3.222 \(\int \frac{(A+B x) \left (b x+c x^2\right )^{5/2}}{x^{13/2}} \, dx\)

Optimal. Leaf size=175 \[ \frac{5 c^2 \sqrt{b x+c x^2} (A c+6 b B)}{8 b \sqrt{x}}-\frac{5 c^2 (A c+6 b B) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{8 \sqrt{b}}-\frac{\left (b x+c x^2\right )^{5/2} (A c+6 b B)}{12 b x^{9/2}}-\frac{5 c \left (b x+c x^2\right )^{3/2} (A c+6 b B)}{24 b x^{5/2}}-\frac{A \left (b x+c x^2\right )^{7/2}}{3 b x^{13/2}} \]

[Out]

(5*c^2*(6*b*B + A*c)*Sqrt[b*x + c*x^2])/(8*b*Sqrt[x]) - (5*c*(6*b*B + A*c)*(b*x
+ c*x^2)^(3/2))/(24*b*x^(5/2)) - ((6*b*B + A*c)*(b*x + c*x^2)^(5/2))/(12*b*x^(9/
2)) - (A*(b*x + c*x^2)^(7/2))/(3*b*x^(13/2)) - (5*c^2*(6*b*B + A*c)*ArcTanh[Sqrt
[b*x + c*x^2]/(Sqrt[b]*Sqrt[x])])/(8*Sqrt[b])

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Rubi [A]  time = 0.366123, antiderivative size = 175, 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 \[ \frac{5 c^2 \sqrt{b x+c x^2} (A c+6 b B)}{8 b \sqrt{x}}-\frac{5 c^2 (A c+6 b B) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{8 \sqrt{b}}-\frac{\left (b x+c x^2\right )^{5/2} (A c+6 b B)}{12 b x^{9/2}}-\frac{5 c \left (b x+c x^2\right )^{3/2} (A c+6 b B)}{24 b x^{5/2}}-\frac{A \left (b x+c x^2\right )^{7/2}}{3 b x^{13/2}} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(b*x + c*x^2)^(5/2))/x^(13/2),x]

[Out]

(5*c^2*(6*b*B + A*c)*Sqrt[b*x + c*x^2])/(8*b*Sqrt[x]) - (5*c*(6*b*B + A*c)*(b*x
+ c*x^2)^(3/2))/(24*b*x^(5/2)) - ((6*b*B + A*c)*(b*x + c*x^2)^(5/2))/(12*b*x^(9/
2)) - (A*(b*x + c*x^2)^(7/2))/(3*b*x^(13/2)) - (5*c^2*(6*b*B + A*c)*ArcTanh[Sqrt
[b*x + c*x^2]/(Sqrt[b]*Sqrt[x])])/(8*Sqrt[b])

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Rubi in Sympy [A]  time = 22.6797, size = 160, normalized size = 0.91 \[ - \frac{A \left (b x + c x^{2}\right )^{\frac{7}{2}}}{3 b x^{\frac{13}{2}}} + \frac{5 c^{2} \left (A c + 6 B b\right ) \sqrt{b x + c x^{2}}}{8 b \sqrt{x}} - \frac{5 c \left (A c + 6 B b\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{24 b x^{\frac{5}{2}}} - \frac{\left (A c + 6 B b\right ) \left (b x + c x^{2}\right )^{\frac{5}{2}}}{12 b x^{\frac{9}{2}}} - \frac{5 c^{2} \left (A c + 6 B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{b x + c x^{2}}}{\sqrt{b} \sqrt{x}} \right )}}{8 \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x)**(5/2)/x**(13/2),x)

[Out]

-A*(b*x + c*x**2)**(7/2)/(3*b*x**(13/2)) + 5*c**2*(A*c + 6*B*b)*sqrt(b*x + c*x**
2)/(8*b*sqrt(x)) - 5*c*(A*c + 6*B*b)*(b*x + c*x**2)**(3/2)/(24*b*x**(5/2)) - (A*
c + 6*B*b)*(b*x + c*x**2)**(5/2)/(12*b*x**(9/2)) - 5*c**2*(A*c + 6*B*b)*atanh(sq
rt(b*x + c*x**2)/(sqrt(b)*sqrt(x)))/(8*sqrt(b))

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Mathematica [A]  time = 0.203398, size = 127, normalized size = 0.73 \[ -\frac{\sqrt{x (b+c x)} \left (\sqrt{b} \sqrt{b+c x} \left (A \left (8 b^2+26 b c x+33 c^2 x^2\right )+6 B x \left (2 b^2+9 b c x-8 c^2 x^2\right )\right )+15 c^2 x^3 (A c+6 b B) \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )\right )}{24 \sqrt{b} x^{7/2} \sqrt{b+c x}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(b*x + c*x^2)^(5/2))/x^(13/2),x]

[Out]

-(Sqrt[x*(b + c*x)]*(Sqrt[b]*Sqrt[b + c*x]*(6*B*x*(2*b^2 + 9*b*c*x - 8*c^2*x^2)
+ A*(8*b^2 + 26*b*c*x + 33*c^2*x^2)) + 15*c^2*(6*b*B + A*c)*x^3*ArcTanh[Sqrt[b +
c*x]/Sqrt[b]]))/(24*Sqrt[b]*x^(7/2)*Sqrt[b + c*x])

