3.1230 \(\int \frac{(A+B x) (d+e x)^{5/2}}{b x+c x^2} \, dx\)

Optimal. Leaf size=173 \[ -\frac{2 (b B-A c) (c d-b e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b c^{7/2}}+\frac{2 \sqrt{d+e x} \left (A c e (2 c d-b e)+B (c d-b e)^2\right )}{c^3}+\frac{2 (d+e x)^{3/2} (A c e-b B e+B c d)}{3 c^2}-\frac{2 A d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b}+\frac{2 B (d+e x)^{5/2}}{5 c} \]

[Out]

(2*(B*(c*d - b*e)^2 + A*c*e*(2*c*d - b*e))*Sqrt[d + e*x])/c^3 + (2*(B*c*d - b*B*
e + A*c*e)*(d + e*x)^(3/2))/(3*c^2) + (2*B*(d + e*x)^(5/2))/(5*c) - (2*A*d^(5/2)
*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/b - (2*(b*B - A*c)*(c*d - b*e)^(5/2)*ArcTanh[(S
qrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b*c^(7/2))

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Rubi [A]  time = 0.855475, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{2 (b B-A c) (c d-b e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b c^{7/2}}+\frac{2 \sqrt{d+e x} \left (A c e (2 c d-b e)+B (c d-b e)^2\right )}{c^3}+\frac{2 (d+e x)^{3/2} (A c e-b B e+B c d)}{3 c^2}-\frac{2 A d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b}+\frac{2 B (d+e x)^{5/2}}{5 c} \]

Antiderivative was successfully verified.

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

[Out]

(2*(B*(c*d - b*e)^2 + A*c*e*(2*c*d - b*e))*Sqrt[d + e*x])/c^3 + (2*(B*c*d - b*B*
e + A*c*e)*(d + e*x)^(3/2))/(3*c^2) + (2*B*(d + e*x)^(5/2))/(5*c) - (2*A*d^(5/2)
*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/b - (2*(b*B - A*c)*(c*d - b*e)^(5/2)*ArcTanh[(S
qrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b*c^(7/2))

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Rubi in Sympy [A]  time = 95.7933, size = 163, normalized size = 0.94 \[ - \frac{2 A d^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{d + e x}}{\sqrt{d}} \right )}}{b} + \frac{2 B \left (d + e x\right )^{\frac{5}{2}}}{5 c} + \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (A c e - B b e + B c d\right )}{3 c^{2}} + \frac{2 \sqrt{d + e x} \left (A c^{2} d e + \left (b e - c d\right ) \left (- A c e + B \left (b e - c d\right )\right )\right )}{c^{3}} + \frac{2 \left (A c - B b\right ) \left (b e - c d\right )^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d + e x}}{\sqrt{b e - c d}} \right )}}{b c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A*d**(5/2)*atanh(sqrt(d + e*x)/sqrt(d))/b + 2*B*(d + e*x)**(5/2)/(5*c) + 2*(d
+ e*x)**(3/2)*(A*c*e - B*b*e + B*c*d)/(3*c**2) + 2*sqrt(d + e*x)*(A*c**2*d*e +
(b*e - c*d)*(-A*c*e + B*(b*e - c*d)))/c**3 + 2*(A*c - B*b)*(b*e - c*d)**(5/2)*at
an(sqrt(c)*sqrt(d + e*x)/sqrt(b*e - c*d))/(b*c**(7/2))

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Mathematica [A]  time = 0.28721, size = 167, normalized size = 0.97 \[ \frac{2 \sqrt{d+e x} \left (5 A c e (-3 b e+7 c d+c e x)+B \left (15 b^2 e^2-5 b c e (7 d+e x)+c^2 \left (23 d^2+11 d e x+3 e^2 x^2\right )\right )\right )}{15 c^3}+\frac{2 (A c-b B) (c d-b e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b c^{7/2}}-\frac{2 A d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d + e*x]*(5*A*c*e*(7*c*d - 3*b*e + c*e*x) + B*(15*b^2*e^2 - 5*b*c*e*(7*d
+ e*x) + c^2*(23*d^2 + 11*d*e*x + 3*e^2*x^2))))/(15*c^3) - (2*A*d^(5/2)*ArcTanh
[Sqrt[d + e*x]/Sqrt[d]])/b + (2*(-(b*B) + A*c)*(c*d - b*e)^(5/2)*ArcTanh[(Sqrt[c
]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b*c^(7/2))

