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

Optimal. Leaf size=170 \[ -\frac{b^5 (5 b B-12 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{512 c^{7/2}}+\frac{b^3 (b+2 c x) \sqrt{b x+c x^2} (5 b B-12 A c)}{512 c^3}-\frac{b (b+2 c x) \left (b x+c x^2\right )^{3/2} (5 b B-12 A c)}{192 c^2}-\frac{\left (b x+c x^2\right )^{5/2} (5 b B-12 A c)}{60 c}+\frac{B \left (b x+c x^2\right )^{7/2}}{6 c x} \]

[Out]

(b^3*(5*b*B - 12*A*c)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(512*c^3) - (b*(5*b*B - 12*
A*c)*(b + 2*c*x)*(b*x + c*x^2)^(3/2))/(192*c^2) - ((5*b*B - 12*A*c)*(b*x + c*x^2
)^(5/2))/(60*c) + (B*(b*x + c*x^2)^(7/2))/(6*c*x) - (b^5*(5*b*B - 12*A*c)*ArcTan
h[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(512*c^(7/2))

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Rubi [A]  time = 0.302385, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{b^5 (5 b B-12 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{512 c^{7/2}}+\frac{b^3 (b+2 c x) \sqrt{b x+c x^2} (5 b B-12 A c)}{512 c^3}-\frac{b (b+2 c x) \left (b x+c x^2\right )^{3/2} (5 b B-12 A c)}{192 c^2}-\frac{\left (b x+c x^2\right )^{5/2} (5 b B-12 A c)}{60 c}+\frac{B \left (b x+c x^2\right )^{7/2}}{6 c x} \]

Antiderivative was successfully verified.

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

[Out]

(b^3*(5*b*B - 12*A*c)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(512*c^3) - (b*(5*b*B - 12*
A*c)*(b + 2*c*x)*(b*x + c*x^2)^(3/2))/(192*c^2) - ((5*b*B - 12*A*c)*(b*x + c*x^2
)^(5/2))/(60*c) + (B*(b*x + c*x^2)^(7/2))/(6*c*x) - (b^5*(5*b*B - 12*A*c)*ArcTan
h[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(512*c^(7/2))

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Rubi in Sympy [A]  time = 20.3595, size = 156, normalized size = 0.92 \[ \frac{B \left (b x + c x^{2}\right )^{\frac{7}{2}}}{6 c x} + \frac{b^{5} \left (12 A c - 5 B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{512 c^{\frac{7}{2}}} - \frac{b^{3} \left (b + 2 c x\right ) \left (12 A c - 5 B b\right ) \sqrt{b x + c x^{2}}}{512 c^{3}} + \frac{b \left (b + 2 c x\right ) \left (12 A c - 5 B b\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{192 c^{2}} + \frac{\left (12 A c - 5 B b\right ) \left (b x + c x^{2}\right )^{\frac{5}{2}}}{60 c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*(b*x + c*x**2)**(7/2)/(6*c*x) + b**5*(12*A*c - 5*B*b)*atanh(sqrt(c)*x/sqrt(b*x
+ c*x**2))/(512*c**(7/2)) - b**3*(b + 2*c*x)*(12*A*c - 5*B*b)*sqrt(b*x + c*x**2
)/(512*c**3) + b*(b + 2*c*x)*(12*A*c - 5*B*b)*(b*x + c*x**2)**(3/2)/(192*c**2) +
(12*A*c - 5*B*b)*(b*x + c*x**2)**(5/2)/(60*c)

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Mathematica [A]  time = 0.328709, size = 168, normalized size = 0.99 \[ \frac{\sqrt{x (b+c x)} \left (\frac{15 b^5 (12 A c-5 b B) \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{\sqrt{x} \sqrt{b+c x}}+\sqrt{c} \left (-10 b^4 c (18 A+5 B x)+40 b^3 c^2 x (3 A+B x)+48 b^2 c^3 x^2 (62 A+45 B x)+64 b c^4 x^3 (63 A+50 B x)+256 c^5 x^4 (6 A+5 B x)+75 b^5 B\right )\right )}{7680 c^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x*(b + c*x)]*(Sqrt[c]*(75*b^5*B + 40*b^3*c^2*x*(3*A + B*x) + 256*c^5*x^4*(
6*A + 5*B*x) - 10*b^4*c*(18*A + 5*B*x) + 48*b^2*c^3*x^2*(62*A + 45*B*x) + 64*b*c
^4*x^3*(63*A + 50*B*x)) + (15*b^5*(-5*b*B + 12*A*c)*Log[c*Sqrt[x] + Sqrt[c]*Sqrt
[b + c*x]])/(Sqrt[x]*Sqrt[b + c*x])))/(7680*c^(7/2))

