3.1239 \(\int \frac{(A+B x) (d+e x)^{7/2}}{\left (b x+c x^2\right )^2} \, dx\)
Optimal. Leaf size=295 \[ -\frac{d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) (7 A b e-4 A c d+2 b B d)}{b^3}+\frac{e (d+e x)^{3/2} \left (-3 b c (A e+B d)+6 A c^2 d+5 b^2 B e\right )}{3 b^2 c^2}+\frac{(d+e x)^{5/2} (b B-2 A c) (c d-b e)}{b^2 c (b+c x)}+\frac{(c d-b e)^{5/2} \left (-3 A b c e-4 A c^2 d+5 b^2 B e+2 b B c d\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 c^{7/2}}+\frac{e \sqrt{d+e x} \left (b^2 c e (3 A e+8 B d)-b c^2 d (2 A e+B d)+2 A c^3 d^2-5 b^3 B e^2\right )}{b^2 c^3}-\frac{A (d+e x)^{7/2}}{b x (b+c x)} \]
[Out]
(e*(2*A*c^3*d^2 - 5*b^3*B*e^2 - b*c^2*d*(B*d + 2*A*e) + b^2*c*e*(8*B*d + 3*A*e))
*Sqrt[d + e*x])/(b^2*c^3) + (e*(6*A*c^2*d + 5*b^2*B*e - 3*b*c*(B*d + A*e))*(d +
e*x)^(3/2))/(3*b^2*c^2) + ((b*B - 2*A*c)*(c*d - b*e)*(d + e*x)^(5/2))/(b^2*c*(b
+ c*x)) - (A*(d + e*x)^(7/2))/(b*x*(b + c*x)) - (d^(5/2)*(2*b*B*d - 4*A*c*d + 7*
A*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/b^3 + ((c*d - b*e)^(5/2)*(2*b*B*c*d - 4*A
*c^2*d + 5*b^2*B*e - 3*A*b*c*e)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]]
)/(b^3*c^(7/2))
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Rubi [A] time = 1.72637, antiderivative size = 295, normalized size of antiderivative = 1.,
number of steps used = 10, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269
\[ -\frac{d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) (7 A b e-4 A c d+2 b B d)}{b^3}+\frac{e (d+e x)^{3/2} \left (-3 b c (A e+B d)+6 A c^2 d+5 b^2 B e\right )}{3 b^2 c^2}+\frac{(d+e x)^{5/2} (b B-2 A c) (c d-b e)}{b^2 c (b+c x)}-\frac{(c d-b e)^{5/2} \left (-b c (2 B d-3 A e)+4 A c^2 d-5 b^2 B e\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 c^{7/2}}+\frac{e \sqrt{d+e x} \left (b^2 c e (3 A e+8 B d)-b c^2 d (2 A e+B d)+2 A c^3 d^2-5 b^3 B e^2\right )}{b^2 c^3}-\frac{A (d+e x)^{7/2}}{b x (b+c x)} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(d + e*x)^(7/2))/(b*x + c*x^2)^2,x]
[Out]
(e*(2*A*c^3*d^2 - 5*b^3*B*e^2 - b*c^2*d*(B*d + 2*A*e) + b^2*c*e*(8*B*d + 3*A*e))
*Sqrt[d + e*x])/(b^2*c^3) + (e*(6*A*c^2*d + 5*b^2*B*e - 3*b*c*(B*d + A*e))*(d +
e*x)^(3/2))/(3*b^2*c^2) + ((b*B - 2*A*c)*(c*d - b*e)*(d + e*x)^(5/2))/(b^2*c*(b
+ c*x)) - (A*(d + e*x)^(7/2))/(b*x*(b + c*x)) - (d^(5/2)*(2*b*B*d - 4*A*c*d + 7*
A*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/b^3 - ((c*d - b*e)^(5/2)*(4*A*c^2*d - 5*b
