3.1254 \(\int \frac{A+B x}{(d+e x)^{5/2} \left (b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=668 \[ -\frac{-7 A b e-8 A c d+4 b B d}{4 b^2 d^2 x (b+c x)^2 (d+e x)^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (-5 b^2 e (4 B d-7 A e)-12 b c d (2 B d-5 A e)+48 A c^2 d^2\right )}{4 b^5 d^{9/2}}+\frac{c \left (b^2 e (4 B d-7 A e)-3 b c d (A e+2 B d)+12 A c^2 d^2\right )}{4 b^3 d^2 (b+c x)^2 (d+e x)^{3/2} (c d-b e)}+\frac{c^{7/2} \left (11 b^2 c e (13 A e+8 B d)-12 b c^2 d (13 A e+2 B d)+48 A c^3 d^2-99 b^3 B e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 (c d-b e)^{9/2}}+\frac{c \left (b^3 \left (-e^2\right ) (4 B d-7 A e)+b^2 c d e (23 B d-2 A e)-12 b c^2 d^2 (3 A e+B d)+24 A c^3 d^3\right )}{4 b^4 d^2 (b+c x) (d+e x)^{3/2} (c d-b e)^2}+\frac{e \left (5 b^4 e^3 (4 B d-7 A e)-9 b^3 c d e^2 (4 B d-5 A e)+3 b^2 c^2 d^2 e (9 A e+29 B d)-36 b c^3 d^3 (4 A e+B d)+72 A c^4 d^4\right )}{12 b^4 d^3 (d+e x)^{3/2} (c d-b e)^3}+\frac{e \left (-5 b^5 e^4 (4 B d-7 A e)+8 b^4 c d e^3 (7 B d-10 A e)-6 b^3 c^2 d^2 e^2 (4 B d-3 A e)+7 b^2 c^3 d^3 e (4 A e+5 B d)-12 b c^4 d^4 (5 A e+B d)+24 A c^5 d^5\right )}{4 b^4 d^4 \sqrt{d+e x} (c d-b e)^4}-\frac{A}{2 b d x^2 (b+c x)^2 (d+e x)^{3/2}} \]

[Out]

(e*(72*A*c^4*d^4 + 5*b^4*e^3*(4*B*d - 7*A*e) - 9*b^3*c*d*e^2*(4*B*d - 5*A*e) - 3
6*b*c^3*d^3*(B*d + 4*A*e) + 3*b^2*c^2*d^2*e*(29*B*d + 9*A*e)))/(12*b^4*d^3*(c*d
- b*e)^3*(d + e*x)^(3/2)) + (c*(12*A*c^2*d^2 + b^2*e*(4*B*d - 7*A*e) - 3*b*c*d*(
2*B*d + A*e)))/(4*b^3*d^2*(c*d - b*e)*(b + c*x)^2*(d + e*x)^(3/2)) - A/(2*b*d*x^
2*(b + c*x)^2*(d + e*x)^(3/2)) - (4*b*B*d - 8*A*c*d - 7*A*b*e)/(4*b^2*d^2*x*(b +
c*x)^2*(d + e*x)^(3/2)) + (c*(24*A*c^3*d^3 - b^3*e^2*(4*B*d - 7*A*e) + b^2*c*d*
e*(23*B*d - 2*A*e) - 12*b*c^2*d^2*(B*d + 3*A*e)))/(4*b^4*d^2*(c*d - b*e)^2*(b +
c*x)*(d + e*x)^(3/2)) + (e*(24*A*c^5*d^5 + 8*b^4*c*d*e^3*(7*B*d - 10*A*e) - 5*b^
5*e^4*(4*B*d - 7*A*e) - 6*b^3*c^2*d^2*e^2*(4*B*d - 3*A*e) + 7*b^2*c^3*d^3*e*(5*B
*d + 4*A*e) - 12*b*c^4*d^4*(B*d + 5*A*e)))/(4*b^4*d^4*(c*d - b*e)^4*Sqrt[d + e*x
]) - ((48*A*c^2*d^2 - 5*b^2*e*(4*B*d - 7*A*e) - 12*b*c*d*(2*B*d - 5*A*e))*ArcTan
h[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5*d^(9/2)) + (c^(7/2)*(48*A*c^3*d^2 - 99*b^3*B*e^
2 - 12*b*c^2*d*(2*B*d + 13*A*e) + 11*b^2*c*e*(8*B*d + 13*A*e))*ArcTanh[(Sqrt[c]*
Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(4*b^5*(c*d - b*e)^(9/2))