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Maple [A]  time = 0.012, size = 274, normalized size = 1.6 \[{\frac{Bx}{6} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}+{\frac{Bb}{12\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{5\,{b}^{2}Bx}{96\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{5\,B{b}^{3}}{192\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{b}^{4}Bx}{256\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{5\,B{b}^{5}}{512\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{5\,B{b}^{6}}{1024}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{7}{2}}}}+{\frac{A}{5} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}+{\frac{Abx}{8} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{{b}^{2}A}{16\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{3\,A{b}^{3}x}{64\,c}\sqrt{c{x}^{2}+bx}}-{\frac{3\,A{b}^{4}}{128\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{3\,A{b}^{5}}{256}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*B*x*(c*x^2+b*x)^(5/2)+1/12*B/c*(c*x^2+b*x)^(5/2)*b-5/96*B*b^2/c*(c*x^2+b*x)^
(3/2)*x-5/192*B*b^3/c^2*(c*x^2+b*x)^(3/2)+5/256*B*b^4/c^2*(c*x^2+b*x)^(1/2)*x+5/
512*B*b^5/c^3*(c*x^2+b*x)^(1/2)-5/1024*B*b^6/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x
^2+b*x)^(1/2))+1/5*A*(c*x^2+b*x)^(5/2)+1/8*A*b*(c*x^2+b*x)^(3/2)*x+1/16*A/c*(c*x
^2+b*x)^(3/2)*b^2-3/64*A*b^3/c*(c*x^2+b*x)^(1/2)*x-3/128*A*b^4/c^2*(c*x^2+b*x)^(
1/2)+3/256*A*b^5/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(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,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.299937, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (1280 \, B c^{5} x^{5} + 75 \, B b^{5} - 180 \, A b^{4} c + 128 \,{\left (25 \, B b c^{4} + 12 \, A c^{5}\right )} x^{4} + 144 \,{\left (15 \, B b^{2} c^{3} + 28 \, A b c^{4}\right )} x^{3} + 8 \,{\left (5 \, B b^{3} c^{2} + 372 \, A b^{2} c^{3}\right )} x^{2} - 10 \,{\left (5 \, B b^{4} c - 12 \, A b^{3} c^{2}\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{c} - 15 \,{\left (5 \, B b^{6} - 12 \, A b^{5} c\right )} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right )}{15360 \, c^{\frac{7}{2}}}, \frac{{\left (1280 \, B c^{5} x^{5} + 75 \, B b^{5} - 180 \, A b^{4} c + 128 \,{\left (25 \, B b c^{4} + 12 \, A c^{5}\right )} x^{4} + 144 \,{\left (15 \, B b^{2} c^{3} + 28 \, A b c^{4}\right )} x^{3} + 8 \,{\left (5 \, B b^{3} c^{2} + 372 \, A b^{2} c^{3}\right )} x^{2} - 10 \,{\left (5 \, B b^{4} c - 12 \, A b^{3} c^{2}\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{-c} - 15 \,{\left (5 \, B b^{6} - 12 \, A b^{5} c\right )} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right )}{7680 \, \sqrt{-c} c^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/15360*(2*(1280*B*c^5*x^5 + 75*B*b^5 - 180*A*b^4*c + 128*(25*B*b*c^4 + 12*A*c^
5)*x^4 + 144*(15*B*b^2*c^3 + 28*A*b*c^4)*x^3 + 8*(5*B*b^3*c^2 + 372*A*b^2*c^3)*x
^2 - 10*(5*B*b^4*c - 12*A*b^3*c^2)*x)*sqrt(c*x^2 + b*x)*sqrt(c) - 15*(5*B*b^6 -
12*A*b^5*c)*log((2*c*x + b)*sqrt(c) + 2*sqrt(c*x^2 + b*x)*c))/c^(7/2), 1/7680*((
1280*B*c^5*x^5 + 75*B*b^5 - 180*A*b^4*c + 128*(25*B*b*c^4 + 12*A*c^5)*x^4 + 144*
(15*B*b^2*c^3 + 28*A*b*c^4)*x^3 + 8*(5*B*b^3*c^2 + 372*A*b^2*c^3)*x^2 - 10*(5*B*
b^4*c - 12*A*b^3*c^2)*x)*sqrt(c*x^2 + b*x)*sqrt(-c) - 15*(5*B*b^6 - 12*A*b^5*c)*
arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)))/(sqrt(-c)*c^3)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x \left (b + c x\right )\right )^{\frac{5}{2}} \left (A + B x\right )}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.282275, size = 267, normalized size = 1.57 \[ \frac{1}{7680} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \, B c^{2} x + \frac{25 \, B b c^{6} + 12 \, A c^{7}}{c^{5}}\right )} x + \frac{9 \,{\left (15 \, B b^{2} c^{5} + 28 \, A b c^{6}\right )}}{c^{5}}\right )} x + \frac{5 \, B b^{3} c^{4} + 372 \, A b^{2} c^{5}}{c^{5}}\right )} x - \frac{5 \,{\left (5 \, B b^{4} c^{3} - 12 \, A b^{3} c^{4}\right )}}{c^{5}}\right )} x + \frac{15 \,{\left (5 \, B b^{5} c^{2} - 12 \, A b^{4} c^{3}\right )}}{c^{5}}\right )} + \frac{{\left (5 \, B b^{6} - 12 \, A b^{5} c\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{1024 \, c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7680*sqrt(c*x^2 + b*x)*(2*(4*(2*(8*(10*B*c^2*x + (25*B*b*c^6 + 12*A*c^7)/c^5)*
x + 9*(15*B*b^2*c^5 + 28*A*b*c^6)/c^5)*x + (5*B*b^3*c^4 + 372*A*b^2*c^5)/c^5)*x
- 5*(5*B*b^4*c^3 - 12*A*b^3*c^4)/c^5)*x + 15*(5*B*b^5*c^2 - 12*A*b^4*c^3)/c^5) +
1/1024*(5*B*b^6 - 12*A*b^5*c)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c)
- b))/c^(7/2)