3.224 \(\int \frac{(A+B x) \left (b x+c x^2\right )^{5/2}}{x^{17/2}} \, dx\)

Optimal. Leaf size=216 \[ \frac{c^4 (10 b B-3 A c) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{128 b^{5/2}}-\frac{c^3 \sqrt{b x+c x^2} (10 b B-3 A c)}{128 b^2 x^{3/2}}-\frac{c^2 \sqrt{b x+c x^2} (10 b B-3 A c)}{64 b x^{5/2}}-\frac{c \left (b x+c x^2\right )^{3/2} (10 b B-3 A c)}{48 b x^{9/2}}-\frac{\left (b x+c x^2\right )^{5/2} (10 b B-3 A c)}{40 b x^{13/2}}-\frac{A \left (b x+c x^2\right )^{7/2}}{5 b x^{17/2}} \]

[Out]

-(c^2*(10*b*B - 3*A*c)*Sqrt[b*x + c*x^2])/(64*b*x^(5/2)) - (c^3*(10*b*B - 3*A*c)
*Sqrt[b*x + c*x^2])/(128*b^2*x^(3/2)) - (c*(10*b*B - 3*A*c)*(b*x + c*x^2)^(3/2))
/(48*b*x^(9/2)) - ((10*b*B - 3*A*c)*(b*x + c*x^2)^(5/2))/(40*b*x^(13/2)) - (A*(b
*x + c*x^2)^(7/2))/(5*b*x^(17/2)) + (c^4*(10*b*B - 3*A*c)*ArcTanh[Sqrt[b*x + c*x
^2]/(Sqrt[b]*Sqrt[x])])/(128*b^(5/2))

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Rubi [A]  time = 0.453146, antiderivative size = 216, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{c^4 (10 b B-3 A c) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{128 b^{5/2}}-\frac{c^3 \sqrt{b x+c x^2} (10 b B-3 A c)}{128 b^2 x^{3/2}}-\frac{c^2 \sqrt{b x+c x^2} (10 b B-3 A c)}{64 b x^{5/2}}-\frac{c \left (b x+c x^2\right )^{3/2} (10 b B-3 A c)}{48 b x^{9/2}}-\frac{\left (b x+c x^2\right )^{5/2} (10 b B-3 A c)}{40 b x^{13/2}}-\frac{A \left (b x+c x^2\right )^{7/2}}{5 b x^{17/2}} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(b*x + c*x^2)^(5/2))/x^(17/2),x]

[Out]

-(c^2*(10*b*B - 3*A*c)*Sqrt[b*x + c*x^2])/(64*b*x^(5/2)) - (c^3*(10*b*B - 3*A*c)
*Sqrt[b*x + c*x^2])/(128*b^2*x^(3/2)) - (c*(10*b*B - 3*A*c)*(b*x + c*x^2)^(3/2))
/(48*b*x^(9/2)) - ((10*b*B - 3*A*c)*(b*x + c*x^2)^(5/2))/(40*b*x^(13/2)) - (A*(b
*x + c*x^2)^(7/2))/(5*b*x^(17/2)) + (c^4*(10*b*B - 3*A*c)*ArcTanh[Sqrt[b*x + c*x
^2]/(Sqrt[b]*Sqrt[x])])/(128*b^(5/2))

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Rubi in Sympy [A]  time = 29.1051, size = 197, normalized size = 0.91 \[ - \frac{A \left (b x + c x^{2}\right )^{\frac{7}{2}}}{5 b x^{\frac{17}{2}}} + \frac{c^{2} \left (3 A c - 10 B b\right ) \sqrt{b x + c x^{2}}}{64 b x^{\frac{5}{2}}} + \frac{c \left (3 A c - 10 B b\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{48 b x^{\frac{9}{2}}} + \frac{\left (3 A c - 10 B b\right ) \left (b x + c x^{2}\right )^{\frac{5}{2}}}{40 b x^{\frac{13}{2}}} + \frac{c^{3} \left (3 A c - 10 B b\right ) \sqrt{b x + c x^{2}}}{128 b^{2} x^{\frac{3}{2}}} - \frac{c^{4} \left (3 A c - 10 B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{b x + c x^{2}}}{\sqrt{b} \sqrt{x}} \right )}}{128 b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x)**(5/2)/x**(17/2),x)

[Out]

