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

Optimal. Leaf size=174 \[ \frac{5 c (4 b B-7 A c) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{4 b^{9/2}}-\frac{5 c \sqrt{x} (4 b B-7 A c)}{4 b^4 \sqrt{b x+c x^2}}-\frac{5 (4 b B-7 A c)}{12 b^3 \sqrt{x} \sqrt{b x+c x^2}}+\frac{\sqrt{x} (4 b B-7 A c)}{6 b^2 \left (b x+c x^2\right )^{3/2}}-\frac{A}{2 b \sqrt{x} \left (b x+c x^2\right )^{3/2}} \]

[Out]

-A/(2*b*Sqrt[x]*(b*x + c*x^2)^(3/2)) + ((4*b*B - 7*A*c)*Sqrt[x])/(6*b^2*(b*x + c
*x^2)^(3/2)) - (5*(4*b*B - 7*A*c))/(12*b^3*Sqrt[x]*Sqrt[b*x + c*x^2]) - (5*c*(4*
b*B - 7*A*c)*Sqrt[x])/(4*b^4*Sqrt[b*x + c*x^2]) + (5*c*(4*b*B - 7*A*c)*ArcTanh[S
qrt[b*x + c*x^2]/(Sqrt[b]*Sqrt[x])])/(4*b^(9/2))

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Rubi [A]  time = 0.340751, antiderivative size = 174, 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 \[ \frac{5 c (4 b B-7 A c) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{4 b^{9/2}}-\frac{5 c \sqrt{x} (4 b B-7 A c)}{4 b^4 \sqrt{b x+c x^2}}-\frac{5 (4 b B-7 A c)}{12 b^3 \sqrt{x} \sqrt{b x+c x^2}}+\frac{\sqrt{x} (4 b B-7 A c)}{6 b^2 \left (b x+c x^2\right )^{3/2}}-\frac{A}{2 b \sqrt{x} \left (b x+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-A/(2*b*Sqrt[x]*(b*x + c*x^2)^(3/2)) + ((4*b*B - 7*A*c)*Sqrt[x])/(6*b^2*(b*x + c
*x^2)^(3/2)) - (5*(4*b*B - 7*A*c))/(12*b^3*Sqrt[x]*Sqrt[b*x + c*x^2]) - (5*c*(4*
b*B - 7*A*c)*Sqrt[x])/(4*b^4*Sqrt[b*x + c*x^2]) + (5*c*(4*b*B - 7*A*c)*ArcTanh[S
qrt[b*x + c*x^2]/(Sqrt[b]*Sqrt[x])])/(4*b^(9/2))

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Rubi in Sympy [A]  time = 23.558, size = 165, normalized size = 0.95 \[ - \frac{A}{2 b \sqrt{x} \left (b x + c x^{2}\right )^{\frac{3}{2}}} - \frac{\sqrt{x} \left (7 A c - 4 B b\right )}{6 b^{2} \left (b x + c x^{2}\right )^{\frac{3}{2}}} + \frac{5 \left (7 A c - 4 B b\right )}{12 b^{3} \sqrt{x} \sqrt{b x + c x^{2}}} + \frac{5 c \sqrt{x} \left (7 A c - 4 B b\right )}{4 b^{4} \sqrt{b x + c x^{2}}} - \frac{5 c \left (7 A c - 4 B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{b x + c x^{2}}}{\sqrt{b} \sqrt{x}} \right )}}{4 b^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/(c*x**2+b*x)**(5/2)/x**(1/2),x)

[Out]

-A/(2*b*sqrt(x)*(b*x + c*x**2)**(3/2)) - sqrt(x)*(7*A*c - 4*B*b)/(6*b**2*(b*x +
c*x**2)**(3/2)) + 5*(7*A*c - 4*B*b)/(12*b**3*sqrt(x)*sqrt(b*x + c*x**2)) + 5*c*s
qrt(x)*(7*A*c - 4*B*b)/(4*b**4*sqrt(b*x + c*x**2)) - 5*c*(7*A*c - 4*B*b)*atanh(s
qrt(b*x + c*x**2)/(sqrt(b)*sqrt(x)))/(4*b**(9/2))

