3.1772 \(\int \frac{(a+b x)^3}{(c+d x) (e+f x)^{9/2}} \, dx\)
Optimal. Leaf size=260 \[ -\frac{2 (b e-a f) \left (a^2 d^2 f^2+a b d f (d e-3 c f)+b^2 \left (3 c^2 f^2-3 c d e f+d^2 e^2\right )\right )}{3 f^3 (e+f x)^{3/2} (d e-c f)^3}+\frac{2 (b e-a f)^2 (a d f-3 b c f+2 b d e)}{5 f^3 (e+f x)^{5/2} (d e-c f)^2}-\frac{2 (b e-a f)^3}{7 f^3 (e+f x)^{7/2} (d e-c f)}-\frac{2 (b c-a d)^3}{\sqrt{e+f x} (d e-c f)^4}+\frac{2 \sqrt{d} (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{(d e-c f)^{9/2}} \]
[Out]
(-2*(b*e - a*f)^3)/(7*f^3*(d*e - c*f)*(e + f*x)^(7/2)) + (2*(b*e - a*f)^2*(2*b*d
*e - 3*b*c*f + a*d*f))/(5*f^3*(d*e - c*f)^2*(e + f*x)^(5/2)) - (2*(b*e - a*f)*(a
^2*d^2*f^2 + a*b*d*f*(d*e - 3*c*f) + b^2*(d^2*e^2 - 3*c*d*e*f + 3*c^2*f^2)))/(3*
f^3*(d*e - c*f)^3*(e + f*x)^(3/2)) - (2*(b*c - a*d)^3)/((d*e - c*f)^4*Sqrt[e + f
*x]) + (2*Sqrt[d]*(b*c - a*d)^3*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d*e - c*f]]
)/(d*e - c*f)^(9/2)
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Rubi [A] time = 0.970684, antiderivative size = 260, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125
\[ -\frac{2 (b e-a f) \left (a^2 d^2 f^2+a b d f (d e-3 c f)+b^2 \left (3 c^2 f^2-3 c d e f+d^2 e^2\right )\right )}{3 f^3 (e+f x)^{3/2} (d e-c f)^3}+\frac{2 (b e-a f)^2 (a d f-3 b c f+2 b d e)}{5 f^3 (e+f x)^{5/2} (d e-c f)^2}-\frac{2 (b e-a f)^3}{7 f^3 (e+f x)^{7/2} (d e-c f)}-\frac{2 (b c-a d)^3}{\sqrt{e+f x} (d e-c f)^4}+\frac{2 \sqrt{d} (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{(d e-c f)^{9/2}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^3/((c + d*x)*(e + f*x)^(9/2)),x]
[Out]
(-2*(b*e - a*f)^3)/(7*f^3*(d*e - c*f)*(e + f*x)^(7/2)) + (2*(b*e - a*f)^2*(2*b*d
*e - 3*b*c*f + a*d*f))/(5*f^3*(d*e - c*f)^2*(e + f*x)^(5/2)) - (2*(b*e - a*f)*(a
^2*d^2*f^2 + a*b*d*f*(d*e - 3*c*f) + b^2*(d^2*e^2 - 3*c*d*e*f + 3*c^2*f^2)))/(3*
f^3*(d*e - c*f)^3*(e + f*x)^(3/2)) - (2*(b*c - a*d)^3)/((d*e - c*f)^4*Sqrt[e + f
*x]) + (2*Sqrt[d]*(b*c - a*d)^3*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d*e - c*f]]
)/(d*e - c*f)^(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)**3/(d*x+c)/(f*x+e)**(9/2),x)
[Out]
Timed out
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Mathematica [A] time = 1.22391, size = 246, normalized size = 0.95 \[ \frac{2 \sqrt{d} (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{(d e-c f)^{9/2}}-\frac{2 \left (35 (e+f x)^2 (b e-a f) (d e-c f) \left (a^2 d^2 f^2+a b d f (d e-3 c f)+b^2 \left (3 c^2 f^2-3 c d e f+d^2 e^2\right )\right )+105 f^3 (e+f x)^3 (b c-a d)^3-21 (e+f x) (b e-a f)^2 (d e-c f)^2 (a d f-3 b c f+2 b d e)+15 (b e-a f)^3 (d e-c f)^3\right )}{105 f^3 (e+f x)^{7/2} (d e-c f)^4} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^3/((c + d*x)*(e + f*x)^(9/2)),x]
