3.211 \(\int \frac{(A+B x) \left (b x+c x^2\right )^{3/2}}{x^{11/2}} \, dx\)

Optimal. Leaf size=142 \[ -\frac{c^2 (6 b B-A c) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{8 b^{3/2}}-\frac{\left (b x+c x^2\right )^{3/2} (6 b B-A c)}{12 b x^{7/2}}-\frac{c \sqrt{b x+c x^2} (6 b B-A c)}{8 b x^{3/2}}-\frac{A \left (b x+c x^2\right )^{5/2}}{3 b x^{11/2}} \]

[Out]

-(c*(6*b*B - A*c)*Sqrt[b*x + c*x^2])/(8*b*x^(3/2)) - ((6*b*B - A*c)*(b*x + c*x^2
)^(3/2))/(12*b*x^(7/2)) - (A*(b*x + c*x^2)^(5/2))/(3*b*x^(11/2)) - (c^2*(6*b*B -
A*c)*ArcTanh[Sqrt[b*x + c*x^2]/(Sqrt[b]*Sqrt[x])])/(8*b^(3/2))

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Rubi [A]  time = 0.294035, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{c^2 (6 b B-A c) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{8 b^{3/2}}-\frac{\left (b x+c x^2\right )^{3/2} (6 b B-A c)}{12 b x^{7/2}}-\frac{c \sqrt{b x+c x^2} (6 b B-A c)}{8 b x^{3/2}}-\frac{A \left (b x+c x^2\right )^{5/2}}{3 b x^{11/2}} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(b*x + c*x^2)^(3/2))/x^(11/2),x]

[Out]

-(c*(6*b*B - A*c)*Sqrt[b*x + c*x^2])/(8*b*x^(3/2)) - ((6*b*B - A*c)*(b*x + c*x^2
)^(3/2))/(12*b*x^(7/2)) - (A*(b*x + c*x^2)^(5/2))/(3*b*x^(11/2)) - (c^2*(6*b*B -
A*c)*ArcTanh[Sqrt[b*x + c*x^2]/(Sqrt[b]*Sqrt[x])])/(8*b^(3/2))

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Rubi in Sympy [A]  time = 18.3881, size = 122, normalized size = 0.86 \[ - \frac{A \left (b x + c x^{2}\right )^{\frac{5}{2}}}{3 b x^{\frac{11}{2}}} + \frac{c \left (A c - 6 B b\right ) \sqrt{b x + c x^{2}}}{8 b x^{\frac{3}{2}}} + \frac{\left (A c - 6 B b\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{12 b x^{\frac{7}{2}}} + \frac{c^{2} \left (A c - 6 B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{b x + c x^{2}}}{\sqrt{b} \sqrt{x}} \right )}}{8 b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x)**(3/2)/x**(11/2),x)

[Out]

-A*(b*x + c*x**2)**(5/2)/(3*b*x**(11/2)) + c*(A*c - 6*B*b)*sqrt(b*x + c*x**2)/(8
*b*x**(3/2)) + (A*c - 6*B*b)*(b*x + c*x**2)**(3/2)/(12*b*x**(7/2)) + c**2*(A*c -
6*B*b)*atanh(sqrt(b*x + c*x**2)/(sqrt(b)*sqrt(x)))/(8*b**(3/2))

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Mathematica [A]  time = 0.219039, size = 118, normalized size = 0.83 \[ \frac{\sqrt{x (b+c x)} \left (3 c^2 x^3 (A c-6 b B) \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )-\sqrt{b} \sqrt{b+c x} \left (A \left (8 b^2+14 b c x+3 c^2 x^2\right )+6 b B x (2 b+5 c x)\right )\right )}{24 b^{3/2} x^{7/2} \sqrt{b+c x}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(b*x + c*x^2)^(3/2))/x^(11/2),x]

[Out]

(Sqrt[x*(b + c*x)]*(-(Sqrt[b]*Sqrt[b + c*x]*(6*b*B*x*(2*b + 5*c*x) + A*(8*b^2 +
14*b*c*x + 3*c^2*x^2))) + 3*c^2*(-6*b*B + A*c)*x^3*ArcTanh[Sqrt[b + c*x]/Sqrt[b]
]))/(24*b^(3/2)*x^(7/2)*Sqrt[b + c*x])