-A*(b*x + c*x**2)**(7/2)/(5*b*x**(17/2)) + c**2*(3*A*c - 10*B*b)*sqrt(b*x + c*x*
*2)/(64*b*x**(5/2)) + c*(3*A*c - 10*B*b)*(b*x + c*x**2)**(3/2)/(48*b*x**(9/2)) +
(3*A*c - 10*B*b)*(b*x + c*x**2)**(5/2)/(40*b*x**(13/2)) + c**3*(3*A*c - 10*B*b)
*sqrt(b*x + c*x**2)/(128*b**2*x**(3/2)) - c**4*(3*A*c - 10*B*b)*atanh(sqrt(b*x +
c*x**2)/(sqrt(b)*sqrt(x)))/(128*b**(5/2))

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Mathematica [A]  time = 0.533654, size = 160, normalized size = 0.74 \[ \frac{15 c^4 x^5 \sqrt{b+c x} (10 b B-3 A c) \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )-\sqrt{b} (b+c x) \left (3 A \left (128 b^4+336 b^3 c x+248 b^2 c^2 x^2+10 b c^3 x^3-15 c^4 x^4\right )+10 b B x \left (48 b^3+136 b^2 c x+118 b c^2 x^2+15 c^3 x^3\right )\right )}{1920 b^{5/2} x^{9/2} \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(b*x + c*x^2)^(5/2))/x^(17/2),x]

[Out]

(-(Sqrt[b]*(b + c*x)*(10*b*B*x*(48*b^3 + 136*b^2*c*x + 118*b*c^2*x^2 + 15*c^3*x^
3) + 3*A*(128*b^4 + 336*b^3*c*x + 248*b^2*c^2*x^2 + 10*b*c^3*x^3 - 15*c^4*x^4)))
+ 15*c^4*(10*b*B - 3*A*c)*x^5*Sqrt[b + c*x]*ArcTanh[Sqrt[b + c*x]/Sqrt[b]])/(19
20*b^(5/2)*x^(9/2)*Sqrt[x*(b + c*x)])

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Maple [A]  time = 0.031, size = 223, normalized size = 1. \[ -{\frac{1}{1920}\sqrt{x \left ( cx+b \right ) } \left ( 45\,A{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){x}^{5}{c}^{5}-150\,B{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){x}^{5}b{c}^{4}-45\,A{x}^{4}{c}^{4}\sqrt{cx+b}\sqrt{b}+150\,B{x}^{4}{b}^{3/2}{c}^{3}\sqrt{cx+b}+30\,A{x}^{3}{b}^{3/2}{c}^{3}\sqrt{cx+b}+1180\,B{x}^{3}{b}^{5/2}{c}^{2}\sqrt{cx+b}+744\,A{x}^{2}{b}^{5/2}{c}^{2}\sqrt{cx+b}+1360\,B{x}^{2}{b}^{7/2}c\sqrt{cx+b}+1008\,Ax{b}^{7/2}c\sqrt{cx+b}+480\,Bx{b}^{9/2}\sqrt{cx+b}+384\,A{b}^{9/2}\sqrt{cx+b} \right ){b}^{-{\frac{5}{2}}}{x}^{-{\frac{11}{2}}}{\frac{1}{\sqrt{cx+b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)^(5/2)/x^(17/2),x)

[Out]

-1/1920*(x*(c*x+b))^(1/2)/b^(5/2)*(45*A*arctanh((c*x+b)^(1/2)/b^(1/2))*x^5*c^5-1
50*B*arctanh((c*x+b)^(1/2)/b^(1/2))*x^5*b*c^4-45*A*x^4*c^4*(c*x+b)^(1/2)*b^(1/2)
+150*B*x^4*b^(3/2)*c^3*(c*x+b)^(1/2)+30*A*x^3*b^(3/2)*c^3*(c*x+b)^(1/2)+1180*B*x
^3*b^(5/2)*c^2*(c*x+b)^(1/2)+744*A*x^2*b^(5/2)*c^2*(c*x+b)^(1/2)+1360*B*x^2*b^(7
/2)*c*(c*x+b)^(1/2)+1008*A*x*b^(7/2)*c*(c*x+b)^(1/2)+480*B*x*b^(9/2)*(c*x+b)^(1/
2)+384*A*b^(9/2)*(c*x+b)^(1/2))/x^(11/2)/(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)^(5/2)*(B*x + A)/x^(17/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.300493, size = 1, normalized size = 0. \[ \left [-\frac{15 \,{\left (10 \, B b c^{4} - 3 \, A c^{5}\right )} x^{6} \log \left (\frac{2 \, \sqrt{c x^{2} + b x} b \sqrt{x} -{\left (c x^{2} + 2 \, b x\right )} \sqrt{b}}{x^{2}}\right ) + 2 \,{\left (384 \, A b^{4} + 15 \,{\left (10 \, B b c^{3} - 3 \, A c^{4}\right )} x^{4} + 10 \,{\left (118 \, B b^{2} c^{2} + 3 \, A b c^{3}\right )} x^{3} + 8 \,{\left (170 \, B b^{3} c + 93 \, A b^{2} c^{2}\right )} x^{2} + 48 \,{\left (10 \, B b^{4} + 21 \, A b^{3} c\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{b} \sqrt{x}}{3840 \, b^{\frac{5}{2}} x^{6}}, \frac{15 \,{\left (10 \, B b c^{4} - 3 \, A c^{5}\right )} x^{6} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right ) -{\left (384 \, A b^{4} + 15 \,{\left (10 \, B b c^{3} - 3 \, A c^{4}\right )} x^{4} + 10 \,{\left (118 \, B b^{2} c^{2} + 3 \, A b c^{3}\right )} x^{3} + 8 \,{\left (170 \, B b^{3} c + 93 \, A b^{2} c^{2}\right )} x^{2} + 48 \,{\left (10 \, B b^{4} + 21 \, A b^{3} c\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{-b} \sqrt{x}}{1920 \, \sqrt{-b} b^{2} x^{6}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)*(B*x + A)/x^(17/2),x, algorithm="fricas")

