3.232 \(\int \frac{A+B x}{x^{7/2} \sqrt{b x+c x^2}} \, dx\)

Optimal. Leaf size=142 \[ -\frac{c^2 (6 b B-5 A c) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{8 b^{7/2}}+\frac{c \sqrt{b x+c x^2} (6 b B-5 A c)}{8 b^3 x^{3/2}}-\frac{\sqrt{b x+c x^2} (6 b B-5 A c)}{12 b^2 x^{5/2}}-\frac{A \sqrt{b x+c x^2}}{3 b x^{7/2}} \]

[Out]

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

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Rubi [A]  time = 0.280801, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{c^2 (6 b B-5 A c) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{8 b^{7/2}}+\frac{c \sqrt{b x+c x^2} (6 b B-5 A c)}{8 b^3 x^{3/2}}-\frac{\sqrt{b x+c x^2} (6 b B-5 A c)}{12 b^2 x^{5/2}}-\frac{A \sqrt{b x+c x^2}}{3 b x^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(x^(7/2)*Sqrt[b*x + c*x^2]),x]

[Out]

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

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Rubi in Sympy [A]  time = 18.6606, size = 131, normalized size = 0.92 \[ - \frac{A \sqrt{b x + c x^{2}}}{3 b x^{\frac{7}{2}}} + \frac{\left (5 A c - 6 B b\right ) \sqrt{b x + c x^{2}}}{12 b^{2} x^{\frac{5}{2}}} - \frac{c \left (5 A c - 6 B b\right ) \sqrt{b x + c x^{2}}}{8 b^{3} x^{\frac{3}{2}}} + \frac{c^{2} \left (5 A c - 6 B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{b x + c x^{2}}}{\sqrt{b} \sqrt{x}} \right )}}{8 b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/x**(7/2)/(c*x**2+b*x)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.14635, size = 115, normalized size = 0.81 \[ \frac{3 c^2 x^3 \sqrt{b+c x} (5 A c-6 b B) \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )-\sqrt{b} (b+c x) \left (A \left (8 b^2-10 b c x+15 c^2 x^2\right )+6 b B x (2 b-3 c x)\right )}{24 b^{7/2} x^{5/2} \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/(x^(7/2)*Sqrt[b*x + c*x^2]),x]

[Out]

(-(Sqrt[b]*(b + c*x)*(6*b*B*x*(2*b - 3*c*x) + A*(8*b^2 - 10*b*c*x + 15*c^2*x^2))
) + 3*c^2*(-6*b*B + 5*A*c)*x^3*Sqrt[b + c*x]*ArcTanh[Sqrt[b + c*x]/Sqrt[b]])/(24
*b^(7/2)*x^(5/2)*Sqrt[x*(b + c*x)])

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Maple [A]  time = 0.027, size = 147, normalized size = 1. \[{\frac{1}{24}\sqrt{x \left ( cx+b \right ) } \left ( 15\,A{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){x}^{3}{c}^{3}-18\,B{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){x}^{3}b{c}^{2}-15\,A{x}^{2}{c}^{2}\sqrt{b}\sqrt{cx+b}+18\,B{x}^{2}{b}^{3/2}c\sqrt{cx+b}+10\,Ax{b}^{3/2}c\sqrt{cx+b}-12\,Bx{b}^{5/2}\sqrt{cx+b}-8\,A{b}^{5/2}\sqrt{cx+b} \right ){b}^{-{\frac{7}{2}}}{x}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{cx+b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^(7/2)/(c*x^2+b*x)^(1/2),x)

[Out]

1/24*(x*(c*x+b))^(1/2)/b^(7/2)*(15*A*arctanh((c*x+b)^(1/2)/b^(1/2))*x^3*c^3-18*B
*arctanh((c*x+b)^(1/2)/b^(1/2))*x^3*b*c^2-15*A*x^2*c^2*b^(1/2)*(c*x+b)^(1/2)+18*
B*x^2*b^(3/2)*c*(c*x+b)^(1/2)+10*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)