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Maple [A]  time = 0.028, size = 147, normalized size = 1. \[{\frac{1}{24}\sqrt{x \left ( cx+b \right ) } \left ( 3\,A{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){x}^{3}{c}^{3}-18\,B{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){x}^{3}b{c}^{2}-3\,A{x}^{2}{c}^{2}\sqrt{b}\sqrt{cx+b}-30\,B{x}^{2}{b}^{3/2}c\sqrt{cx+b}-14\,Ax{b}^{3/2}c\sqrt{cx+b}-12\,Bx{b}^{5/2}\sqrt{cx+b}-8\,A{b}^{5/2}\sqrt{cx+b} \right ){b}^{-{\frac{3}{2}}}{x}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{cx+b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)^(3/2)/x^(11/2),x)

[Out]

1/24*(x*(c*x+b))^(1/2)/b^(3/2)*(3*A*arctanh((c*x+b)^(1/2)/b^(1/2))*x^3*c^3-18*B*
arctanh((c*x+b)^(1/2)/b^(1/2))*x^3*b*c^2-3*A*x^2*c^2*b^(1/2)*(c*x+b)^(1/2)-30*B*
x^2*b^(3/2)*c*(c*x+b)^(1/2)-14*A*x*b^(3/2)*c*(c*x+b)^(1/2)-12*B*x*b^(5/2)*(c*x+b
)^(1/2)-8*A*b^(5/2)*(c*x+b)^(1/2))/x^(7/2)/(c*x+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)^(3/2)*(B*x + A)/x^(11/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.303639, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (6 \, B b c^{2} - A c^{3}\right )} x^{4} \log \left (-\frac{2 \, \sqrt{c x^{2} + b x} b \sqrt{x} +{\left (c x^{2} + 2 \, b x\right )} \sqrt{b}}{x^{2}}\right ) + 2 \,{\left (8 \, A b^{2} + 3 \,{\left (10 \, B b c + A c^{2}\right )} x^{2} + 2 \,{\left (6 \, B b^{2} + 7 \, A b c\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{b} \sqrt{x}}{48 \, b^{\frac{3}{2}} x^{4}}, -\frac{3 \,{\left (6 \, B b c^{2} - A c^{3}\right )} x^{4} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right ) +{\left (8 \, A b^{2} + 3 \,{\left (10 \, B b c + A c^{2}\right )} x^{2} + 2 \,{\left (6 \, B b^{2} + 7 \, A b c\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{-b} \sqrt{x}}{24 \, \sqrt{-b} b x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)*(B*x + A)/x^(11/2),x, algorithm="fricas")

[Out]

[-1/48*(3*(6*B*b*c^2 - A*c^3)*x^4*log(-(2*sqrt(c*x^2 + b*x)*b*sqrt(x) + (c*x^2 +
2*b*x)*sqrt(b))/x^2) + 2*(8*A*b^2 + 3*(10*B*b*c + A*c^2)*x^2 + 2*(6*B*b^2 + 7*A
*b*c)*x)*sqrt(c*x^2 + b*x)*sqrt(b)*sqrt(x))/(b^(3/2)*x^4), -1/24*(3*(6*B*b*c^2 -
A*c^3)*x^4*arctan(sqrt(-b)*sqrt(x)/sqrt(c*x^2 + b*x)) + (8*A*b^2 + 3*(10*B*b*c
+ A*c^2)*x^2 + 2*(6*B*b^2 + 7*A*b*c)*x)*sqrt(c*x^2 + b*x)*sqrt(-b)*sqrt(x))/(sqr
t(-b)*b*x^4)]

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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)**(3/2)/x**(11/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.348734, size = 196, normalized size = 1.38 \[ \frac{\frac{3 \,{\left (6 \, B b c^{3} - A c^{4}\right )} \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b} - \frac{30 \,{\left (c x + b\right )}^{\frac{5}{2}} B b c^{3} - 48 \,{\left (c x + b\right )}^{\frac{3}{2}} B b^{2} c^{3} + 18 \, \sqrt{c x + b} B b^{3} c^{3} + 3 \,{\left (c x + b\right )}^{\frac{5}{2}} A c^{4} + 8 \,{\left (c x + b\right )}^{\frac{3}{2}} A b c^{4} - 3 \, \sqrt{c x + b} A b^{2} c^{4}}{b c^{3} x^{3}}}{24 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)*(B*x + A)/x^(11/2),x, algorithm="giac")

[Out]

1/24*(3*(6*B*b*c^3 - A*c^4)*arctan(sqrt(c*x + b)/sqrt(-b))/(sqrt(-b)*b) - (30*(c
*x + b)^(5/2)*B*b*c^3 - 48*(c*x + b)^(3/2)*B*b^2*c^3 + 18*sqrt(c*x + b)*B*b^3*c^
3 + 3*(c*x + b)^(5/2)*A*c^4 + 8*(c*x + b)^(3/2)*A*b*c^4 - 3*sqrt(c*x + b)*A*b^2*
c^4)/(b*c^3*x^3))/c