^2*B*e - b*c*(2*B*d - 3*A*e))*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/
(b^3*c^(7/2))
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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)**(7/2)/(c*x**2+b*x)**2,x)
[Out]
Timed out
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Mathematica [A] time = 0.600911, size = 214, normalized size = 0.73 \[ -\frac{d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) (7 A b e-4 A c d+2 b B d)}{b^3}+\sqrt{d+e x} \left (-\frac{(b B-A c) (b e-c d)^3}{b^2 c^3 (b+c x)}-\frac{A d^3}{b^2 x}+\frac{2 e^2 (3 A c e-6 b B e+10 B c d)}{3 c^3}+\frac{2 B e^3 x}{3 c^2}\right )-\frac{(c d-b e)^{5/2} \left (3 A b c e+4 A c^2 d-5 b^2 B e-2 b B c d\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 c^{7/2}} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(d + e*x)^(7/2))/(b*x + c*x^2)^2,x]
[Out]
Sqrt[d + e*x]*((2*e^2*(10*B*c*d - 6*b*B*e + 3*A*c*e))/(3*c^3) - (A*d^3)/(b^2*x)
+ (2*B*e^3*x)/(3*c^2) - ((b*B - A*c)*(-(c*d) + b*e)^3)/(b^2*c^3*(b + c*x))) - (d
^(5/2)*(2*b*B*d - 4*A*c*d + 7*A*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/b^3 - ((c*d
- b*e)^(5/2)*(-2*b*B*c*d + 4*A*c^2*d - 5*b^2*B*e + 3*A*b*c*e)*ArcTanh[(Sqrt[c]*
Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b^3*c^(7/2))
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Maple [B] time = 0.039, size = 823, 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)^(7/2)/(c*x^2+b*x)^2,x)
[Out]
2/3*e^2/c^2*B*(e*x+d)^(3/2)+2*e^3/c^2*A*(e*x+d)^(1/2)-2*d^(7/2)/b^2*arctanh((e*x
+d)^(1/2)/d^(1/2))*B-d^3/b^2*A*(e*x+d)^(1/2)/x+4/b^3*c^2/((b*e-c*d)*c)^(1/2)*arc
tan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*A*d^4-2/b^2*c/((b*e-c*d)*c)^(1/2)*arcta
n(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*B*d^4+e^4*b/c^2*(e*x+d)^(1/2)/(c*e*x+b*e)
*A-e^4*b^2/c^3*(e*x+d)^(1/2)/(c*e*x+b*e)*B-3*e^4*b/c^2/((b*e-c*d)*c)^(1/2)*arcta
n(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*A+5*e^4*b^2/c^3/((b*e-c*d)*c)^(1/2)*arcta
n(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*B+e/b*(e*x+d)^(1/2)/(c*e*x+b*e)*B*d^3+5*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+3*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^2+9*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^2+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^3-3*e^3/c*(e*x+d)^(
