3.1185 \(\int (A+B x) \left (b x+c x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=171 \[ \frac{5 b^6 (b B-2 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{1024 c^{9/2}}-\frac{5 b^4 (b+2 c x) \sqrt{b x+c x^2} (b B-2 A c)}{1024 c^4}+\frac{5 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2} (b B-2 A c)}{384 c^3}-\frac{(b+2 c x) \left (b x+c x^2\right )^{5/2} (b B-2 A c)}{24 c^2}+\frac{B \left (b x+c x^2\right )^{7/2}}{7 c} \]

[Out]

(-5*b^4*(b*B - 2*A*c)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(1024*c^4) + (5*b^2*(b*B -
2*A*c)*(b + 2*c*x)*(b*x + c*x^2)^(3/2))/(384*c^3) - ((b*B - 2*A*c)*(b + 2*c*x)*(
b*x + c*x^2)^(5/2))/(24*c^2) + (B*(b*x + c*x^2)^(7/2))/(7*c) + (5*b^6*(b*B - 2*A
*c)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(1024*c^(9/2))

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Rubi [A]  time = 0.172867, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{5 b^6 (b B-2 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{1024 c^{9/2}}-\frac{5 b^4 (b+2 c x) \sqrt{b x+c x^2} (b B-2 A c)}{1024 c^4}+\frac{5 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2} (b B-2 A c)}{384 c^3}-\frac{(b+2 c x) \left (b x+c x^2\right )^{5/2} (b B-2 A c)}{24 c^2}+\frac{B \left (b x+c x^2\right )^{7/2}}{7 c} \]

Antiderivative was successfully verified.

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

[Out]

(-5*b^4*(b*B - 2*A*c)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(1024*c^4) + (5*b^2*(b*B -
2*A*c)*(b + 2*c*x)*(b*x + c*x^2)^(3/2))/(384*c^3) - ((b*B - 2*A*c)*(b + 2*c*x)*(
b*x + c*x^2)^(5/2))/(24*c^2) + (B*(b*x + c*x^2)^(7/2))/(7*c) + (5*b^6*(b*B - 2*A
*c)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(1024*c^(9/2))

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Rubi in Sympy [A]  time = 20.7077, size = 163, normalized size = 0.95 \[ \frac{B \left (b x + c x^{2}\right )^{\frac{7}{2}}}{7 c} - \frac{5 b^{6} \left (2 A c - B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{1024 c^{\frac{9}{2}}} + \frac{5 b^{4} \left (b + 2 c x\right ) \left (2 A c - B b\right ) \sqrt{b x + c x^{2}}}{1024 c^{4}} - \frac{5 b^{2} \left (b + 2 c x\right ) \left (2 A c - B b\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{384 c^{3}} + \frac{\left (b + 2 c x\right ) \left (2 A c - B b\right ) \left (b x + c x^{2}\right )^{\frac{5}{2}}}{24 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*(b*x + c*x**2)**(7/2)/(7*c) - 5*b**6*(2*A*c - B*b)*atanh(sqrt(c)*x/sqrt(b*x +
c*x**2))/(1024*c**(9/2)) + 5*b**4*(b + 2*c*x)*(2*A*c - B*b)*sqrt(b*x + c*x**2)/(
1024*c**4) - 5*b**2*(b + 2*c*x)*(2*A*c - B*b)*(b*x + c*x**2)**(3/2)/(384*c**3) +
(b + 2*c*x)*(2*A*c - B*b)*(b*x + c*x**2)**(5/2)/(24*c**2)