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Maple [B]  time = 0.022, size = 516, normalized size = 3. \[{\frac{2\,B}{5\,c} \left ( ex+d \right ) ^{{\frac{5}{2}}}}+{\frac{2\,Ae}{3\,c} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{2\,bBe}{3\,{c}^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{2\,Bd}{3\,c} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-2\,{\frac{Ab{e}^{2}\sqrt{ex+d}}{{c}^{2}}}+4\,{\frac{dAe\sqrt{ex+d}}{c}}+2\,{\frac{B{b}^{2}{e}^{2}\sqrt{ex+d}}{{c}^{3}}}-4\,{\frac{Bbde\sqrt{ex+d}}{{c}^{2}}}+2\,{\frac{B{d}^{2}\sqrt{ex+d}}{c}}+2\,{\frac{A{b}^{2}{e}^{3}}{{c}^{2}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-6\,{\frac{Abd{e}^{2}}{c\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+6\,{\frac{{d}^{2}Ae}{\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-2\,{\frac{A{d}^{3}c}{b\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-2\,{\frac{B{e}^{3}{b}^{3}}{{c}^{3}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+6\,{\frac{Bd{b}^{2}{e}^{2}}{{c}^{2}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-6\,{\frac{B{d}^{2}be}{c\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+2\,{\frac{B{d}^{3}}{\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-2\,{\frac{A{d}^{5/2}}{b}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*B*(e*x+d)^(5/2)/c+2/3/c*A*(e*x+d)^(3/2)*e-2/3/c^2*B*(e*x+d)^(3/2)*b*e+2/3/c*
B*(e*x+d)^(3/2)*d-2/c^2*A*b*e^2*(e*x+d)^(1/2)+4/c*A*d*e*(e*x+d)^(1/2)+2/c^3*B*e^
2*b^2*(e*x+d)^(1/2)-4/c^2*B*b*d*e*(e*x+d)^(1/2)+2/c*B*d^2*(e*x+d)^(1/2)+2/c^2*b^
2/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*A*e^3-6/c*b/((
b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*A*d*e^2+6/((b*e-c*
d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*A*d^2*e-2*c/b/((b*e-c*d)
*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*A*d^3-2/c^3*b^3/((b*e-c*d)
*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*B*e^3+6/c^2*b^2/((b*e-c*d)
*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*B*d*e^2-6/c*b/((b*e-c*d)*c
)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*B*d^2*e+2/((b*e-c*d)*c)^(1/2
)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*B*d^3-2*A*d^(5/2)*arctanh((e*x+d)^
(1/2)/d^(1/2))/b

