3.1238 \(\int \frac{(A+B x) (d+e x)^{9/2}}{\left (b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=389 \[ -\frac{d^{7/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) (9 A b e-4 A c d+2 b B d)}{b^3}+\frac{e (d+e x)^{5/2} \left (-5 b c (A e+B d)+10 A c^2 d+7 b^2 B e\right )}{5 b^2 c^2}+\frac{(d+e x)^{7/2} (b B-2 A c) (c d-b e)}{b^2 c (b+c x)}-\frac{(c d-b e)^{7/2} \left (-b c (2 B d-5 A e)+4 A c^2 d-7 b^2 B e\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 c^{9/2}}+\frac{e (d+e x)^{3/2} \left (b^2 c e (5 A e+12 B d)-3 b c^2 d (2 A e+B d)+6 A c^3 d^2-7 b^3 B e^2\right )}{3 b^2 c^3}+\frac{e \sqrt{d+e x} \left (-b^3 c e^2 (5 A e+19 B d)+b^2 c^2 d e (11 A e+15 B d)-b c^3 d^2 (3 A e+B d)+2 A c^4 d^3+7 b^4 B e^3\right )}{b^2 c^4}-\frac{A (d+e x)^{9/2}}{b x (b+c x)} \]

[Out]

(e*(2*A*c^4*d^3 + 7*b^4*B*e^3 - b*c^3*d^2*(B*d + 3*A*e) - b^3*c*e^2*(19*B*d + 5*
A*e) + b^2*c^2*d*e*(15*B*d + 11*A*e))*Sqrt[d + e*x])/(b^2*c^4) + (e*(6*A*c^3*d^2
- 7*b^3*B*e^2 - 3*b*c^2*d*(B*d + 2*A*e) + b^2*c*e*(12*B*d + 5*A*e))*(d + e*x)^(
3/2))/(3*b^2*c^3) + (e*(10*A*c^2*d + 7*b^2*B*e - 5*b*c*(B*d + A*e))*(d + e*x)^(5
/2))/(5*b^2*c^2) + ((b*B - 2*A*c)*(c*d - b*e)*(d + e*x)^(7/2))/(b^2*c*(b + c*x))
- (A*(d + e*x)^(9/2))/(b*x*(b + c*x)) - (d^(7/2)*(2*b*B*d - 4*A*c*d + 9*A*b*e)*
ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/b^3 - ((c*d - b*e)^(7/2)*(4*A*c^2*d - 7*b^2*B*e
- b*c*(2*B*d - 5*A*e))*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b^3*c^
(9/2))

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Rubi [A]  time = 2.34197, antiderivative size = 389, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ -\frac{d^{7/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) (9 A b e-4 A c d+2 b B d)}{b^3}+\frac{e (d+e x)^{5/2} \left (-5 b c (A e+B d)+10 A c^2 d+7 b^2 B e\right )}{5 b^2 c^2}+\frac{(d+e x)^{7/2} (b B-2 A c) (c d-b e)}{b^2 c (b+c x)}+\frac{(c d-b e)^{7/2} \left (-5 A b c e-4 A c^2 d+7 b^2 B e+2 b B c d\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 c^{9/2}}+\frac{e (d+e x)^{3/2} \left (b^2 c e (5 A e+12 B d)-3 b c^2 d (2 A e+B d)+6 A c^3 d^2-7 b^3 B e^2\right )}{3 b^2 c^3}+\frac{e \sqrt{d+e x} \left (-b^3 c e^2 (5 A e+19 B d)+b^2 c^2 d e (11 A e+15 B d)-b c^3 d^2 (3 A e+B d)+2 A c^4 d^3+7 b^4 B e^3\right )}{b^2 c^4}-\frac{A (d+e x)^{9/2}}{b x (b+c x)} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(d + e*x)^(9/2))/(b*x + c*x^2)^2,x]

[Out]

(e*(2*A*c^4*d^3 + 7*b^4*B*e^3 - b*c^3*d^2*(B*d + 3*A*e) - b^3*c*e^2*(19*B*d + 5*
A*e) + b^2*c^2*d*e*(15*B*d + 11*A*e))*Sqrt[d + e*x])/(b^2*c^4) + (e*(6*A*c^3*d^2
- 7*b^3*B*e^2 - 3*b*c^2*d*(B*d + 2*A*e) + b^2*c*e*(12*B*d + 5*A*e))*(d + e*x)^(
3/2))/(3*b^2*c^3) + (e*(10*A*c^2*d + 7*b^2*B*e - 5*b*c*(B*d + A*e))*(d + e*x)^(5
/2))/(5*b^2*c^2) + ((b*B - 2*A*c)*(c*d - b*e)*(d + e*x)^(7/2))/(b^2*c*(b + c*x))
- (A*(d + e*x)^(9/2))/(b*x*(b + c*x)) - (d^(7/2)*(2*b*B*d - 4*A*c*d + 9*A*b*e)*
ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/b^3 + ((c*d - b*e)^(7/2)*(2*b*B*c*d - 4*A*c^2*d
+ 7*b^2*B*e - 5*A*b*c*e)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b^3*
c^(9/2))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**(9/2)/(c*x**2+b*x)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 0.767487, size = 265, normalized size = 0.68 \[ -\frac{d^{7/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) (9 A b e-4 A c d+2 b B d)}{b^3}+\sqrt{d+e x} \left (\frac{(b B-A c) (c d-b e)^4}{b^2 c^4 (b+c x)}+\frac{2 e^2 \left (5 A c e (13 c d-6 b e)+B \left (45 b^2 e^2-130 b c d e+108 c^2 d^2\right )\right )}{15 c^4}-\frac{A d^4}{b^2 x}+\frac{2 e^3 x (5 A c e-10 b B e+21 B c d)}{15 c^3}+\frac{2 B e^4 x^2}{5 c^2}\right )+\frac{(c d-b e)^{7/2} \left (b c (2 B d-5 A e)-4 A c^2 d+7 b^2 B e\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 c^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(d + e*x)^(9/2))/(b*x + c*x^2)^2,x]

