^2*B*(a/c)^(1/4)*2^(1/2)*arctan(2
^(1/2)/(a/c)^(1/4)*x^(1/2)-1)+1/4*a/c^2*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/
c^2*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/c^2*A/(a/c)^(1/4)*2^(1/2)*arctan(2^(
1/2)/(a/c)^(1/4)*x^(1/2)+1)-1/2*a/c^2*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)*x^(5/2)/(c*x^2 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.298847, size = 1164, normalized size = 3.99 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^(5/2)/(c*x^2 + a),x, algorithm="fricas")

[Out]

-1/30*(15*c^2*sqrt((c^4*sqrt(-(B^4*a^5 - 2*A^2*B^2*a^4*c + A^4*a^3*c^2)/c^9) + 2
*A*B*a^2)/c^4)*log(-(B^4*a^4 - A^4*a^2*c^2)*sqrt(x) + (A*c^7*sqrt(-(B^4*a^5 - 2*
A^2*B^2*a^4*c + A^4*a^3*c^2)/c^9) + B^3*a^3*c^2 - A^2*B*a^2*c^3)*sqrt((c^4*sqrt(
-(B^4*a^5 - 2*A^2*B^2*a^4*c + A^4*a^3*c^2)/c^9) + 2*A*B*a^2)/c^4)) - 15*c^2*sqrt
((c^4*sqrt(-(B^4*a^5 - 2*A^2*B^2*a^4*c + A^4*a^3*c^2)/c^9) + 2*A*B*a^2)/c^4)*log
(-(B^4*a^4 - A^4*a^2*c^2)*sqrt(x) - (A*c^7*sqrt(-(B^4*a^5 - 2*A^2*B^2*a^4*c + A^
4*a^3*c^2)/c^9) + B^3*a^3*c^2 - A^2*B*a^2*c^3)*sqrt((c^4*sqrt(-(B^4*a^5 - 2*A^2*
B^2*a^4*c + A^4*a^3*c^2)/c^9) + 2*A*B*a^2)/c^4)) - 15*c^2*sqrt(-(c^4*sqrt(-(B^4*
a^5 - 2*A^2*B^2*a^4*c + A^4*a^3*c^2)/c^9) - 2*A*B*a^2)/c^4)*log(-(B^4*a^4 - A^4*
a^2*c^2)*sqrt(x) + (A*c^7*sqrt(-(B^4*a^5 - 2*A^2*B^2*a^4*c + A^4*a^3*c^2)/c^9) -
B^3*a^3*c^2 + A^2*B*a^2*c^3)*sqrt(-(c^4*sqrt(-(B^4*a^5 - 2*A^2*B^2*a^4*c + A^4*
a^3*c^2)/c^9) - 2*A*B*a^2)/c^4)) + 15*c^2*sqrt(-(c^4*sqrt(-(B^4*a^5 - 2*A^2*B^2*
a^4*c + A^4*a^3*c^2)/c^9) - 2*A*B*a^2)/c^4)*log(-(B^4*a^4 - A^4*a^2*c^2)*sqrt(x)
- (A*c^7*sqrt(-(B^4*a^5 - 2*A^2*B^2*a^4*c + A^4*a^3*c^2)/c^9) - B^3*a^3*c^2 + A
^2*B*a^2*c^3)*sqrt(-(c^4*sqrt(-(B^4*a^5 - 2*A^2*B^2*a^4*c + A^4*a^3*c^2)/c^9) -
2*A*B*a^2)/c^4)) - 4*(3*B*c*x^2 + 5*A*c*x - 15*B*a)*sqrt(x))/c^2

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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**(5/2)*(B*x+A)/(c*x**2+a),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.283081, size = 354, normalized size = 1.21 \[ \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} B a c - \left (a c^{3}\right )^{\frac{3}{4}} A\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{2 \, c^{4}} + \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} B a c - \left (a c^{3}\right )^{\frac{3}{4}} A\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{2 \, c^{4}} + \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} B a c + \left (a c^{3}\right )^{\frac{3}{4}} A\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{c}}\right )}{4 \, c^{4}} - \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} B a c + \left (a c^{3}\right )^{\frac{3}{4}} A\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{c}}\right )}{4 \, c^{4}} + \frac{2 \,{\left (3 \, B c^{4} x^{\frac{5}{2}} + 5 \, A c^{4} x^{\frac{3}{2}} - 15 \, B a c^{3} \sqrt{x}\right )}}{15 \, c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^(5/2)/(c*x^2 + a),x, algorithm="giac")

[Out]

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))/c^4 + 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
))/c^4 + 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))/c^4 - 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))/c^4 + 2/15*(3*B*c^4*x^(5
/2) + 5*A*c^4*x^(3/2) - 15*B*a*c^3*sqrt(x))/c^5