3.212 \(\int \frac{(A+B x) \left (b x+c x^2\right )^{3/2}}{x^{13/2}} \, dx\)

Optimal. Leaf size=179 \[ \frac{c^3 (8 b B-3 A c) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{64 b^{5/2}}-\frac{c^2 \sqrt{b x+c x^2} (8 b B-3 A c)}{64 b^2 x^{3/2}}-\frac{c \sqrt{b x+c x^2} (8 b B-3 A c)}{32 b x^{5/2}}-\frac{\left (b x+c x^2\right )^{3/2} (8 b B-3 A c)}{24 b x^{9/2}}-\frac{A \left (b x+c x^2\right )^{5/2}}{4 b x^{13/2}} \]

[Out]

-(c*(8*b*B - 3*A*c)*Sqrt[b*x + c*x^2])/(32*b*x^(5/2)) - (c^2*(8*b*B - 3*A*c)*Sqr
t[b*x + c*x^2])/(64*b^2*x^(3/2)) - ((8*b*B - 3*A*c)*(b*x + c*x^2)^(3/2))/(24*b*x
^(9/2)) - (A*(b*x + c*x^2)^(5/2))/(4*b*x^(13/2)) + (c^3*(8*b*B - 3*A*c)*ArcTanh[
Sqrt[b*x + c*x^2]/(Sqrt[b]*Sqrt[x])])/(64*b^(5/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.375497, antiderivative size = 179, 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 \[ \frac{c^3 (8 b B-3 A c) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{64 b^{5/2}}-\frac{c^2 \sqrt{b x+c x^2} (8 b B-3 A c)}{64 b^2 x^{3/2}}-\frac{c \sqrt{b x+c x^2} (8 b B-3 A c)}{32 b x^{5/2}}-\frac{\left (b x+c x^2\right )^{3/2} (8 b B-3 A c)}{24 b x^{9/2}}-\frac{A \left (b x+c x^2\right )^{5/2}}{4 b x^{13/2}} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(b*x + c*x^2)^(3/2))/x^(13/2),x]

[Out]

-(c*(8*b*B - 3*A*c)*Sqrt[b*x + c*x^2])/(32*b*x^(5/2)) - (c^2*(8*b*B - 3*A*c)*Sqr
t[b*x + c*x^2])/(64*b^2*x^(3/2)) - ((8*b*B - 3*A*c)*(b*x + c*x^2)^(3/2))/(24*b*x
^(9/2)) - (A*(b*x + c*x^2)^(5/2))/(4*b*x^(13/2)) + (c^3*(8*b*B - 3*A*c)*ArcTanh[
Sqrt[b*x + c*x^2]/(Sqrt[b]*Sqrt[x])])/(64*b^(5/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 23.5606, size = 163, normalized size = 0.91 \[ - \frac{A \left (b x + c x^{2}\right )^{\frac{5}{2}}}{4 b x^{\frac{13}{2}}} + \frac{c \left (3 A c - 8 B b\right ) \sqrt{b x + c x^{2}}}{32 b x^{\frac{5}{2}}} + \frac{\left (3 A c - 8 B b\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{24 b x^{\frac{9}{2}}} + \frac{c^{2} \left (3 A c - 8 B b\right ) \sqrt{b x + c x^{2}}}{64 b^{2} x^{\frac{3}{2}}} - \frac{c^{3} \left (3 A c - 8 B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{b x + c x^{2}}}{\sqrt{b} \sqrt{x}} \right )}}{64 b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x)**(3/2)/x**(13/2),x)

[Out]

