3.220 \(\int \frac{(A+B x) \left (b x+c x^2\right )^{5/2}}{x^{9/2}} \, dx\)

Optimal. Leaf size=160 \[ -b^{3/2} (5 A c+2 b B) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )+\frac{b \sqrt{b x+c x^2} (5 A c+2 b B)}{\sqrt{x}}+\frac{\left (b x+c x^2\right )^{5/2} (5 A c+2 b B)}{5 b x^{5/2}}+\frac{\left (b x+c x^2\right )^{3/2} (5 A c+2 b B)}{3 x^{3/2}}-\frac{A \left (b x+c x^2\right )^{7/2}}{b x^{9/2}} \]

[Out]

(b*(2*b*B + 5*A*c)*Sqrt[b*x + c*x^2])/Sqrt[x] + ((2*b*B + 5*A*c)*(b*x + c*x^2)^(
3/2))/(3*x^(3/2)) + ((2*b*B + 5*A*c)*(b*x + c*x^2)^(5/2))/(5*b*x^(5/2)) - (A*(b*
x + c*x^2)^(7/2))/(b*x^(9/2)) - b^(3/2)*(2*b*B + 5*A*c)*ArcTanh[Sqrt[b*x + c*x^2
]/(Sqrt[b]*Sqrt[x])]

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Rubi [A]  time = 0.352858, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -b^{3/2} (5 A c+2 b B) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )+\frac{b \sqrt{b x+c x^2} (5 A c+2 b B)}{\sqrt{x}}+\frac{\left (b x+c x^2\right )^{5/2} (5 A c+2 b B)}{5 b x^{5/2}}+\frac{\left (b x+c x^2\right )^{3/2} (5 A c+2 b B)}{3 x^{3/2}}-\frac{A \left (b x+c x^2\right )^{7/2}}{b x^{9/2}} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(b*x + c*x^2)^(5/2))/x^(9/2),x]

[Out]

(b*(2*b*B + 5*A*c)*Sqrt[b*x + c*x^2])/Sqrt[x] + ((2*b*B + 5*A*c)*(b*x + c*x^2)^(
3/2))/(3*x^(3/2)) + ((2*b*B + 5*A*c)*(b*x + c*x^2)^(5/2))/(5*b*x^(5/2)) - (A*(b*
x + c*x^2)^(7/2))/(b*x^(9/2)) - b^(3/2)*(2*b*B + 5*A*c)*ArcTanh[Sqrt[b*x + c*x^2
]/(Sqrt[b]*Sqrt[x])]

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Rubi in Sympy [A]  time = 21.9958, size = 151, normalized size = 0.94 \[ - \frac{A \left (b x + c x^{2}\right )^{\frac{7}{2}}}{b x^{\frac{9}{2}}} - 2 b^{\frac{3}{2}} \left (\frac{5 A c}{2} + B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{b x + c x^{2}}}{\sqrt{b} \sqrt{x}} \right )} + \frac{b \left (5 A c + 2 B b\right ) \sqrt{b x + c x^{2}}}{\sqrt{x}} + \frac{\left (\frac{5 A c}{3} + \frac{2 B b}{3}\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{x^{\frac{3}{2}}} + \frac{2 \left (\frac{5 A c}{2} + B b\right ) \left (b x + c x^{2}\right )^{\frac{5}{2}}}{5 b x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x)**(5/2)/x**(9/2),x)

[Out]

-A*(b*x + c*x**2)**(7/2)/(b*x**(9/2)) - 2*b**(3/2)*(5*A*c/2 + B*b)*atanh(sqrt(b*
x + c*x**2)/(sqrt(b)*sqrt(x))) + b*(5*A*c + 2*B*b)*sqrt(b*x + c*x**2)/sqrt(x) +
(5*A*c/3 + 2*B*b/3)*(b*x + c*x**2)**(3/2)/x**(3/2) + 2*(5*A*c/2 + B*b)*(b*x + c*
x**2)**(5/2)/(5*b*x**(5/2))

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Mathematica [A]  time = 0.183673, size = 118, normalized size = 0.74 \[ \frac{\sqrt{x (b+c x)} \left (\sqrt{b+c x} \left (A \left (-15 b^2+70 b c x+10 c^2 x^2\right )+2 B x \left (23 b^2+11 b c x+3 c^2 x^2\right )\right )-15 b^{3/2} x (5 A c+2 b B) \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )\right )}{15 x^{3/2} \sqrt{b+c x}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(b*x + c*x^2)^(5/2))/x^(9/2),x]

[Out]

(Sqrt[x*(b + c*x)]*(Sqrt[b + c*x]*(2*B*x*(23*b^2 + 11*b*c*x + 3*c^2*x^2) + A*(-1
5*b^2 + 70*b*c*x + 10*c^2*x^2)) - 15*b^(3/2)*(2*b*B + 5*A*c)*x*ArcTanh[Sqrt[b +
c*x]/Sqrt[b]]))/(15*x^(3/2)*Sqrt[b + c*x])

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Maple [A]  time = 0.026, size = 162, normalized size = 1. \[ -{\frac{1}{15}\sqrt{x \left ( cx+b \right ) } \left ( -6\,B{x}^{3}{c}^{2}\sqrt{b}\sqrt{cx+b}-10\,A{x}^{2}{c}^{2}\sqrt{b}\sqrt{cx+b}-22\,B{x}^{2}{b}^{3/2}c\sqrt{cx+b}+75\,A{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ) x{b}^{2}c-70\,Ax{b}^{3/2}c\sqrt{cx+b}+30\,B{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ) x{b}^{3}-46\,Bx{b}^{5/2}\sqrt{cx+b}+15\,A{b}^{5/2}\sqrt{cx+b} \right ){x}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{cx+b}}}{\frac{1}{\sqrt{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)^(5/2)/x^(9/2),x)