[Out]

[-1/3840*(15*(10*B*b*c^4 - 3*A*c^5)*x^6*log((2*sqrt(c*x^2 + b*x)*b*sqrt(x) - (c*
x^2 + 2*b*x)*sqrt(b))/x^2) + 2*(384*A*b^4 + 15*(10*B*b*c^3 - 3*A*c^4)*x^4 + 10*(
118*B*b^2*c^2 + 3*A*b*c^3)*x^3 + 8*(170*B*b^3*c + 93*A*b^2*c^2)*x^2 + 48*(10*B*b
^4 + 21*A*b^3*c)*x)*sqrt(c*x^2 + b*x)*sqrt(b)*sqrt(x))/(b^(5/2)*x^6), 1/1920*(15
*(10*B*b*c^4 - 3*A*c^5)*x^6*arctan(sqrt(-b)*sqrt(x)/sqrt(c*x^2 + b*x)) - (384*A*
b^4 + 15*(10*B*b*c^3 - 3*A*c^4)*x^4 + 10*(118*B*b^2*c^2 + 3*A*b*c^3)*x^3 + 8*(17
0*B*b^3*c + 93*A*b^2*c^2)*x^2 + 48*(10*B*b^4 + 21*A*b^3*c)*x)*sqrt(c*x^2 + b*x)*
sqrt(-b)*sqrt(x))/(sqrt(-b)*b^2*x^6)]

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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**(17/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.436696, size = 281, normalized size = 1.3 \[ -\frac{\frac{15 \,{\left (10 \, B b c^{5} - 3 \, A c^{6}\right )} \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{2}} + \frac{150 \,{\left (c x + b\right )}^{\frac{9}{2}} B b c^{5} + 580 \,{\left (c x + b\right )}^{\frac{7}{2}} B b^{2} c^{5} - 1280 \,{\left (c x + b\right )}^{\frac{5}{2}} B b^{3} c^{5} + 700 \,{\left (c x + b\right )}^{\frac{3}{2}} B b^{4} c^{5} - 150 \, \sqrt{c x + b} B b^{5} c^{5} - 45 \,{\left (c x + b\right )}^{\frac{9}{2}} A c^{6} + 210 \,{\left (c x + b\right )}^{\frac{7}{2}} A b c^{6} + 384 \,{\left (c x + b\right )}^{\frac{5}{2}} A b^{2} c^{6} - 210 \,{\left (c x + b\right )}^{\frac{3}{2}} A b^{3} c^{6} + 45 \, \sqrt{c x + b} A b^{4} c^{6}}{b^{2} c^{5} x^{5}}}{1920 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)*(B*x + A)/x^(17/2),x, algorithm="giac")

[Out]

-1/1920*(15*(10*B*b*c^5 - 3*A*c^6)*arctan(sqrt(c*x + b)/sqrt(-b))/(sqrt(-b)*b^2)
+ (150*(c*x + b)^(9/2)*B*b*c^5 + 580*(c*x + b)^(7/2)*B*b^2*c^5 - 1280*(c*x + b)
^(5/2)*B*b^3*c^5 + 700*(c*x + b)^(3/2)*B*b^4*c^5 - 150*sqrt(c*x + b)*B*b^5*c^5 -
45*(c*x + b)^(9/2)*A*c^6 + 210*(c*x + b)^(7/2)*A*b*c^6 + 384*(c*x + b)^(5/2)*A*
b^2*c^6 - 210*(c*x + b)^(3/2)*A*b^3*c^6 + 45*sqrt(c*x + b)*A*b^4*c^6)/(b^2*c^5*x
^5))/c