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 4.37589, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/15*(15*A*c^3*d^(5/2)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) + 15*((B*b*
c^2 - A*c^3)*d^2 - 2*(B*b^2*c - A*b*c^2)*d*e + (B*b^3 - A*b^2*c)*e^2)*sqrt((c*d
- b*e)/c)*log((c*e*x + 2*c*d - b*e - 2*sqrt(e*x + d)*c*sqrt((c*d - b*e)/c))/(c*x
+ b)) + 2*(3*B*b*c^2*e^2*x^2 + 23*B*b*c^2*d^2 - 35*(B*b^2*c - A*b*c^2)*d*e + 15
*(B*b^3 - A*b^2*c)*e^2 + (11*B*b*c^2*d*e - 5*(B*b^2*c - A*b*c^2)*e^2)*x)*sqrt(e*
x + d))/(b*c^3), 1/15*(15*A*c^3*d^(5/2)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d
)/x) - 30*((B*b*c^2 - A*c^3)*d^2 - 2*(B*b^2*c - A*b*c^2)*d*e + (B*b^3 - A*b^2*c)
*e^2)*sqrt(-(c*d - b*e)/c)*arctan(sqrt(e*x + d)/sqrt(-(c*d - b*e)/c)) + 2*(3*B*b
*c^2*e^2*x^2 + 23*B*b*c^2*d^2 - 35*(B*b^2*c - A*b*c^2)*d*e + 15*(B*b^3 - A*b^2*c
)*e^2 + (11*B*b*c^2*d*e - 5*(B*b^2*c - A*b*c^2)*e^2)*x)*sqrt(e*x + d))/(b*c^3),
-1/15*(30*A*c^3*sqrt(-d)*d^2*arctan(sqrt(e*x + d)/sqrt(-d)) - 15*((B*b*c^2 - A*c
^3)*d^2 - 2*(B*b^2*c - A*b*c^2)*d*e + (B*b^3 - A*b^2*c)*e^2)*sqrt((c*d - b*e)/c)
*log((c*e*x + 2*c*d - b*e - 2*sqrt(e*x + d)*c*sqrt((c*d - b*e)/c))/(c*x + b)) -
2*(3*B*b*c^2*e^2*x^2 + 23*B*b*c^2*d^2 - 35*(B*b^2*c - A*b*c^2)*d*e + 15*(B*b^3 -
A*b^2*c)*e^2 + (11*B*b*c^2*d*e - 5*(B*b^2*c - A*b*c^2)*e^2)*x)*sqrt(e*x + d))/(
b*c^3), -2/15*(15*A*c^3*sqrt(-d)*d^2*arctan(sqrt(e*x + d)/sqrt(-d)) + 15*((B*b*c
^2 - A*c^3)*d^2 - 2*(B*b^2*c - A*b*c^2)*d*e + (B*b^3 - A*b^2*c)*e^2)*sqrt(-(c*d
- b*e)/c)*arctan(sqrt(e*x + d)/sqrt(-(c*d - b*e)/c)) - (3*B*b*c^2*e^2*x^2 + 23*B
*b*c^2*d^2 - 35*(B*b^2*c - A*b*c^2)*d*e + 15*(B*b^3 - A*b^2*c)*e^2 + (11*B*b*c^2
*d*e - 5*(B*b^2*c - A*b*c^2)*e^2)*x)*sqrt(e*x + d))/(b*c^3)]

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Sympy [A]  time = 122.577, size = 374, normalized size = 2.16 \[ - \frac{2 A d^{3} \left (\begin{cases} - \frac{\operatorname{atan}{\left (\frac{\sqrt{d + e x}}{\sqrt{- d}} \right )}}{\sqrt{- d}} & \text{for}\: - d > 0 \\\frac{\operatorname{acoth}{\left (\frac{\sqrt{d + e x}}{\sqrt{d}} \right )}}{\sqrt{d}} & \text{for}\: - d < 0 \wedge d < d + e x \\\frac{\operatorname{atanh}{\left (\frac{\sqrt{d + e x}}{\sqrt{d}} \right )}}{\sqrt{d}} & \text{for}\: d > d + e x \wedge - d < 0 \end{cases}\right )}{b} + \frac{2 B \left (d + e x\right )^{\frac{5}{2}}}{5 c} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (2 A c e - 2 B b e + 2 B c d\right )}{3 c^{2}} + \frac{\sqrt{d + e x} \left (- 2 A b c e^{2} + 4 A c^{2} d e + 2 B b^{2} e^{2} - 4 B b c d e + 2 B c^{2} d^{2}\right )}{c^{3}} - \frac{2 \left (- A c + B b\right ) \left (b e - c d\right )^{3} \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{b e - c d}{c}}} \right )}}{c \sqrt{\frac{b e - c d}{c}}} & \text{for}\: \frac{b e - c d}{c} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{- b e + c d}{c}}} \right )}}{c \sqrt{\frac{- b e + c d}{c}}} & \text{for}\: d + e x > \frac{- b e + c d}{c} \wedge \frac{b e - c d}{c} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{- b e + c d}{c}}} \right )}}{c \sqrt{\frac{- b e + c d}{c}}} & \text{for}\: \frac{b e - c d}{c} < 0 \wedge d + e x < \frac{- b e + c d}{c} \end{cases}\right )}{b c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(e*x+d)**(5/2)/(c*x**2+b*x),x)