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Mathematica [A]  time = 0.336658, size = 186, normalized size = 1.09 \[ \frac{\sqrt{x (b+c x)} \left (\frac{105 b^6 (b B-2 A c) \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{\sqrt{x} \sqrt{b+c x}}+\sqrt{c} \left (70 b^5 c (3 A+B x)-28 b^4 c^2 x (5 A+2 B x)+16 b^3 c^3 x^2 (7 A+3 B x)+32 b^2 c^4 x^3 (189 A+148 B x)+256 b c^5 x^4 (35 A+29 B x)+512 c^6 x^5 (7 A+6 B x)-105 b^6 B\right )\right )}{21504 c^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x*(b + c*x)]*(Sqrt[c]*(-105*b^6*B + 70*b^5*c*(3*A + B*x) - 28*b^4*c^2*x*(5
*A + 2*B*x) + 16*b^3*c^3*x^2*(7*A + 3*B*x) + 512*c^6*x^5*(7*A + 6*B*x) + 256*b*c
^5*x^4*(35*A + 29*B*x) + 32*b^2*c^4*x^3*(189*A + 148*B*x)) + (105*b^6*(b*B - 2*A
*c)*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[b + c*x]])/(Sqrt[x]*Sqrt[b + c*x])))/(21504*c^(
9/2))