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Maple [A]  time = 0.03, size = 166, normalized size = 1. \[ -{\frac{1}{24}\sqrt{x \left ( cx+b \right ) } \left ( 15\,A{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){x}^{3}{c}^{3}+90\,B{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){x}^{3}b{c}^{2}-48\,B{x}^{3}{c}^{2}\sqrt{b}\sqrt{cx+b}+33\,A{x}^{2}{c}^{2}\sqrt{b}\sqrt{cx+b}+54\,B{x}^{2}{b}^{3/2}c\sqrt{cx+b}+26\,Ax{b}^{3/2}c\sqrt{cx+b}+12\,Bx{b}^{5/2}\sqrt{cx+b}+8\,A{b}^{5/2}\sqrt{cx+b} \right ){x}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{cx+b}}}{\frac{1}{\sqrt{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)^(5/2)/x^(13/2),x)

[Out]

-1/24*(x*(c*x+b))^(1/2)*(15*A*arctanh((c*x+b)^(1/2)/b^(1/2))*x^3*c^3+90*B*arctan
h((c*x+b)^(1/2)/b^(1/2))*x^3*b*c^2-48*B*x^3*c^2*b^(1/2)*(c*x+b)^(1/2)+33*A*x^2*c
^2*b^(1/2)*(c*x+b)^(1/2)+54*B*x^2*b^(3/2)*c*(c*x+b)^(1/2)+26*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)/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^(13/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.296621, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (6 \, B b c^{2} + A c^{3}\right )} \sqrt{c x^{2} + b x} x^{\frac{5}{2}} \log \left (\frac{2 \, \sqrt{c x^{2} + b x} b \sqrt{x} -{\left (c x^{2} + 2 \, b x\right )} \sqrt{b}}{x^{2}}\right ) + 2 \,{\left (48 \, B c^{3} x^{4} - 8 \, A b^{3} - 3 \,{\left (2 \, B b c^{2} + 11 \, A c^{3}\right )} x^{3} -{\left (66 \, B b^{2} c + 59 \, A b c^{2}\right )} x^{2} - 2 \,{\left (6 \, B b^{3} + 17 \, A b^{2} c\right )} x\right )} \sqrt{b}}{48 \, \sqrt{c x^{2} + b x} \sqrt{b} x^{\frac{5}{2}}}, -\frac{15 \,{\left (6 \, B b c^{2} + A c^{3}\right )} \sqrt{c x^{2} + b x} x^{\frac{5}{2}} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right ) -{\left (48 \, B c^{3} x^{4} - 8 \, A b^{3} - 3 \,{\left (2 \, B b c^{2} + 11 \, A c^{3}\right )} x^{3} -{\left (66 \, B b^{2} c + 59 \, A b c^{2}\right )} x^{2} - 2 \,{\left (6 \, B b^{3} + 17 \, A b^{2} c\right )} x\right )} \sqrt{-b}}{24 \, \sqrt{c x^{2} + b x} \sqrt{-b} x^{\frac{5}{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)*(B*x + A)/x^(13/2),x, algorithm="fricas")

[Out]

[1/48*(15*(6*B*b*c^2 + A*c^3)*sqrt(c*x^2 + b*x)*x^(5/2)*log((2*sqrt(c*x^2 + b*x)
*b*sqrt(x) - (c*x^2 + 2*b*x)*sqrt(b))/x^2) + 2*(48*B*c^3*x^4 - 8*A*b^3 - 3*(2*B*
b*c^2 + 11*A*c^3)*x^3 - (66*B*b^2*c + 59*A*b*c^2)*x^2 - 2*(6*B*b^3 + 17*A*b^2*c)
*x)*sqrt(b))/(sqrt(c*x^2 + b*x)*sqrt(b)*x^(5/2)), -1/24*(15*(6*B*b*c^2 + A*c^3)*
sqrt(c*x^2 + b*x)*x^(5/2)*arctan(sqrt(-b)*sqrt(x)/sqrt(c*x^2 + b*x)) - (48*B*c^3
*x^4 - 8*A*b^3 - 3*(2*B*b*c^2 + 11*A*c^3)*x^3 - (66*B*b^2*c + 59*A*b*c^2)*x^2 -
2*(6*B*b^3 + 17*A*b^2*c)*x)*sqrt(-b))/(sqrt(c*x^2 + b*x)*sqrt(-b)*x^(5/2))]

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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**(13/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.372179, size = 204, normalized size = 1.17 \[ \frac{48 \, \sqrt{c x + b} B c^{3} + \frac{15 \,{\left (6 \, B b c^{3} + A c^{4}\right )} \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} - \frac{54 \,{\left (c x + b\right )}^{\frac{5}{2}} B b c^{3} - 96 \,{\left (c x + b\right )}^{\frac{3}{2}} B b^{2} c^{3} + 42 \, \sqrt{c x + b} B b^{3} c^{3} + 33 \,{\left (c x + b\right )}^{\frac{5}{2}} A c^{4} - 40 \,{\left (c x + b\right )}^{\frac{3}{2}} A b c^{4} + 15 \, \sqrt{c x + b} A b^{2} c^{4}}{c^{3} x^{3}}}{24 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)*(B*x + A)/x^(13/2),x, algorithm="giac")

[Out]

1/24*(48*sqrt(c*x + b)*B*c^3 + 15*(6*B*b*c^3 + A*c^4)*arctan(sqrt(c*x + b)/sqrt(
-b))/sqrt(-b) - (54*(c*x + b)^(5/2)*B*b*c^3 - 96*(c*x + b)^(3/2)*B*b^2*c^3 + 42*
sqrt(c*x + b)*B*b^3*c^3 + 33*(c*x + b)^(5/2)*A*c^4 - 40*(c*x + b)^(3/2)*A*b*c^4
+ 15*sqrt(c*x + b)*A*b^2*c^4)/(c^3*x^3))/c