[Out]

-2*A*d**3*Piecewise((-atan(sqrt(d + e*x)/sqrt(-d))/sqrt(-d), -d > 0), (acoth(sqr
t(d + e*x)/sqrt(d))/sqrt(d), (-d < 0) & (d < d + e*x)), (atanh(sqrt(d + e*x)/sqr
t(d))/sqrt(d), (-d < 0) & (d > d + e*x)))/b + 2*B*(d + e*x)**(5/2)/(5*c) + (d +
e*x)**(3/2)*(2*A*c*e - 2*B*b*e + 2*B*c*d)/(3*c**2) + sqrt(d + e*x)*(-2*A*b*c*e**
2 + 4*A*c**2*d*e + 2*B*b**2*e**2 - 4*B*b*c*d*e + 2*B*c**2*d**2)/c**3 - 2*(-A*c +
B*b)*(b*e - c*d)**3*Piecewise((atan(sqrt(d + e*x)/sqrt((b*e - c*d)/c))/(c*sqrt(
(b*e - c*d)/c)), (b*e - c*d)/c > 0), (-acoth(sqrt(d + e*x)/sqrt((-b*e + c*d)/c))
/(c*sqrt((-b*e + c*d)/c)), ((b*e - c*d)/c < 0) & (d + e*x > (-b*e + c*d)/c)), (-
atanh(sqrt(d + e*x)/sqrt((-b*e + c*d)/c))/(c*sqrt((-b*e + c*d)/c)), ((b*e - c*d)
/c < 0) & (d + e*x < (-b*e + c*d)/c)))/(b*c**3)

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GIAC/XCAS [A]  time = 0.28936, size = 427, normalized size = 2.47 \[ \frac{2 \, A d^{3} \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{b \sqrt{-d}} + \frac{2 \,{\left (B b c^{3} d^{3} - A c^{4} d^{3} - 3 \, B b^{2} c^{2} d^{2} e + 3 \, A b c^{3} d^{2} e + 3 \, B b^{3} c d e^{2} - 3 \, A b^{2} c^{2} d e^{2} - B b^{4} e^{3} + A b^{3} c e^{3}\right )} \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{\sqrt{-c^{2} d + b c e} b c^{3}} + \frac{2 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} B c^{4} + 5 \,{\left (x e + d\right )}^{\frac{3}{2}} B c^{4} d + 15 \, \sqrt{x e + d} B c^{4} d^{2} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} B b c^{3} e + 5 \,{\left (x e + d\right )}^{\frac{3}{2}} A c^{4} e - 30 \, \sqrt{x e + d} B b c^{3} d e + 30 \, \sqrt{x e + d} A c^{4} d e + 15 \, \sqrt{x e + d} B b^{2} c^{2} e^{2} - 15 \, \sqrt{x e + d} A b c^{3} e^{2}\right )}}{15 \, c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*d^3*arctan(sqrt(x*e + d)/sqrt(-d))/(b*sqrt(-d)) + 2*(B*b*c^3*d^3 - A*c^4*d^3
- 3*B*b^2*c^2*d^2*e + 3*A*b*c^3*d^2*e + 3*B*b^3*c*d*e^2 - 3*A*b^2*c^2*d*e^2 - B
*b^4*e^3 + A*b^3*c*e^3)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/(sqrt(-c^2*
d + b*c*e)*b*c^3) + 2/15*(3*(x*e + d)^(5/2)*B*c^4 + 5*(x*e + d)^(3/2)*B*c^4*d +
15*sqrt(x*e + d)*B*c^4*d^2 - 5*(x*e + d)^(3/2)*B*b*c^3*e + 5*(x*e + d)^(3/2)*A*c
^4*e - 30*sqrt(x*e + d)*B*b*c^3*d*e + 30*sqrt(x*e + d)*A*c^4*d*e + 15*sqrt(x*e +
d)*B*b^2*c^2*e^2 - 15*sqrt(x*e + d)*A*b*c^3*e^2)/c^5