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Maple [B]  time = 0.007, size = 321, normalized size = 1.9 \[{\frac{Ax}{6} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}+{\frac{Ab}{12\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{5\,Ax{b}^{2}}{96\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{5\,A{b}^{3}}{192\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{5\,Ax{b}^{4}}{256\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{5\,A{b}^{5}}{512\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{5\,A{b}^{6}}{1024}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{7}{2}}}}+{\frac{B}{7\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}-{\frac{Bbx}{12\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{{b}^{2}B}{24\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}+{\frac{5\,B{b}^{3}x}{192\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{b}^{4}B}{384\,{c}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{5\,B{b}^{5}x}{512\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{5\,B{b}^{6}}{1024\,{c}^{4}}\sqrt{c{x}^{2}+bx}}+{\frac{5\,B{b}^{7}}{2048}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*A*x*(c*x^2+b*x)^(5/2)+1/12*A/c*(c*x^2+b*x)^(5/2)*b-5/96*A*b^2/c*(c*x^2+b*x)^
(3/2)*x-5/192*A*b^3/c^2*(c*x^2+b*x)^(3/2)+5/256*A*b^4/c^2*(c*x^2+b*x)^(1/2)*x+5/
512*A*b^5/c^3*(c*x^2+b*x)^(1/2)-5/1024*A*b^6/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x
^2+b*x)^(1/2))+1/7*B*(c*x^2+b*x)^(7/2)/c-1/12*B*b/c*(c*x^2+b*x)^(5/2)*x-1/24*B*b
^2/c^2*(c*x^2+b*x)^(5/2)+5/192*B*b^3/c^2*(c*x^2+b*x)^(3/2)*x+5/384*B*b^4/c^3*(c*
x^2+b*x)^(3/2)-5/512*B*b^5/c^3*(c*x^2+b*x)^(1/2)*x-5/1024*B*b^6/c^4*(c*x^2+b*x)^
(1/2)+5/2048*B*b^7/c^(9/2)*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, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.296758, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (3072 \, B c^{6} x^{6} - 105 \, B b^{6} + 210 \, A b^{5} c + 256 \,{\left (29 \, B b c^{5} + 14 \, A c^{6}\right )} x^{5} + 128 \,{\left (37 \, B b^{2} c^{4} + 70 \, A b c^{5}\right )} x^{4} + 48 \,{\left (B b^{3} c^{3} + 126 \, A b^{2} c^{4}\right )} x^{3} - 56 \,{\left (B b^{4} c^{2} - 2 \, A b^{3} c^{3}\right )} x^{2} + 70 \,{\left (B b^{5} c - 2 \, A b^{4} c^{2}\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{c} - 105 \,{\left (B b^{7} - 2 \, A b^{6} c\right )} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} - 2 \, \sqrt{c x^{2} + b x} c\right )}{43008 \, c^{\frac{9}{2}}}, \frac{{\left (3072 \, B c^{6} x^{6} - 105 \, B b^{6} + 210 \, A b^{5} c + 256 \,{\left (29 \, B b c^{5} + 14 \, A c^{6}\right )} x^{5} + 128 \,{\left (37 \, B b^{2} c^{4} + 70 \, A b c^{5}\right )} x^{4} + 48 \,{\left (B b^{3} c^{3} + 126 \, A b^{2} c^{4}\right )} x^{3} - 56 \,{\left (B b^{4} c^{2} - 2 \, A b^{3} c^{3}\right )} x^{2} + 70 \,{\left (B b^{5} c - 2 \, A b^{4} c^{2}\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{-c} + 105 \,{\left (B b^{7} - 2 \, A b^{6} c\right )} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right )}{21504 \, \sqrt{-c} c^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/43008*(2*(3072*B*c^6*x^6 - 105*B*b^6 + 210*A*b^5*c + 256*(29*B*b*c^5 + 14*A*c
^6)*x^5 + 128*(37*B*b^2*c^4 + 70*A*b*c^5)*x^4 + 48*(B*b^3*c^3 + 126*A*b^2*c^4)*x
^3 - 56*(B*b^4*c^2 - 2*A*b^3*c^3)*x^2 + 70*(B*b^5*c - 2*A*b^4*c^2)*x)*sqrt(c*x^2
+ b*x)*sqrt(c) - 105*(B*b^7 - 2*A*b^6*c)*log((2*c*x + b)*sqrt(c) - 2*sqrt(c*x^2
+ b*x)*c))/c^(9/2), 1/21504*((3072*B*c^6*x^6 - 105*B*b^6 + 210*A*b^5*c + 256*(2
9*B*b*c^5 + 14*A*c^6)*x^5 + 128*(37*B*b^2*c^4 + 70*A*b*c^5)*x^4 + 48*(B*b^3*c^3
+ 126*A*b^2*c^4)*x^3 - 56*(B*b^4*c^2 - 2*A*b^3*c^3)*x^2 + 70*(B*b^5*c - 2*A*b^4*
c^2)*x)*sqrt(c*x^2 + b*x)*sqrt(-c) + 105*(B*b^7 - 2*A*b^6*c)*arctan(sqrt(c*x^2 +
b*x)*sqrt(-c)/(c*x)))/(sqrt(-c)*c^4)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.284901, size = 298, normalized size = 1.74 \[ \frac{1}{21504} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (2 \,{\left (12 \, B c^{2} x + \frac{29 \, B b c^{7} + 14 \, A c^{8}}{c^{6}}\right )} x + \frac{37 \, B b^{2} c^{6} + 70 \, A b c^{7}}{c^{6}}\right )} x + \frac{3 \,{\left (B b^{3} c^{5} + 126 \, A b^{2} c^{6}\right )}}{c^{6}}\right )} x - \frac{7 \,{\left (B b^{4} c^{4} - 2 \, A b^{3} c^{5}\right )}}{c^{6}}\right )} x + \frac{35 \,{\left (B b^{5} c^{3} - 2 \, A b^{4} c^{4}\right )}}{c^{6}}\right )} x - \frac{105 \,{\left (B b^{6} c^{2} - 2 \, A b^{5} c^{3}\right )}}{c^{6}}\right )} - \frac{5 \,{\left (B b^{7} - 2 \, A b^{6} c\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{2048 \, c^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/21504*sqrt(c*x^2 + b*x)*(2*(4*(2*(8*(2*(12*B*c^2*x + (29*B*b*c^7 + 14*A*c^8)/c
^6)*x + (37*B*b^2*c^6 + 70*A*b*c^7)/c^6)*x + 3*(B*b^3*c^5 + 126*A*b^2*c^6)/c^6)*
x - 7*(B*b^4*c^4 - 2*A*b^3*c^5)/c^6)*x + 35*(B*b^5*c^3 - 2*A*b^4*c^4)/c^6)*x - 1
05*(B*b^6*c^2 - 2*A*b^5*c^3)/c^6) - 5/2048*(B*b^7 - 2*A*b^6*c)*ln(abs(-2*(sqrt(c
)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/c^(9/2)