-A*(b*x + c*x**2)**(5/2)/(4*b*x**(13/2)) + c*(3*A*c - 8*B*b)*sqrt(b*x + c*x**2)/
(32*b*x**(5/2)) + (3*A*c - 8*B*b)*(b*x + c*x**2)**(3/2)/(24*b*x**(9/2)) + c**2*(
3*A*c - 8*B*b)*sqrt(b*x + c*x**2)/(64*b**2*x**(3/2)) - c**3*(3*A*c - 8*B*b)*atan
h(sqrt(b*x + c*x**2)/(sqrt(b)*sqrt(x)))/(64*b**(5/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.323949, size = 140, normalized size = 0.78 \[ -\frac{\sqrt{x (b+c x)} \left (\sqrt{b} \sqrt{b+c x} \left (A \left (48 b^3+72 b^2 c x+6 b c^2 x^2-9 c^3 x^3\right )+8 b B x \left (8 b^2+14 b c x+3 c^2 x^2\right )\right )+3 c^3 x^4 (3 A c-8 b B) \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )\right )}{192 b^{5/2} x^{9/2} \sqrt{b+c x}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(b*x + c*x^2)^(3/2))/x^(13/2),x]

[Out]

-(Sqrt[x*(b + c*x)]*(Sqrt[b]*Sqrt[b + c*x]*(8*b*B*x*(8*b^2 + 14*b*c*x + 3*c^2*x^
2) + A*(48*b^3 + 72*b^2*c*x + 6*b*c^2*x^2 - 9*c^3*x^3)) + 3*c^3*(-8*b*B + 3*A*c)
*x^4*ArcTanh[Sqrt[b + c*x]/Sqrt[b]]))/(192*b^(5/2)*x^(9/2)*Sqrt[b + c*x])

_______________________________________________________________________________________

Maple [A]  time = 0.029, size = 185, normalized size = 1. \[ -{\frac{1}{192}\sqrt{x \left ( cx+b \right ) } \left ( 9\,A{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){x}^{4}{c}^{4}-24\,B{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){x}^{4}b{c}^{3}-9\,A{x}^{3}{c}^{3}\sqrt{cx+b}\sqrt{b}+24\,B{x}^{3}{b}^{3/2}{c}^{2}\sqrt{cx+b}+6\,A{x}^{2}{b}^{3/2}{c}^{2}\sqrt{cx+b}+112\,B{x}^{2}{b}^{5/2}c\sqrt{cx+b}+72\,Ax{b}^{5/2}c\sqrt{cx+b}+64\,Bx{b}^{7/2}\sqrt{cx+b}+48\,A{b}^{7/2}\sqrt{cx+b} \right ){b}^{-{\frac{5}{2}}}{x}^{-{\frac{9}{2}}}{\frac{1}{\sqrt{cx+b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)^(3/2)/x^(13/2),x)

[Out]

-1/192*(x*(c*x+b))^(1/2)/b^(5/2)*(9*A*arctanh((c*x+b)^(1/2)/b^(1/2))*x^4*c^4-24*
B*arctanh((c*x+b)^(1/2)/b^(1/2))*x^4*b*c^3-9*A*x^3*c^3*(c*x+b)^(1/2)*b^(1/2)+24*
B*x^3*b^(3/2)*c^2*(c*x+b)^(1/2)+6*A*x^2*b^(3/2)*c^2*(c*x+b)^(1/2)+112*B*x^2*b^(5
/2)*c*(c*x+b)^(1/2)+72*A*x*b^(5/2)*c*(c*x+b)^(1/2)+64*B*x*b^(7/2)*(c*x+b)^(1/2)+
48*A*b^(7/2)*(c*x+b)^(1/2))/x^(9/2)/(c*x+b)^(1/2)