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Rubi [A]  time = 4.84591, antiderivative size = 668, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ -\frac{-7 A b e-8 A c d+4 b B d}{4 b^2 d^2 x (b+c x)^2 (d+e x)^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (-5 b^2 e (4 B d-7 A e)-12 b c d (2 B d-5 A e)+48 A c^2 d^2\right )}{4 b^5 d^{9/2}}+\frac{c \left (b^2 e (4 B d-7 A e)-3 b c d (A e+2 B d)+12 A c^2 d^2\right )}{4 b^3 d^2 (b+c x)^2 (d+e x)^{3/2} (c d-b e)}+\frac{c^{7/2} \left (11 b^2 c e (13 A e+8 B d)-12 b c^2 d (13 A e+2 B d)+48 A c^3 d^2-99 b^3 B e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 (c d-b e)^{9/2}}+\frac{c \left (b^3 \left (-e^2\right ) (4 B d-7 A e)+b^2 c d e (23 B d-2 A e)-12 b c^2 d^2 (3 A e+B d)+24 A c^3 d^3\right )}{4 b^4 d^2 (b+c x) (d+e x)^{3/2} (c d-b e)^2}+\frac{e \left (5 b^4 e^3 (4 B d-7 A e)-9 b^3 c d e^2 (4 B d-5 A e)+3 b^2 c^2 d^2 e (9 A e+29 B d)-36 b c^3 d^3 (4 A e+B d)+72 A c^4 d^4\right )}{12 b^4 d^3 (d+e x)^{3/2} (c d-b e)^3}+\frac{e \left (-5 b^5 e^4 (4 B d-7 A e)+8 b^4 c d e^3 (7 B d-10 A e)-6 b^3 c^2 d^2 e^2 (4 B d-3 A e)+7 b^2 c^3 d^3 e (4 A e+5 B d)-12 b c^4 d^4 (5 A e+B d)+24 A c^5 d^5\right )}{4 b^4 d^4 \sqrt{d+e x} (c d-b e)^4}-\frac{A}{2 b d x^2 (b+c x)^2 (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(e*(72*A*c^4*d^4 + 5*b^4*e^3*(4*B*d - 7*A*e) - 9*b^3*c*d*e^2*(4*B*d - 5*A*e) - 3
6*b*c^3*d^3*(B*d + 4*A*e) + 3*b^2*c^2*d^2*e*(29*B*d + 9*A*e)))/(12*b^4*d^3*(c*d
- b*e)^3*(d + e*x)^(3/2)) + (c*(12*A*c^2*d^2 + b^2*e*(4*B*d - 7*A*e) - 3*b*c*d*(
2*B*d + A*e)))/(4*b^3*d^2*(c*d - b*e)*(b + c*x)^2*(d + e*x)^(3/2)) - A/(2*b*d*x^
2*(b + c*x)^2*(d + e*x)^(3/2)) - (4*b*B*d - 8*A*c*d - 7*A*b*e)/(4*b^2*d^2*x*(b +
c*x)^2*(d + e*x)^(3/2)) + (c*(24*A*c^3*d^3 - b^3*e^2*(4*B*d - 7*A*e) + b^2*c*d*
e*(23*B*d - 2*A*e) - 12*b*c^2*d^2*(B*d + 3*A*e)))/(4*b^4*d^2*(c*d - b*e)^2*(b +
c*x)*(d + e*x)^(3/2)) + (e*(24*A*c^5*d^5 + 8*b^4*c*d*e^3*(7*B*d - 10*A*e) - 5*b^
5*e^4*(4*B*d - 7*A*e) - 6*b^3*c^2*d^2*e^2*(4*B*d - 3*A*e) + 7*b^2*c^3*d^3*e*(5*B
