3.221 \(\int \frac{(A+B x) \left (b x+c x^2\right )^{5/2}}{x^{11/2}} \, dx\)

Optimal. Leaf size=172 \[ \frac{5 c \sqrt{b x+c x^2} (3 A c+4 b B)}{4 \sqrt{x}}-\frac{5}{4} \sqrt{b} c (3 A c+4 b B) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )-\frac{\left (b x+c x^2\right )^{5/2} (3 A c+4 b B)}{4 b x^{7/2}}+\frac{5 c \left (b x+c x^2\right )^{3/2} (3 A c+4 b B)}{12 b x^{3/2}}-\frac{A \left (b x+c x^2\right )^{7/2}}{2 b x^{11/2}} \]

[Out]

(5*c*(4*b*B + 3*A*c)*Sqrt[b*x + c*x^2])/(4*Sqrt[x]) + (5*c*(4*b*B + 3*A*c)*(b*x
+ c*x^2)^(3/2))/(12*b*x^(3/2)) - ((4*b*B + 3*A*c)*(b*x + c*x^2)^(5/2))/(4*b*x^(7
/2)) - (A*(b*x + c*x^2)^(7/2))/(2*b*x^(11/2)) - (5*Sqrt[b]*c*(4*b*B + 3*A*c)*Arc
Tanh[Sqrt[b*x + c*x^2]/(Sqrt[b]*Sqrt[x])])/4

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Rubi [A]  time = 0.359577, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{5 c \sqrt{b x+c x^2} (3 A c+4 b B)}{4 \sqrt{x}}-\frac{5}{4} \sqrt{b} c (3 A c+4 b B) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )-\frac{\left (b x+c x^2\right )^{5/2} (3 A c+4 b B)}{4 b x^{7/2}}+\frac{5 c \left (b x+c x^2\right )^{3/2} (3 A c+4 b B)}{12 b x^{3/2}}-\frac{A \left (b x+c x^2\right )^{7/2}}{2 b x^{11/2}} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(b*x + c*x^2)^(5/2))/x^(11/2),x]

[Out]

(5*c*(4*b*B + 3*A*c)*Sqrt[b*x + c*x^2])/(4*Sqrt[x]) + (5*c*(4*b*B + 3*A*c)*(b*x
+ c*x^2)^(3/2))/(12*b*x^(3/2)) - ((4*b*B + 3*A*c)*(b*x + c*x^2)^(5/2))/(4*b*x^(7
/2)) - (A*(b*x + c*x^2)^(7/2))/(2*b*x^(11/2)) - (5*Sqrt[b]*c*(4*b*B + 3*A*c)*Arc
Tanh[Sqrt[b*x + c*x^2]/(Sqrt[b]*Sqrt[x])])/4

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Rubi in Sympy [A]  time = 21.8764, size = 162, normalized size = 0.94 \[ - \frac{A \left (b x + c x^{2}\right )^{\frac{7}{2}}}{2 b x^{\frac{11}{2}}} - \frac{5 \sqrt{b} c \left (3 A c + 4 B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{b x + c x^{2}}}{\sqrt{b} \sqrt{x}} \right )}}{4} + \frac{5 c \left (3 A c + 4 B b\right ) \sqrt{b x + c x^{2}}}{4 \sqrt{x}} + \frac{5 c \left (3 A c + 4 B b\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{12 b x^{\frac{3}{2}}} - \frac{\left (3 A c + 4 B b\right ) \left (b x + c x^{2}\right )^{\frac{5}{2}}}{4 b x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x)**(5/2)/x**(11/2),x)

[Out]

-A*(b*x + c*x**2)**(7/2)/(2*b*x**(11/2)) - 5*sqrt(b)*c*(3*A*c + 4*B*b)*atanh(sqr
t(b*x + c*x**2)/(sqrt(b)*sqrt(x)))/4 + 5*c*(3*A*c + 4*B*b)*sqrt(b*x + c*x**2)/(4
*sqrt(x)) + 5*c*(3*A*c + 4*B*b)*(b*x + c*x**2)**(3/2)/(12*b*x**(3/2)) - (3*A*c +
4*B*b)*(b*x + c*x**2)**(5/2)/(4*b*x**(7/2))

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Mathematica [A]  time = 0.244327, size = 122, normalized size = 0.71 \[ \frac{\sqrt{x (b+c x)} \left (\sqrt{b+c x} \left (4 B x \left (-3 b^2+14 b c x+2 c^2 x^2\right )-3 A \left (2 b^2+9 b c x-8 c^2 x^2\right )\right )-15 \sqrt{b} c x^2 (3 A c+4 b B) \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )\right )}{12 x^{5/2} \sqrt{b+c x}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(b*x + c*x^2)^(5/2))/x^(11/2),x]

[Out]

(Sqrt[x*(b + c*x)]*(Sqrt[b + c*x]*(-3*A*(2*b^2 + 9*b*c*x - 8*c^2*x^2) + 4*B*x*(-
3*b^2 + 14*b*c*x + 2*c^2*x^2)) - 15*Sqrt[b]*c*(4*b*B + 3*A*c)*x^2*ArcTanh[Sqrt[b
+ c*x]/Sqrt[b]]))/(12*x^(5/2)*Sqrt[b + c*x])

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Maple [A]  time = 0.028, size = 167, normalized size = 1. \[ -{\frac{1}{12}\sqrt{x \left ( cx+b \right ) } \left ( -8\,B{x}^{3}{c}^{2}\sqrt{b}\sqrt{cx+b}+45\,A{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){x}^{2}b{c}^{2}-24\,A{x}^{2}{c}^{2}\sqrt{b}\sqrt{cx+b}+60\,B{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){x}^{2}{b}^{2}c-56\,B{x}^{2}{b}^{3/2}c\sqrt{cx+b}+27\,Ax{b}^{3/2}c\sqrt{cx+b}+12\,Bx{b}^{5/2}\sqrt{cx+b}+6\,A{b}^{5/2}\sqrt{cx+b} \right ){x}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{cx+b}}}{\frac{1}{\sqrt{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)^(5/2)/x^(11/2),x)