1/2)/(c*e*x+b*e)*A*d+3*e^2/b*(e*x+d)^(1/2)/(c*e*x+b*e)*A*d^2-3*e^2/c*(e*x+d)^(1/
2)/(c*e*x+b*e)*B*d^2-13*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-e/b^2*c*(e*x+d)^(1/2)/(c*e*x+b*e)*A*d^3+3*e^3*b/c^2*(e*x+d
)^(1/2)/(c*e*x+b*e)*B*d+4*d^(7/2)/b^3*arctanh((e*x+d)^(1/2)/d^(1/2))*A*c-4*e^3/c
^3*b*B*(e*x+d)^(1/2)-7*e*d^(5/2)/b^2*arctanh((e*x+d)^(1/2)/d^(1/2))*A+6*e^2/c^2*
B*d*(e*x+d)^(1/2)-9*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^3
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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)^(7/2)/(c*x^2 + b*x)^2,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 75.0988, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)^(7/2)/(c*x^2 + b*x)^2,x, algorithm="fricas")
[Out]
[-1/6*(3*((2*(B*b*c^4 - 2*A*c^5)*d^3 + (B*b^2*c^3 + 5*A*b*c^4)*d^2*e - 2*(4*B*b^
3*c^2 - A*b^2*c^3)*d*e^2 + (5*B*b^4*c - 3*A*b^3*c^2)*e^3)*x^2 + (2*(B*b^2*c^3 -
2*A*b*c^4)*d^3 + (B*b^3*c^2 + 5*A*b^2*c^3)*d^2*e - 2*(4*B*b^4*c - A*b^3*c^2)*d*e
^2 + (5*B*b^5 - 3*A*b^4*c)*e^3)*x)*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)) - 3*((7*A*b*c^4*d^2*e + 2*(B
*b*c^4 - 2*A*c^5)*d^3)*x^2 + (7*A*b^2*c^3*d^2*e + 2*(B*b^2*c^3 - 2*A*b*c^4)*d^3)
*x)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 2*(2*B*b^3*c^2*e^3*x^
3 - 3*A*b^2*c^3*d^3 + 2*(10*B*b^3*c^2*d*e^2 - (5*B*b^4*c - 3*A*b^3*c^2)*e^3)*x^2
+ (3*(B*b^2*c^3 - 2*A*b*c^4)*d^3 - 9*(B*b^3*c^2 - A*b^2*c^3)*d^2*e + (29*B*b^4*
c - 9*A*b^3*c^2)*d*e^2 - 3*(5*B*b^5 - 3*A*b^4*c)*e^3)*x)*sqrt(e*x + d))/(b^3*c^4
*x^2 + b^4*c^3*x), 1/6*(6*((2*(B*b*c^4 - 2*A*c^5)*d^3 + (B*b^2*c^3 + 5*A*b*c^4)*
d^2*e - 2*(4*B*b^3*c^2 - A*b^2*c^3)*d*e^2 + (5*B*b^4*c - 3*A*b^3*c^2)*e^3)*x^2 +
(2*(B*b^2*c^3 - 2*A*b*c^4)*d^3 + (B*b^3*c^2 + 5*A*b^2*c^3)*d^2*e - 2*(4*B*b^4*c
- A*b^3*c^2)*d*e^2 + (5*B*b^5 - 3*A*b^4*c)*e^3)*x)*sqrt(-(c*d - b*e)/c)*arctan(
sqrt(e*x + d)/sqrt(-(c*d - b*e)/c)) + 3*((7*A*b*c^4*d^2*e + 2*(B*b*c^4 - 2*A*c^5
)*d^3)*x^2 + (7*A*b^2*c^3*d^2*e + 2*(B*b^2*c^3 - 2*A*b*c^4)*d^3)*x)*sqrt(d)*log(
(e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) + 2*(2*B*b^3*c^2*e^3*x^3 - 3*A*b^2*c^3*
d^3 + 2*(10*B*b^3*c^2*d*e^2 - (5*B*b^4*c - 3*A*b^3*c^2)*e^3)*x^2 + (3*(B*b^2*c^3
- 2*A*b*c^4)*d^3 - 9*(B*b^3*c^2 - A*b^2*c^3)*d^2*e + (29*B*b^4*c - 9*A*b^3*c^2)
*d*e^2 - 3*(5*B*b^5 - 3*A*b^4*c)*e^3)*x)*sqrt(e*x + d))/(b^3*c^4*x^2 + b^4*c^3*x
), -1/6*(6*((7*A*b*c^4*d^2*e + 2*(B*b*c^4 - 2*A*c^5)*d^3)*x^2 + (7*A*b^2*c^3*d^2