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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((B*x + A)/(sqrt(c*x^2 + b*x)*x^(7/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.289036, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (6 \, B b c^{2} - 5 \, A c^{3}\right )} x^{4} \log \left (-\frac{2 \, \sqrt{c x^{2} + b x} b \sqrt{x} +{\left (c x^{2} + 2 \, b x\right )} \sqrt{b}}{x^{2}}\right ) + 2 \,{\left (8 \, A b^{2} - 3 \,{\left (6 \, B b c - 5 \, A c^{2}\right )} x^{2} + 2 \,{\left (6 \, B b^{2} - 5 \, A b c\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{b} \sqrt{x}}{48 \, b^{\frac{7}{2}} x^{4}}, -\frac{3 \,{\left (6 \, B b c^{2} - 5 \, A c^{3}\right )} x^{4} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right ) +{\left (8 \, A b^{2} - 3 \,{\left (6 \, B b c - 5 \, A c^{2}\right )} x^{2} + 2 \,{\left (6 \, B b^{2} - 5 \, A b c\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{-b} \sqrt{x}}{24 \, \sqrt{-b} b^{3} x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(sqrt(c*x^2 + b*x)*x^(7/2)),x, algorithm="fricas")

[Out]

[-1/48*(3*(6*B*b*c^2 - 5*A*c^3)*x^4*log(-(2*sqrt(c*x^2 + b*x)*b*sqrt(x) + (c*x^2
+ 2*b*x)*sqrt(b))/x^2) + 2*(8*A*b^2 - 3*(6*B*b*c - 5*A*c^2)*x^2 + 2*(6*B*b^2 -
5*A*b*c)*x)*sqrt(c*x^2 + b*x)*sqrt(b)*sqrt(x))/(b^(7/2)*x^4), -1/24*(3*(6*B*b*c^
2 - 5*A*c^3)*x^4*arctan(sqrt(-b)*sqrt(x)/sqrt(c*x^2 + b*x)) + (8*A*b^2 - 3*(6*B*
b*c - 5*A*c^2)*x^2 + 2*(6*B*b^2 - 5*A*b*c)*x)*sqrt(c*x^2 + b*x)*sqrt(-b)*sqrt(x)
)/(sqrt(-b)*b^3*x^4)]

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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)/x**(7/2)/(c*x**2+b*x)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.345696, size = 194, normalized size = 1.37 \[ \frac{\frac{3 \,{\left (6 \, B b c^{3} - 5 \, A c^{4}\right )} \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{3}} + \frac{18 \,{\left (c x + b\right )}^{\frac{5}{2}} B b c^{3} - 48 \,{\left (c x + b\right )}^{\frac{3}{2}} B b^{2} c^{3} + 30 \, \sqrt{c x + b} B b^{3} c^{3} - 15 \,{\left (c x + b\right )}^{\frac{5}{2}} A c^{4} + 40 \,{\left (c x + b\right )}^{\frac{3}{2}} A b c^{4} - 33 \, \sqrt{c x + b} A b^{2} c^{4}}{b^{3} c^{3} x^{3}}}{24 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(sqrt(c*x^2 + b*x)*x^(7/2)),x, algorithm="giac")

[Out]

1/24*(3*(6*B*b*c^3 - 5*A*c^4)*arctan(sqrt(c*x + b)/sqrt(-b))/(sqrt(-b)*b^3) + (1
8*(c*x + b)^(5/2)*B*b*c^3 - 48*(c*x + b)^(3/2)*B*b^2*c^3 + 30*sqrt(c*x + b)*B*b^
3*c^3 - 15*(c*x + b)^(5/2)*A*c^4 + 40*(c*x + b)^(3/2)*A*b*c^4 - 33*sqrt(c*x + b)
*A*b^2*c^4)/(b^3*c^3*x^3))/c