[Out]

-1/12*(x*(c*x+b))^(1/2)*(-8*B*x^3*c^2*b^(1/2)*(c*x+b)^(1/2)+45*A*arctanh((c*x+b)
^(1/2)/b^(1/2))*x^2*b*c^2-24*A*x^2*c^2*b^(1/2)*(c*x+b)^(1/2)+60*B*arctanh((c*x+b
)^(1/2)/b^(1/2))*x^2*b^2*c-56*B*x^2*b^(3/2)*c*(c*x+b)^(1/2)+27*A*x*b^(3/2)*c*(c*
x+b)^(1/2)+12*B*x*b^(5/2)*(c*x+b)^(1/2)+6*A*b^(5/2)*(c*x+b)^(1/2))/x^(5/2)/(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^(11/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.306133, size = 1, normalized size = 0.01 \[ \left [\frac{16 \, B c^{3} x^{4} + 15 \,{\left (4 \, B b c + 3 \, A c^{2}\right )} \sqrt{c x^{2} + b x} \sqrt{b} x^{\frac{3}{2}} \log \left (-\frac{c x^{2} + 2 \, b x - 2 \, \sqrt{c x^{2} + b x} \sqrt{b} \sqrt{x}}{x^{2}}\right ) - 12 \, A b^{3} + 16 \,{\left (8 \, B b c^{2} + 3 \, A c^{3}\right )} x^{3} + 2 \,{\left (44 \, B b^{2} c - 3 \, A b c^{2}\right )} x^{2} - 6 \,{\left (4 \, B b^{3} + 11 \, A b^{2} c\right )} x}{24 \, \sqrt{c x^{2} + b x} x^{\frac{3}{2}}}, \frac{8 \, B c^{3} x^{4} - 15 \,{\left (4 \, B b c + 3 \, A c^{2}\right )} \sqrt{c x^{2} + b x} \sqrt{-b} x^{\frac{3}{2}} \arctan \left (\frac{b \sqrt{x}}{\sqrt{c x^{2} + b x} \sqrt{-b}}\right ) - 6 \, A b^{3} + 8 \,{\left (8 \, B b c^{2} + 3 \, A c^{3}\right )} x^{3} +{\left (44 \, B b^{2} c - 3 \, A b c^{2}\right )} x^{2} - 3 \,{\left (4 \, B b^{3} + 11 \, A b^{2} c\right )} x}{12 \, \sqrt{c x^{2} + b x} x^{\frac{3}{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)*(B*x + A)/x^(11/2),x, algorithm="fricas")

[Out]

[1/24*(16*B*c^3*x^4 + 15*(4*B*b*c + 3*A*c^2)*sqrt(c*x^2 + b*x)*sqrt(b)*x^(3/2)*l
og(-(c*x^2 + 2*b*x - 2*sqrt(c*x^2 + b*x)*sqrt(b)*sqrt(x))/x^2) - 12*A*b^3 + 16*(
8*B*b*c^2 + 3*A*c^3)*x^3 + 2*(44*B*b^2*c - 3*A*b*c^2)*x^2 - 6*(4*B*b^3 + 11*A*b^
2*c)*x)/(sqrt(c*x^2 + b*x)*x^(3/2)), 1/12*(8*B*c^3*x^4 - 15*(4*B*b*c + 3*A*c^2)*
sqrt(c*x^2 + b*x)*sqrt(-b)*x^(3/2)*arctan(b*sqrt(x)/(sqrt(c*x^2 + b*x)*sqrt(-b))
) - 6*A*b^3 + 8*(8*B*b*c^2 + 3*A*c^3)*x^3 + (44*B*b^2*c - 3*A*b*c^2)*x^2 - 3*(4*
B*b^3 + 11*A*b^2*c)*x)/(sqrt(c*x^2 + b*x)*x^(3/2))]

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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((B*x+A)*(c*x**2+b*x)**(5/2)/x**(11/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.333663, size = 209, normalized size = 1.22 \[ \frac{8 \,{\left (c x + b\right )}^{\frac{3}{2}} B c^{2} + 48 \, \sqrt{c x + b} B b c^{2} + 24 \, \sqrt{c x + b} A c^{3} + \frac{15 \,{\left (4 \, B b^{2} c^{2} + 3 \, A b c^{3}\right )} \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} - \frac{3 \,{\left (4 \,{\left (c x + b\right )}^{\frac{3}{2}} B b^{2} c^{2} - 4 \, \sqrt{c x + b} B b^{3} c^{2} + 9 \,{\left (c x + b\right )}^{\frac{3}{2}} A b c^{3} - 7 \, \sqrt{c x + b} A b^{2} c^{3}\right )}}{c^{2} x^{2}}}{12 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)*(B*x + A)/x^(11/2),x, algorithm="giac")

[Out]

1/12*(8*(c*x + b)^(3/2)*B*c^2 + 48*sqrt(c*x + b)*B*b*c^2 + 24*sqrt(c*x + b)*A*c^
3 + 15*(4*B*b^2*c^2 + 3*A*b*c^3)*arctan(sqrt(c*x + b)/sqrt(-b))/sqrt(-b) - 3*(4*
(c*x + b)^(3/2)*B*b^2*c^2 - 4*sqrt(c*x + b)*B*b^3*c^2 + 9*(c*x + b)^(3/2)*A*b*c^
3 - 7*sqrt(c*x + b)*A*b^2*c^3)/(c^2*x^2))/c