*d + 4*A*e) - 12*b*c^4*d^4*(B*d + 5*A*e)))/(4*b^4*d^4*(c*d - b*e)^4*Sqrt[d + e*x
]) - ((48*A*c^2*d^2 - 5*b^2*e*(4*B*d - 7*A*e) - 12*b*c*d*(2*B*d - 5*A*e))*ArcTan
h[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5*d^(9/2)) + (c^(7/2)*(48*A*c^3*d^2 - 99*b^3*B*e^
2 - 12*b*c^2*d*(2*B*d + 13*A*e) + 11*b^2*c*e*(8*B*d + 13*A*e))*ArcTanh[(Sqrt[c]*
Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(4*b^5*(c*d - b*e)^(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)**(5/2)/(c*x**2+b*x)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 7.13572, size = 390, normalized size = 0.58 \[ \frac{1}{12} \left (\frac{3 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (5 b^2 e (4 B d-7 A e)+12 b c d (2 B d-5 A e)-48 A c^2 d^2\right )}{b^5 d^{9/2}}+\frac{3 c^{7/2} \left (11 b^2 c e (13 A e+8 B d)-12 b c^2 d (13 A e+2 B d)+48 A c^3 d^2-99 b^3 B e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^5 (c d-b e)^{9/2}}+\sqrt{d+e x} \left (\frac{3 (11 A b e+12 A c d-4 b B d)}{b^4 d^4 x}+\frac{6 c^4 (b B-A c)}{b^3 (b+c x)^2 (b e-c d)^3}-\frac{6 A}{b^3 d^3 x^2}+\frac{3 c^4 \left (-b c (23 A e+8 B d)+12 A c^2 d+19 b^2 B e\right )}{b^4 (b+c x) (c d-b e)^4}+\frac{24 e^4 (3 A e (b e-2 c d)+B d (5 c d-2 b e))}{d^4 (d+e x) (c d-b e)^4}+\frac{8 e^4 (B d-A e)}{d^3 (d+e x)^2 (c d-b e)^3}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d + e*x]*((-6*A)/(b^3*d^3*x^2) + (3*(-4*b*B*d + 12*A*c*d + 11*A*b*e))/(b^4
*d^4*x) + (6*c^4*(b*B - A*c))/(b^3*(-(c*d) + b*e)^3*(b + c*x)^2) + (3*c^4*(12*A*
c^2*d + 19*b^2*B*e - b*c*(8*B*d + 23*A*e)))/(b^4*(c*d - b*e)^4*(b + c*x)) + (8*e
^4*(B*d - A*e))/(d^3*(c*d - b*e)^3*(d + e*x)^2) + (24*e^4*(B*d*(5*c*d - 2*b*e) +
3*A*e*(-2*c*d + b*e)))/(d^4*(c*d - b*e)^4*(d + e*x))) + (3*(-48*A*c^2*d^2 + 5*b
^2*e*(4*B*d - 7*A*e) + 12*b*c*d*(2*B*d - 5*A*e))*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])
/(b^5*d^(9/2)) + (3*c^(7/2)*(48*A*c^3*d^2 - 99*b^3*B*e^2 - 12*b*c^2*d*(2*B*d + 1
3*A*e) + 11*b^2*c*e*(8*B*d + 13*A*e))*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d -
b*e]])/(b^5*(c*d - b*e)^(9/2)))/12