*e + 2*(B*b^2*c^3 - 2*A*b*c^4)*d^3)*x)*sqrt(-d)*arctan(sqrt(e*x + d)/sqrt(-d)) +
3*((2*(B*b*c^4 - 2*A*c^5)*d^3 + (B*b^2*c^3 + 5*A*b*c^4)*d^2*e - 2*(4*B*b^3*c^2
- A*b^2*c^3)*d*e^2 + (5*B*b^4*c - 3*A*b^3*c^2)*e^3)*x^2 + (2*(B*b^2*c^3 - 2*A*b*
c^4)*d^3 + (B*b^3*c^2 + 5*A*b^2*c^3)*d^2*e - 2*(4*B*b^4*c - A*b^3*c^2)*d*e^2 + (
5*B*b^5 - 3*A*b^4*c)*e^3)*x)*sqrt((c*d - b*e)/c)*log((c*e*x + 2*c*d - b*e - 2*sq
rt(e*x + d)*c*sqrt((c*d - b*e)/c))/(c*x + b)) - 2*(2*B*b^3*c^2*e^3*x^3 - 3*A*b^2
*c^3*d^3 + 2*(10*B*b^3*c^2*d*e^2 - (5*B*b^4*c - 3*A*b^3*c^2)*e^3)*x^2 + (3*(B*b^
2*c^3 - 2*A*b*c^4)*d^3 - 9*(B*b^3*c^2 - A*b^2*c^3)*d^2*e + (29*B*b^4*c - 9*A*b^3
*c^2)*d*e^2 - 3*(5*B*b^5 - 3*A*b^4*c)*e^3)*x)*sqrt(e*x + d))/(b^3*c^4*x^2 + b^4*
c^3*x), -1/3*(3*((7*A*b*c^4*d^2*e + 2*(B*b*c^4 - 2*A*c^5)*d^3)*x^2 + (7*A*b^2*c^
3*d^2*e + 2*(B*b^2*c^3 - 2*A*b*c^4)*d^3)*x)*sqrt(-d)*arctan(sqrt(e*x + d)/sqrt(-
d)) - 3*((2*(B*b*c^4 - 2*A*c^5)*d^3 + (B*b^2*c^3 + 5*A*b*c^4)*d^2*e - 2*(4*B*b^3
*c^2 - A*b^2*c^3)*d*e^2 + (5*B*b^4*c - 3*A*b^3*c^2)*e^3)*x^2 + (2*(B*b^2*c^3 - 2
*A*b*c^4)*d^3 + (B*b^3*c^2 + 5*A*b^2*c^3)*d^2*e - 2*(4*B*b^4*c - A*b^3*c^2)*d*e^
2 + (5*B*b^5 - 3*A*b^4*c)*e^3)*x)*sqrt(-(c*d - b*e)/c)*arctan(sqrt(e*x + d)/sqrt
(-(c*d - b*e)/c)) - (2*B*b^3*c^2*e^3*x^3 - 3*A*b^2*c^3*d^3 + 2*(10*B*b^3*c^2*d*e
^2 - (5*B*b^4*c - 3*A*b^3*c^2)*e^3)*x^2 + (3*(B*b^2*c^3 - 2*A*b*c^4)*d^3 - 9*(B*
b^3*c^2 - A*b^2*c^3)*d^2*e + (29*B*b^4*c - 9*A*b^3*c^2)*d*e^2 - 3*(5*B*b^5 - 3*A
*b^4*c)*e^3)*x)*sqrt(e*x + d))/(b^3*c^4*x^2 + b^4*c^3*x)]
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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)**(7/2)/(c*x**2+b*x)**2,x)
[Out]
Timed out
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GIAC/XCAS [A] time = 0.317725, size = 863, normalized size = 2.93 \[ \frac{{\left (2 \, B b d^{4} - 4 \, A c d^{4} + 7 \, A b d^{3} e\right )} \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{b^{3} \sqrt{-d}} - \frac{{\left (2 \, B b c^{4} d^{4} - 4 \, A c^{5} d^{4} - B b^{2} c^{3} d^{3} e + 9 \, A b c^{4} d^{3} e - 9 \, B b^{3} c^{2} d^{2} e^{2} - 3 \, A b^{2} c^{3} d^{2} e^{2} + 13 \, B b^{4} c d e^{3} - 5 \, A b^{3} c^{2} d e^{3} - 5 \, B b^{5} e^{4} + 3 \, A b^{4} c e^{4}\right )} \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{\sqrt{-c^{2} d + b c e} b^{3} c^{3}} + \frac{2 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} B