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Mathematica [A]  time = 0.214014, size = 129, normalized size = 0.74 \[ \frac{\sqrt{b} \left (A \left (-6 b^3+21 b^2 c x+140 b c^2 x^2+105 c^3 x^3\right )-4 b B x \left (3 b^2+20 b c x+15 c^2 x^2\right )\right )+15 c x^2 (b+c x)^{3/2} (4 b B-7 A c) \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )}{12 b^{9/2} \sqrt{x} (x (b+c x))^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[b]*(-4*b*B*x*(3*b^2 + 20*b*c*x + 15*c^2*x^2) + A*(-6*b^3 + 21*b^2*c*x + 14
0*b*c^2*x^2 + 105*c^3*x^3)) + 15*c*(4*b*B - 7*A*c)*x^2*(b + c*x)^(3/2)*ArcTanh[S
qrt[b + c*x]/Sqrt[b]])/(12*b^(9/2)*Sqrt[x]*(x*(b + c*x))^(3/2))

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Maple [A]  time = 0.035, size = 208, normalized size = 1.2 \[ -{\frac{1}{12\, \left ( cx+b \right ) ^{2}}\sqrt{x \left ( cx+b \right ) } \left ( 105\,A{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ) \sqrt{cx+b}{x}^{3}{c}^{3}-60\,B{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ) \sqrt{cx+b}{x}^{3}b{c}^{2}+105\,A{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){x}^{2}b{c}^{2}\sqrt{cx+b}-105\,A\sqrt{b}{x}^{3}{c}^{3}-60\,B{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){x}^{2}{b}^{2}c\sqrt{cx+b}+60\,B{b}^{3/2}{x}^{3}{c}^{2}-140\,A{b}^{3/2}{x}^{2}{c}^{2}+80\,B{b}^{5/2}{x}^{2}c-21\,A{b}^{5/2}xc+12\,B{b}^{7/2}x+6\,A{b}^{7/2} \right ){x}^{-{\frac{5}{2}}}{b}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(c*x^2+b*x)^(5/2)/x^(1/2),x)

[Out]

-1/12*(x*(c*x+b))^(1/2)*(105*A*arctanh((c*x+b)^(1/2)/b^(1/2))*(c*x+b)^(1/2)*x^3*
c^3-60*B*arctanh((c*x+b)^(1/2)/b^(1/2))*(c*x+b)^(1/2)*x^3*b*c^2+105*A*arctanh((c
*x+b)^(1/2)/b^(1/2))*x^2*b*c^2*(c*x+b)^(1/2)-105*A*b^(1/2)*x^3*c^3-60*B*arctanh(
(c*x+b)^(1/2)/b^(1/2))*x^2*b^2*c*(c*x+b)^(1/2)+60*B*b^(3/2)*x^3*c^2-140*A*b^(3/2
)*x^2*c^2+80*B*b^(5/2)*x^2*c-21*A*b^(5/2)*x*c+12*B*b^(7/2)*x+6*A*b^(7/2))/x^(5/2
)/(c*x+b)^2/b^(9/2)