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Maple [A]  time = 0.064, size = 1130, normalized size = 1.7 \[ \text{result too large to display} \]

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)^3,x)

[Out]

5*e/b^3/d^(7/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*B+2/3*e^5/d^3/(b*e-c*d)^3/(e*x+d)
^(3/2)*A-35/4*e^2/b^3/d^(9/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*A-12/b^5/d^(5/2)*ar
ctanh((e*x+d)^(1/2)/d^(1/2))*A*c^2+6/b^4/d^(5/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*
B*c+11/4/b^3/d^4/x^2*A*(e*x+d)^(3/2)-13/4/b^3/d^3/x^2*(e*x+d)^(1/2)*A-2/3*e^4/d^
2/(b*e-c*d)^3/(e*x+d)^(3/2)*B-25/4*e^3*c^5/(b*e-c*d)^4/b^2/(c*e*x+b*e)^2*A*(e*x+
d)^(1/2)+21/4*e^3*c^4/(b*e-c*d)^4/b/(c*e*x+b*e)^2*B*(e*x+d)^(1/2)-143/4*e^2*c^5/
(b*e-c*d)^4/b^3/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*
A+99/4*e^2*c^4/(b*e-c*d)^4/b^2/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-
c*d)*c)^(1/2))*B-12*c^7/(b*e-c*d)^4/b^5/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/
2)/((b*e-c*d)*c)^(1/2))*A*d^2+6*c^6/(b*e-c*d)^4/b^4/((b*e-c*d)*c)^(1/2)*arctan(c
*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*B*d^2+3/e/b^4/d^3/x^2*A*(e*x+d)^(3/2)*c-3/e/
b^4/d^2/x^2*(e*x+d)^(1/2)*A*c-23/4*e^2*c^6/(b*e-c*d)^4/b^3/(c*e*x+b*e)^2*(e*x+d)
^(3/2)*A+19/4*e^2*c^5/(b*e-c*d)^4/b^2/(c*e*x+b*e)^2*(e*x+d)^(3/2)*B+1/e/b^3/d^2/
x^2*(e*x+d)^(1/2)*B-15*e/b^4/d^(7/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*A*c-12*e^5/d
^3/(b*e-c*d)^4/(e*x+d)^(1/2)*A*c-4*e^5/d^3/(b*e-c*d)^4/(e*x+d)^(1/2)*B*b+6*e^6/d
^4/(b*e-c*d)^4/(e*x+d)^(1/2)*A*b+10*e^4/d^2/(b*e-c*d)^4/(e*x+d)^(1/2)*B*c-1/e/b^
3/d^3/x^2*B*(e*x+d)^(3/2)+37/4*e^2*c^6/(b*e-c*d)^4/b^3/(c*e*x+b*e)^2*A*(e*x+d)^(
1/2)*d-3*e*c^7/(b*e-c*d)^4/b^4/(c*e*x+b*e)^2*A*(e*x+d)^(1/2)*d^2-29/4*e^2*c^5/(b
*e-c*d)^4/b^2/(c*e*x+b*e)^2*B*(e*x+d)^(1/2)*d+2*e*c^6/(b*e-c*d)^4/b^3/(c*e*x+b*e
)^2*B*(e*x+d)^(1/2)*d^2+39*e*c^6/(b*e-c*d)^4/b^4/((b*e-c*d)*c)^(1/2)*arctan(c*(e
*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*A*d-22*e*c^5/(b*e-c*d)^4/b^3/((b*e-c*d)*c)^(1/2
)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*B*d+3*e*c^7/(b*e-c*d)^4/b^4/(c*e*x
+b*e)^2*(e*x+d)^(3/2)*A*d-2*e*c^6/(b*e-c*d)^4/b^3/(c*e*x+b*e)^2*(e*x+d)^(3/2)*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)/((c*x^2 + b*x)^3*(e*x + d)^(5/2)),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)/((c*x^2 + b*x)^3*(e*x + d)^(5/2)),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)**(5/2)/(c*x**2+b*x)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.988359, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done