3.249 \(\int \frac{A+B x}{x^{3/2} \left (b x+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=214 \[ -\frac{35 c^2 (2 b B-3 A c) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{8 b^{11/2}}+\frac{35 c^2 \sqrt{x} (2 b B-3 A c)}{8 b^5 \sqrt{b x+c x^2}}+\frac{35 c (2 b B-3 A c)}{24 b^4 \sqrt{x} \sqrt{b x+c x^2}}-\frac{7 c \sqrt{x} (2 b B-3 A c)}{12 b^3 \left (b x+c x^2\right )^{3/2}}-\frac{2 b B-3 A c}{4 b^2 \sqrt{x} \left (b x+c x^2\right )^{3/2}}-\frac{A}{3 b x^{3/2} \left (b x+c x^2\right )^{3/2}} \]

[Out]

-A/(3*b*x^(3/2)*(b*x + c*x^2)^(3/2)) - (2*b*B - 3*A*c)/(4*b^2*Sqrt[x]*(b*x + c*x
^2)^(3/2)) - (7*c*(2*b*B - 3*A*c)*Sqrt[x])/(12*b^3*(b*x + c*x^2)^(3/2)) + (35*c*
(2*b*B - 3*A*c))/(24*b^4*Sqrt[x]*Sqrt[b*x + c*x^2]) + (35*c^2*(2*b*B - 3*A*c)*Sq
rt[x])/(8*b^5*Sqrt[b*x + c*x^2]) - (35*c^2*(2*b*B - 3*A*c)*ArcTanh[Sqrt[b*x + c*
x^2]/(Sqrt[b]*Sqrt[x])])/(8*b^(11/2))

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Rubi [A]  time = 0.400425, antiderivative size = 214, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{35 c^2 (2 b B-3 A c) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{8 b^{11/2}}+\frac{35 c^2 \sqrt{x} (2 b B-3 A c)}{8 b^5 \sqrt{b x+c x^2}}+\frac{35 c (2 b B-3 A c)}{24 b^4 \sqrt{x} \sqrt{b x+c x^2}}-\frac{7 c \sqrt{x} (2 b B-3 A c)}{12 b^3 \left (b x+c x^2\right )^{3/2}}-\frac{2 b B-3 A c}{4 b^2 \sqrt{x} \left (b x+c x^2\right )^{3/2}}-\frac{A}{3 b x^{3/2} \left (b x+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(x^(3/2)*(b*x + c*x^2)^(5/2)),x]

[Out]

-A/(3*b*x^(3/2)*(b*x + c*x^2)^(3/2)) - (2*b*B - 3*A*c)/(4*b^2*Sqrt[x]*(b*x + c*x
^2)^(3/2)) - (7*c*(2*b*B - 3*A*c)*Sqrt[x])/(12*b^3*(b*x + c*x^2)^(3/2)) + (35*c*
(2*b*B - 3*A*c))/(24*b^4*Sqrt[x]*Sqrt[b*x + c*x^2]) + (35*c^2*(2*b*B - 3*A*c)*Sq
rt[x])/(8*b^5*Sqrt[b*x + c*x^2]) - (35*c^2*(2*b*B - 3*A*c)*ArcTanh[Sqrt[b*x + c*
x^2]/(Sqrt[b]*Sqrt[x])])/(8*b^(11/2))

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Rubi in Sympy [A]  time = 30.6437, size = 207, normalized size = 0.97 \[ - \frac{A}{3 b x^{\frac{3}{2}} \left (b x + c x^{2}\right )^{\frac{3}{2}}} + \frac{\frac{3 A c}{2} - B b}{2 b^{2} \sqrt{x} \left (b x + c x^{2}\right )^{\frac{3}{2}}} + \frac{7 c \sqrt{x} \left (3 A c - 2 B b\right )}{12 b^{3} \left (b x + c x^{2}\right )^{\frac{3}{2}}} - \frac{35 c \left (3 A c - 2 B b\right )}{24 b^{4} \sqrt{x} \sqrt{b x + c x^{2}}} - \frac{35 c^{2} \sqrt{x} \left (\frac{3 A c}{2} - B b\right )}{4 b^{5} \sqrt{b x + c x^{2}}} + \frac{35 c^{2} \left (\frac{3 A c}{2} - B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{b x + c x^{2}}}{\sqrt{b} \sqrt{x}} \right )}}{4 b^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/x**(3/2)/(c*x**2+b*x)**(5/2),x)

[Out]

-A/(3*b*x**(3/2)*(b*x + c*x**2)**(3/2)) + (3*A*c/2 - B*b)/(2*b**2*sqrt(x)*(b*x +
c*x**2)**(3/2)) + 7*c*sqrt(x)*(3*A*c - 2*B*b)/(12*b**3*(b*x + c*x**2)**(3/2)) -
35*c*(3*A*c - 2*B*b)/(24*b**4*sqrt(x)*sqrt(b*x + c*x**2)) - 35*c**2*sqrt(x)*(3*
A*c/2 - B*b)/(4*b**5*sqrt(b*x + c*x**2)) + 35*c**2*(3*A*c/2 - B*b)*atanh(sqrt(b*
x + c*x**2)/(sqrt(b)*sqrt(x)))/(4*b**(11/2))