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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)/((c*x^2 + b*x)^(5/2)*sqrt(x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.317788, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (6 \, A b^{3} + 15 \,{\left (4 \, B b c^{2} - 7 \, A c^{3}\right )} x^{3} + 20 \,{\left (4 \, B b^{2} c - 7 \, A b c^{2}\right )} x^{2} + 3 \,{\left (4 \, B b^{3} - 7 \, A b^{2} c\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{b} \sqrt{x} + 15 \,{\left ({\left (4 \, B b c^{3} - 7 \, A c^{4}\right )} x^{5} + 2 \,{\left (4 \, B b^{2} c^{2} - 7 \, A b c^{3}\right )} x^{4} +{\left (4 \, B b^{3} c - 7 \, A b^{2} c^{2}\right )} x^{3}\right )} \log \left (\frac{2 \, \sqrt{c x^{2} + b x} b \sqrt{x} -{\left (c x^{2} + 2 \, b x\right )} \sqrt{b}}{x^{2}}\right )}{24 \,{\left (b^{4} c^{2} x^{5} + 2 \, b^{5} c x^{4} + b^{6} x^{3}\right )} \sqrt{b}}, -\frac{{\left (6 \, A b^{3} + 15 \,{\left (4 \, B b c^{2} - 7 \, A c^{3}\right )} x^{3} + 20 \,{\left (4 \, B b^{2} c - 7 \, A b c^{2}\right )} x^{2} + 3 \,{\left (4 \, B b^{3} - 7 \, A b^{2} c\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{-b} \sqrt{x} - 15 \,{\left ({\left (4 \, B b c^{3} - 7 \, A c^{4}\right )} x^{5} + 2 \,{\left (4 \, B b^{2} c^{2} - 7 \, A b c^{3}\right )} x^{4} +{\left (4 \, B b^{3} c - 7 \, A b^{2} c^{2}\right )} x^{3}\right )} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right )}{12 \,{\left (b^{4} c^{2} x^{5} + 2 \, b^{5} c x^{4} + b^{6} x^{3}\right )} \sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/24*(2*(6*A*b^3 + 15*(4*B*b*c^2 - 7*A*c^3)*x^3 + 20*(4*B*b^2*c - 7*A*b*c^2)*x
^2 + 3*(4*B*b^3 - 7*A*b^2*c)*x)*sqrt(c*x^2 + b*x)*sqrt(b)*sqrt(x) + 15*((4*B*b*c
^3 - 7*A*c^4)*x^5 + 2*(4*B*b^2*c^2 - 7*A*b*c^3)*x^4 + (4*B*b^3*c - 7*A*b^2*c^2)*
x^3)*log((2*sqrt(c*x^2 + b*x)*b*sqrt(x) - (c*x^2 + 2*b*x)*sqrt(b))/x^2))/((b^4*c
^2*x^5 + 2*b^5*c*x^4 + b^6*x^3)*sqrt(b)), -1/12*((6*A*b^3 + 15*(4*B*b*c^2 - 7*A*
c^3)*x^3 + 20*(4*B*b^2*c - 7*A*b*c^2)*x^2 + 3*(4*B*b^3 - 7*A*b^2*c)*x)*sqrt(c*x^
2 + b*x)*sqrt(-b)*sqrt(x) - 15*((4*B*b*c^3 - 7*A*c^4)*x^5 + 2*(4*B*b^2*c^2 - 7*A
*b*c^3)*x^4 + (4*B*b^3*c - 7*A*b^2*c^2)*x^3)*arctan(sqrt(-b)*sqrt(x)/sqrt(c*x^2
+ b*x)))/((b^4*c^2*x^5 + 2*b^5*c*x^4 + b^6*x^3)*sqrt(-b))]

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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**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.426408, size = 201, normalized size = 1.16 \[ -\frac{5 \,{\left (4 \, B b c - 7 \, A c^{2}\right )} \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{4 \, \sqrt{-b} b^{4}} - \frac{2 \,{\left (6 \,{\left (c x + b\right )} B b c + B b^{2} c - 9 \,{\left (c x + b\right )} A c^{2} - A b c^{2}\right )}}{3 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{4}} - \frac{4 \,{\left (c x + b\right )}^{\frac{3}{2}} B b c - 4 \, \sqrt{c x + b} B b^{2} c - 11 \,{\left (c x + b\right )}^{\frac{3}{2}} A c^{2} + 13 \, \sqrt{c x + b} A b c^{2}}{4 \, b^{4} c^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/4*(4*B*b*c - 7*A*c^2)*arctan(sqrt(c*x + b)/sqrt(-b))/(sqrt(-b)*b^4) - 2/3*(6*
(c*x + b)*B*b*c + B*b^2*c - 9*(c*x + b)*A*c^2 - A*b*c^2)/((c*x + b)^(3/2)*b^4) -
1/4*(4*(c*x + b)^(3/2)*B*b*c - 4*sqrt(c*x + b)*B*b^2*c - 11*(c*x + b)^(3/2)*A*c
^2 + 13*sqrt(c*x + b)*A*b*c^2)/(b^4*c^2*x^2)