[Out]

Sqrt[d + e*x]*((2*e^2*(5*A*c*e*(13*c*d - 6*b*e) + B*(108*c^2*d^2 - 130*b*c*d*e +
45*b^2*e^2)))/(15*c^4) - (A*d^4)/(b^2*x) + (2*e^3*(21*B*c*d - 10*b*B*e + 5*A*c*
e)*x)/(15*c^3) + (2*B*e^4*x^2)/(5*c^2) + ((b*B - A*c)*(c*d - b*e)^4)/(b^2*c^4*(b
+ c*x))) - (d^(7/2)*(2*b*B*d - 4*A*c*d + 9*A*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]
])/b^3 + ((c*d - b*e)^(7/2)*(-4*A*c^2*d + 7*b^2*B*e + b*c*(2*B*d - 5*A*e))*ArcTa
nh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b^3*c^(9/2))

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Maple [B]  time = 0.047, size = 1075, normalized size = 2.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^(9/2)/(c*x^2+b*x)^2,x)

[Out]

6*e^4/c^4*B*b^2*(e*x+d)^(1/2)-d^4/b^2*A*(e*x+d)^(1/2)/x+4*d^(9/2)/b^3*arctanh((e
*x+d)^(1/2)/d^(1/2))*A*c+12*e^2/c^2*B*d^2*(e*x+d)^(1/2)-9*e*d^(7/2)/b^2*arctanh(
(e*x+d)^(1/2)/d^(1/2))*A-4/3*e^3/c^3*B*(e*x+d)^(3/2)*b-4*e^4/c^3*A*b*(e*x+d)^(1/
2)+8*e^3/c^2*A*d*(e*x+d)^(1/2)+2*e^2/c^2*B*(e*x+d)^(3/2)*d-2*d^(9/2)/b^2*arctanh
((e*x+d)^(1/2)/d^(1/2))*B+2/3*e^3/c^2*A*(e*x+d)^(3/2)+2/5*e^2/c^2*B*(e*x+d)^(5/2
)-16*e^3/c^3*B*b*d*(e*x+d)^(1/2)+14*e^3/c/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(
1/2)/((b*e-c*d)*c)^(1/2))*A*d^2+4*e^2/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+e/b/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b
*e-c*d)*c)^(1/2))*B*d^4+4/b^3*c^2/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b
*e-c*d)*c)^(1/2))*A*d^5-2/b^2*c/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e
-c*d)*c)^(1/2))*B*d^5+16*e^2/c/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-
c*d)*c)^(1/2))*B*d^3-6*e^3/c*(e*x+d)^(1/2)/(c*e*x+b*e)*A*d^2+4*e^2/b*(e*x+d)^(1/
2)/(c*e*x+b*e)*A*d^3-4*e^2/c*(e*x+d)^(1/2)/(c*e*x+b*e)*B*d^3+e/b*(e*x+d)^(1/2)/(
c*e*x+b*e)*B*d^4-e^5*b^2/c^3*(e*x+d)^(1/2)/(c*e*x+b*e)*A+e^5*b^3/c^4*(e*x+d)^(1/
2)/(c*e*x+b*e)*B+5*e^5*b^2/c^3/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-
c*d)*c)^(1/2))*A-7*e^5*b^3/c^4/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-
c*d)*c)^(1/2))*B+6*e^3*b/c^2*(e*x+d)^(1/2)/(c*e*x+b*e)*B*d^2-16*e^4*b/c^2/((b*e-
c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*A*d-11*e/b^2*c/((b*e-c
*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*A*d^4+26*e^4*b^2/c^3/((
b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*B*d-34*e^3*b/c^2/(
(b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*B*d^2+4*e^4*b/c^2
*(e*x+d)^(1/2)/(c*e*x+b*e)*A*d-e/b^2*c*(e*x+d)^(1/2)/(c*e*x+b*e)*A*d^4-4*e^4*b^2
/c^3*(e*x+d)^(1/2)/(c*e*x+b*e)*B*d

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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)^(9/2)/(c*x^2 + b*x)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [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)*(e*x + d)^(9/2)/(c*x^2 + b*x)^2,x, algorithm="fricas")

[Out]

Timed out

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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)*(e*x+d)**(9/2)/(c*x**2+b*x)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.338961, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^(9/2)/(c*x^2 + b*x)^2,x, algorithm="giac")

[Out]

Done