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Mathematica [A]  time = 0.236134, size = 154, normalized size = 0.72 \[ \frac{\sqrt{b} \left (2 b B x \left (-6 b^3+21 b^2 c x+140 b c^2 x^2+105 c^3 x^3\right )-A \left (8 b^4-18 b^3 c x+63 b^2 c^2 x^2+420 b c^3 x^3+315 c^4 x^4\right )\right )+105 c^2 x^3 (b+c x)^{3/2} (3 A c-2 b B) \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )}{24 b^{11/2} x^{3/2} (x (b+c x))^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/(x^(3/2)*(b*x + c*x^2)^(5/2)),x]

[Out]

(Sqrt[b]*(2*b*B*x*(-6*b^3 + 21*b^2*c*x + 140*b*c^2*x^2 + 105*c^3*x^3) - A*(8*b^4
- 18*b^3*c*x + 63*b^2*c^2*x^2 + 420*b*c^3*x^3 + 315*c^4*x^4)) + 105*c^2*(-2*b*B
+ 3*A*c)*x^3*(b + c*x)^(3/2)*ArcTanh[Sqrt[b + c*x]/Sqrt[b]])/(24*b^(11/2)*x^(3/
2)*(x*(b + c*x))^(3/2))

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Maple [A]  time = 0.039, size = 234, normalized size = 1.1 \[{\frac{1}{24\, \left ( cx+b \right ) ^{2}}\sqrt{x \left ( cx+b \right ) } \left ( 315\,A{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ) \sqrt{cx+b}{x}^{4}{c}^{4}-210\,B{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ) \sqrt{cx+b}{x}^{4}b{c}^{3}+315\,A{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){x}^{3}b{c}^{3}\sqrt{cx+b}-315\,A\sqrt{b}{x}^{4}{c}^{4}-210\,B{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){x}^{3}{b}^{2}{c}^{2}\sqrt{cx+b}+210\,B{b}^{3/2}{x}^{4}{c}^{3}-420\,A{b}^{3/2}{x}^{3}{c}^{3}+280\,B{b}^{5/2}{x}^{3}{c}^{2}-63\,A{b}^{5/2}{x}^{2}{c}^{2}+42\,B{b}^{7/2}{x}^{2}c+18\,A{b}^{7/2}xc-12\,B{b}^{9/2}x-8\,A{b}^{9/2} \right ){x}^{-{\frac{7}{2}}}{b}^{-{\frac{11}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^(3/2)/(c*x^2+b*x)^(5/2),x)

[Out]

1/24*(x*(c*x+b))^(1/2)*(315*A*arctanh((c*x+b)^(1/2)/b^(1/2))*(c*x+b)^(1/2)*x^4*c
^4-210*B*arctanh((c*x+b)^(1/2)/b^(1/2))*(c*x+b)^(1/2)*x^4*b*c^3+315*A*arctanh((c
*x+b)^(1/2)/b^(1/2))*x^3*b*c^3*(c*x+b)^(1/2)-315*A*b^(1/2)*x^4*c^4-210*B*arctanh
((c*x+b)^(1/2)/b^(1/2))*x^3*b^2*c^2*(c*x+b)^(1/2)+210*B*b^(3/2)*x^4*c^3-420*A*b^
(3/2)*x^3*c^3+280*B*b^(5/2)*x^3*c^2-63*A*b^(5/2)*x^2*c^2+42*B*b^(7/2)*x^2*c+18*A
*b^(7/2)*x*c-12*B*b^(9/2)*x-8*A*b^(9/2))/x^(7/2)/(c*x+b)^2/b^(11/2)