[Out]

-1/15*(x*(c*x+b))^(1/2)*(-6*B*x^3*c^2*b^(1/2)*(c*x+b)^(1/2)-10*A*x^2*c^2*b^(1/2)
*(c*x+b)^(1/2)-22*B*x^2*b^(3/2)*c*(c*x+b)^(1/2)+75*A*arctanh((c*x+b)^(1/2)/b^(1/
2))*x*b^2*c-70*A*x*b^(3/2)*c*(c*x+b)^(1/2)+30*B*arctanh((c*x+b)^(1/2)/b^(1/2))*x
*b^3-46*B*x*b^(5/2)*(c*x+b)^(1/2)+15*A*b^(5/2)*(c*x+b)^(1/2))/x^(3/2)/(c*x+b)^(1
/2)/b^(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((c*x^2 + b*x)^(5/2)*(B*x + A)/x^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.308742, size = 1, normalized size = 0.01 \[ \left [\frac{12 \, B c^{3} x^{4} - 30 \, A b^{3} + 4 \,{\left (14 \, B b c^{2} + 5 \, A c^{3}\right )} x^{3} + 15 \,{\left (2 \, B b^{2} + 5 \, A b c\right )} \sqrt{c x^{2} + b x} \sqrt{b} \sqrt{x} \log \left (-\frac{c x^{2} + 2 \, b x - 2 \, \sqrt{c x^{2} + b x} \sqrt{b} \sqrt{x}}{x^{2}}\right ) + 8 \,{\left (17 \, B b^{2} c + 20 \, A b c^{2}\right )} x^{2} + 2 \,{\left (46 \, B b^{3} + 55 \, A b^{2} c\right )} x}{30 \, \sqrt{c x^{2} + b x} \sqrt{x}}, \frac{6 \, B c^{3} x^{4} - 15 \, A b^{3} + 2 \,{\left (14 \, B b c^{2} + 5 \, A c^{3}\right )} x^{3} - 15 \,{\left (2 \, B b^{2} + 5 \, A b c\right )} \sqrt{c x^{2} + b x} \sqrt{-b} \sqrt{x} \arctan \left (\frac{b \sqrt{x}}{\sqrt{c x^{2} + b x} \sqrt{-b}}\right ) + 4 \,{\left (17 \, B b^{2} c + 20 \, A b c^{2}\right )} x^{2} +{\left (46 \, B b^{3} + 55 \, A b^{2} c\right )} x}{15 \, \sqrt{c x^{2} + b x} \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)*(B*x + A)/x^(9/2),x, algorithm="fricas")

[Out]

[1/30*(12*B*c^3*x^4 - 30*A*b^3 + 4*(14*B*b*c^2 + 5*A*c^3)*x^3 + 15*(2*B*b^2 + 5*
A*b*c)*sqrt(c*x^2 + b*x)*sqrt(b)*sqrt(x)*log(-(c*x^2 + 2*b*x - 2*sqrt(c*x^2 + b*
x)*sqrt(b)*sqrt(x))/x^2) + 8*(17*B*b^2*c + 20*A*b*c^2)*x^2 + 2*(46*B*b^3 + 55*A*
b^2*c)*x)/(sqrt(c*x^2 + b*x)*sqrt(x)), 1/15*(6*B*c^3*x^4 - 15*A*b^3 + 2*(14*B*b*
c^2 + 5*A*c^3)*x^3 - 15*(2*B*b^2 + 5*A*b*c)*sqrt(c*x^2 + b*x)*sqrt(-b)*sqrt(x)*a
rctan(b*sqrt(x)/(sqrt(c*x^2 + b*x)*sqrt(-b))) + 4*(17*B*b^2*c + 20*A*b*c^2)*x^2
+ (46*B*b^3 + 55*A*b^2*c)*x)/(sqrt(c*x^2 + b*x)*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)*(c*x**2+b*x)**(5/2)/x**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.359382, size = 169, normalized size = 1.06 \[ \frac{6 \,{\left (c x + b\right )}^{\frac{5}{2}} B c + 10 \,{\left (c x + b\right )}^{\frac{3}{2}} B b c + 30 \, \sqrt{c x + b} B b^{2} c + 10 \,{\left (c x + b\right )}^{\frac{3}{2}} A c^{2} + 60 \, \sqrt{c x + b} A b c^{2} - \frac{15 \, \sqrt{c x + b} A b^{2} c}{x} + \frac{15 \,{\left (2 \, B b^{3} c + 5 \, A b^{2} c^{2}\right )} \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b}}}{15 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)*(B*x + A)/x^(9/2),x, algorithm="giac")

[Out]

1/15*(6*(c*x + b)^(5/2)*B*c + 10*(c*x + b)^(3/2)*B*b*c + 30*sqrt(c*x + b)*B*b^2*
c + 10*(c*x + b)^(3/2)*A*c^2 + 60*sqrt(c*x + b)*A*b*c^2 - 15*sqrt(c*x + b)*A*b^2
*c/x + 15*(2*B*b^3*c + 5*A*b^2*c^2)*arctan(sqrt(c*x + b)/sqrt(-b))/sqrt(-b))/c