c^{4} e^{2} + 9 \, \sqrt{x e + d} B c^{4} d e^{2} - 6 \, \sqrt{x e + d} B b c^{3} e^{3} + 3 \, \sqrt{x e + d} A c^{4} e^{3}\right )}}{3 \, c^{6}} + \frac{{\left (x e + d\right )}^{\frac{3}{2}} B b c^{3} d^{3} e - 2 \,{\left (x e + d\right )}^{\frac{3}{2}} A c^{4} d^{3} e - \sqrt{x e + d} B b c^{3} d^{4} e + 2 \, \sqrt{x e + d} A c^{4} d^{4} e - 3 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{2} c^{2} d^{2} e^{2} + 3 \,{\left (x e + d\right )}^{\frac{3}{2}} A b c^{3} d^{2} e^{2} + 3 \, \sqrt{x e + d} B b^{2} c^{2} d^{3} e^{2} - 4 \, \sqrt{x e + d} A b c^{3} d^{3} e^{2} + 3 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{3} c d e^{3} - 3 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{2} c^{2} d e^{3} - 3 \, \sqrt{x e + d} B b^{3} c d^{2} e^{3} + 3 \, \sqrt{x e + d} A b^{2} c^{2} d^{2} e^{3} -{\left (x e + d\right )}^{\frac{3}{2}} B b^{4} e^{4} +{\left (x e + d\right )}^{\frac{3}{2}} A b^{3} c e^{4} + \sqrt{x e + d} B b^{4} d e^{4} - \sqrt{x e + d} A b^{3} c d e^{4}}{{\left ({\left (x e + d\right )}^{2} c - 2 \,{\left (x e + d\right )} c d + c d^{2} +{\left (x e + d\right )} b e - b d e\right )} b^{2} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)^(7/2)/(c*x^2 + b*x)^2,x, algorithm="giac")
[Out]
(2*B*b*d^4 - 4*A*c*d^4 + 7*A*b*d^3*e)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^3*sqrt(-
d)) - (2*B*b*c^4*d^4 - 4*A*c^5*d^4 - B*b^2*c^3*d^3*e + 9*A*b*c^4*d^3*e - 9*B*b^3
*c^2*d^2*e^2 - 3*A*b^2*c^3*d^2*e^2 + 13*B*b^4*c*d*e^3 - 5*A*b^3*c^2*d*e^3 - 5*B*
b^5*e^4 + 3*A*b^4*c*e^4)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/(sqrt(-c^2
*d + b*c*e)*b^3*c^3) + 2/3*((x*e + d)^(3/2)*B*c^4*e^2 + 9*sqrt(x*e + d)*B*c^4*d*
e^2 - 6*sqrt(x*e + d)*B*b*c^3*e^3 + 3*sqrt(x*e + d)*A*c^4*e^3)/c^6 + ((x*e + d)^
(3/2)*B*b*c^3*d^3*e - 2*(x*e + d)^(3/2)*A*c^4*d^3*e - sqrt(x*e + d)*B*b*c^3*d^4*
e + 2*sqrt(x*e + d)*A*c^4*d^4*e - 3*(x*e + d)^(3/2)*B*b^2*c^2*d^2*e^2 + 3*(x*e +
d)^(3/2)*A*b*c^3*d^2*e^2 + 3*sqrt(x*e + d)*B*b^2*c^2*d^3*e^2 - 4*sqrt(x*e + d)*
A*b*c^3*d^3*e^2 + 3*(x*e + d)^(3/2)*B*b^3*c*d*e^3 - 3*(x*e + d)^(3/2)*A*b^2*c^2*
d*e^3 - 3*sqrt(x*e + d)*B*b^3*c*d^2*e^3 + 3*sqrt(x*e + d)*A*b^2*c^2*d^2*e^3 - (x
*e + d)^(3/2)*B*b^4*e^4 + (x*e + d)^(3/2)*A*b^3*c*e^4 + sqrt(x*e + d)*B*b^4*d*e^
4 - sqrt(x*e + d)*A*b^3*c*d*e^4)/(((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 + (x*
e + d)*b*e - b*d*e)*b^2*c^3)