_______________________________________________________________________________________

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^(13/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.311936, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (8 \, B b c^{3} - 3 \, A c^{4}\right )} x^{5} \log \left (\frac{2 \, \sqrt{c x^{2} + b x} b \sqrt{x} -{\left (c x^{2} + 2 \, b x\right )} \sqrt{b}}{x^{2}}\right ) + 2 \,{\left (48 \, A b^{3} + 3 \,{\left (8 \, B b c^{2} - 3 \, A c^{3}\right )} x^{3} + 2 \,{\left (56 \, B b^{2} c + 3 \, A b c^{2}\right )} x^{2} + 8 \,{\left (8 \, B b^{3} + 9 \, A b^{2} c\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{b} \sqrt{x}}{384 \, b^{\frac{5}{2}} x^{5}}, \frac{3 \,{\left (8 \, B b c^{3} - 3 \, A c^{4}\right )} x^{5} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right ) -{\left (48 \, A b^{3} + 3 \,{\left (8 \, B b c^{2} - 3 \, A c^{3}\right )} x^{3} + 2 \,{\left (56 \, B b^{2} c + 3 \, A b c^{2}\right )} x^{2} + 8 \,{\left (8 \, B b^{3} + 9 \, A b^{2} c\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{-b} \sqrt{x}}{192 \, \sqrt{-b} b^{2} x^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)*(B*x + A)/x^(13/2),x, algorithm="fricas")

[Out]

[-1/384*(3*(8*B*b*c^3 - 3*A*c^4)*x^5*log((2*sqrt(c*x^2 + b*x)*b*sqrt(x) - (c*x^2
+ 2*b*x)*sqrt(b))/x^2) + 2*(48*A*b^3 + 3*(8*B*b*c^2 - 3*A*c^3)*x^3 + 2*(56*B*b^
2*c + 3*A*b*c^2)*x^2 + 8*(8*B*b^3 + 9*A*b^2*c)*x)*sqrt(c*x^2 + b*x)*sqrt(b)*sqrt
(x))/(b^(5/2)*x^5), 1/192*(3*(8*B*b*c^3 - 3*A*c^4)*x^5*arctan(sqrt(-b)*sqrt(x)/s
qrt(c*x^2 + b*x)) - (48*A*b^3 + 3*(8*B*b*c^2 - 3*A*c^3)*x^3 + 2*(56*B*b^2*c + 3*
A*b*c^2)*x^2 + 8*(8*B*b^3 + 9*A*b^2*c)*x)*sqrt(c*x^2 + b*x)*sqrt(-b)*sqrt(x))/(s
qrt(-b)*b^2*x^5)]

_______________________________________________________________________________________

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**(13/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.385626, size = 238, normalized size = 1.33 \[ -\frac{\frac{3 \,{\left (8 \, B b c^{4} - 3 \, A c^{5}\right )} \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{2}} + \frac{24 \,{\left (c x + b\right )}^{\frac{7}{2}} B b c^{4} + 40 \,{\left (c x + b\right )}^{\frac{5}{2}} B b^{2} c^{4} - 88 \,{\left (c x + b\right )}^{\frac{3}{2}} B b^{3} c^{4} + 24 \, \sqrt{c x + b} B b^{4} c^{4} - 9 \,{\left (c x + b\right )}^{\frac{7}{2}} A c^{5} + 33 \,{\left (c x + b\right )}^{\frac{5}{2}} A b c^{5} + 33 \,{\left (c x + b\right )}^{\frac{3}{2}} A b^{2} c^{5} - 9 \, \sqrt{c x + b} A b^{3} c^{5}}{b^{2} c^{4} x^{4}}}{192 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)*(B*x + A)/x^(13/2),x, algorithm="giac")

[Out]

-1/192*(3*(8*B*b*c^4 - 3*A*c^5)*arctan(sqrt(c*x + b)/sqrt(-b))/(sqrt(-b)*b^2) +
(24*(c*x + b)^(7/2)*B*b*c^4 + 40*(c*x + b)^(5/2)*B*b^2*c^4 - 88*(c*x + b)^(3/2)*
B*b^3*c^4 + 24*sqrt(c*x + b)*B*b^4*c^4 - 9*(c*x + b)^(7/2)*A*c^5 + 33*(c*x + b)^
(5/2)*A*b*c^5 + 33*(c*x + b)^(3/2)*A*b^2*c^5 - 9*sqrt(c*x + b)*A*b^3*c^5)/(b^2*c
^4*x^4))/c