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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)*x^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.317594, size = 1, normalized size = 0. \[ \left [-\frac{2 \,{\left (8 \, A b^{4} - 105 \,{\left (2 \, B b c^{3} - 3 \, A c^{4}\right )} x^{4} - 140 \,{\left (2 \, B b^{2} c^{2} - 3 \, A b c^{3}\right )} x^{3} - 21 \,{\left (2 \, B b^{3} c - 3 \, A b^{2} c^{2}\right )} x^{2} + 6 \,{\left (2 \, B b^{4} - 3 \, A b^{3} c\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{b} \sqrt{x} + 105 \,{\left ({\left (2 \, B b c^{4} - 3 \, A c^{5}\right )} x^{6} + 2 \,{\left (2 \, B b^{2} c^{3} - 3 \, A b c^{4}\right )} x^{5} +{\left (2 \, B b^{3} c^{2} - 3 \, A b^{2} c^{3}\right )} x^{4}\right )} \log \left (-\frac{2 \, \sqrt{c x^{2} + b x} b \sqrt{x} +{\left (c x^{2} + 2 \, b x\right )} \sqrt{b}}{x^{2}}\right )}{48 \,{\left (b^{5} c^{2} x^{6} + 2 \, b^{6} c x^{5} + b^{7} x^{4}\right )} \sqrt{b}}, -\frac{{\left (8 \, A b^{4} - 105 \,{\left (2 \, B b c^{3} - 3 \, A c^{4}\right )} x^{4} - 140 \,{\left (2 \, B b^{2} c^{2} - 3 \, A b c^{3}\right )} x^{3} - 21 \,{\left (2 \, B b^{3} c - 3 \, A b^{2} c^{2}\right )} x^{2} + 6 \,{\left (2 \, B b^{4} - 3 \, A b^{3} c\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{-b} \sqrt{x} + 105 \,{\left ({\left (2 \, B b c^{4} - 3 \, A c^{5}\right )} x^{6} + 2 \,{\left (2 \, B b^{2} c^{3} - 3 \, A b c^{4}\right )} x^{5} +{\left (2 \, B b^{3} c^{2} - 3 \, A b^{2} c^{3}\right )} x^{4}\right )} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right )}{24 \,{\left (b^{5} c^{2} x^{6} + 2 \, b^{6} c x^{5} + b^{7} x^{4}\right )} \sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((c*x^2 + b*x)^(5/2)*x^(3/2)),x, algorithm="fricas")

[Out]

[-1/48*(2*(8*A*b^4 - 105*(2*B*b*c^3 - 3*A*c^4)*x^4 - 140*(2*B*b^2*c^2 - 3*A*b*c^
3)*x^3 - 21*(2*B*b^3*c - 3*A*b^2*c^2)*x^2 + 6*(2*B*b^4 - 3*A*b^3*c)*x)*sqrt(c*x^
2 + b*x)*sqrt(b)*sqrt(x) + 105*((2*B*b*c^4 - 3*A*c^5)*x^6 + 2*(2*B*b^2*c^3 - 3*A
*b*c^4)*x^5 + (2*B*b^3*c^2 - 3*A*b^2*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))/((b^5*c^2*x^6 + 2*b^6*c*x^5 + b^7*x^4)*sqrt(
b)), -1/24*((8*A*b^4 - 105*(2*B*b*c^3 - 3*A*c^4)*x^4 - 140*(2*B*b^2*c^2 - 3*A*b*
c^3)*x^3 - 21*(2*B*b^3*c - 3*A*b^2*c^2)*x^2 + 6*(2*B*b^4 - 3*A*b^3*c)*x)*sqrt(c*
x^2 + b*x)*sqrt(-b)*sqrt(x) + 105*((2*B*b*c^4 - 3*A*c^5)*x^6 + 2*(2*B*b^2*c^3 -
3*A*b*c^4)*x^5 + (2*B*b^3*c^2 - 3*A*b^2*c^3)*x^4)*arctan(sqrt(-b)*sqrt(x)/sqrt(c
*x^2 + b*x)))/((b^5*c^2*x^6 + 2*b^6*c*x^5 + b^7*x^4)*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)/x**(3/2)/(c*x**2+b*x)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.89187, size = 270, normalized size = 1.26 \[ \frac{35 \,{\left (2 \, B b c^{2} - 3 \, A c^{3}\right )} \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{8 \, \sqrt{-b} b^{5}} + \frac{210 \,{\left (c x + b\right )}^{4} B b c^{2} - 560 \,{\left (c x + b\right )}^{3} B b^{2} c^{2} + 462 \,{\left (c x + b\right )}^{2} B b^{3} c^{2} - 96 \,{\left (c x + b\right )} B b^{4} c^{2} - 16 \, B b^{5} c^{2} - 315 \,{\left (c x + b\right )}^{4} A c^{3} + 840 \,{\left (c x + b\right )}^{3} A b c^{3} - 693 \,{\left (c x + b\right )}^{2} A b^{2} c^{3} + 144 \,{\left (c x + b\right )} A b^{3} c^{3} + 16 \, A b^{4} c^{3}}{24 \,{\left ({\left (c x + b\right )}^{\frac{3}{2}} - \sqrt{c x + b} b\right )}^{3} b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((c*x^2 + b*x)^(5/2)*x^(3/2)),x, algorithm="giac")

[Out]

35/8*(2*B*b*c^2 - 3*A*c^3)*arctan(sqrt(c*x + b)/sqrt(-b))/(sqrt(-b)*b^5) + 1/24*
(210*(c*x + b)^4*B*b*c^2 - 560*(c*x + b)^3*B*b^2*c^2 + 462*(c*x + b)^2*B*b^3*c^2
- 96*(c*x + b)*B*b^4*c^2 - 16*B*b^5*c^2 - 315*(c*x + b)^4*A*c^3 + 840*(c*x + b)
^3*A*b*c^3 - 693*(c*x + b)^2*A*b^2*c^3 + 144*(c*x + b)*A*b^3*c^3 + 16*A*b^4*c^3)
/(((c*x + b)^(3/2) - sqrt(c*x + b)*b)^3*b^5)