[Out]
(-2*(15*(b*e - a*f)^3*(d*e - c*f)^3 - 21*(b*e - a*f)^2*(d*e - c*f)^2*(2*b*d*e -
3*b*c*f + a*d*f)*(e + f*x) + 35*(b*e - a*f)*(d*e - c*f)*(a^2*d^2*f^2 + a*b*d*f*(
d*e - 3*c*f) + b^2*(d^2*e^2 - 3*c*d*e*f + 3*c^2*f^2))*(e + f*x)^2 + 105*(b*c - a
*d)^3*f^3*(e + f*x)^3))/(105*f^3*(d*e - c*f)^4*(e + f*x)^(7/2)) + (2*Sqrt[d]*(b*
c - a*d)^3*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d*e - c*f]])/(d*e - c*f)^(9/2)
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Maple [B] time = 0.032, size = 756, normalized size = 2.9 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^3/(d*x+c)/(f*x+e)^(9/2),x)
[Out]
-2/7/(c*f-d*e)/(f*x+e)^(7/2)*a^3+6/7/f/(c*f-d*e)/(f*x+e)^(7/2)*a^2*b*e-6/7/f^2/(
c*f-d*e)/(f*x+e)^(7/2)*a*b^2*e^2+2/7/f^3/(c*f-d*e)/(f*x+e)^(7/2)*b^3*e^3+2/5/(c*
f-d*e)^2/(f*x+e)^(5/2)*a^3*d-6/5/(c*f-d*e)^2/(f*x+e)^(5/2)*a^2*b*c+12/5/f/(c*f-d
*e)^2/(f*x+e)^(5/2)*a*b^2*c*e-6/5/f^2/(c*f-d*e)^2/(f*x+e)^(5/2)*a*b^2*d*e^2-6/5/
f^2/(c*f-d*e)^2/(f*x+e)^(5/2)*b^3*c*e^2+4/5/f^3/(c*f-d*e)^2/(f*x+e)^(5/2)*b^3*d*
e^3-2/3/(c*f-d*e)^3/(f*x+e)^(3/2)*a^3*d^2+2/(c*f-d*e)^3/(f*x+e)^(3/2)*a^2*b*c*d-
2/(c*f-d*e)^3/(f*x+e)^(3/2)*a*b^2*c^2+2/f/(c*f-d*e)^3/(f*x+e)^(3/2)*b^3*c^2*e-2/
f^2/(c*f-d*e)^3/(f*x+e)^(3/2)*b^3*c*d*e^2+2/3/f^3/(c*f-d*e)^3/(f*x+e)^(3/2)*b^3*
d^2*e^3+2/(c*f-d*e)^4/(f*x+e)^(1/2)*a^3*d^3-6/(c*f-d*e)^4/(f*x+e)^(1/2)*a^2*c*b*
d^2+6/(c*f-d*e)^4/(f*x+e)^(1/2)*a*b^2*c^2*d-2/(c*f-d*e)^4/(f*x+e)^(1/2)*b^3*c^3+
2*d^4/(c*f-d*e)^4/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)*d/((c*f-d*e)*d)^(1/2)
)*a^3-6*d^3/(c*f-d*e)^4/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)*d/((c*f-d*e)*d)
^(1/2))*a^2*c*b+6*d^2/(c*f-d*e)^4/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)*d/((c
*f-d*e)*d)^(1/2))*a*b^2*c^2-2*d/(c*f-d*e)^4/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(
1/2)*d/((c*f-d*e)*d)^(1/2))*b^3*c^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)^3/((d*x + c)*(f*x + e)^(9/2)),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.23939, 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)^3/((d*x + c)*(f*x + e)^(9/2)),x, algorithm="fricas")
[Out]
[-1/105*(16*b^3*d^3*e^6 + 30*a^3*c^3*f^6 + 210*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*
b*c*d^2 - a^3*d^3)*f^6*x^3 - 4*(19*b^3*c*d^2 - 9*a*b^2*d^3)*e^5*f + 6*(29*b^3*c^
2*d - 39*a*b^2*c*d^2 + 15*a^2*b*d^3)*e^4*f^2 + 8*(12*b^3*c^3 - 60*a*b^2*c^2*d +
87*a^2*b*c*d^2 - 44*a^3*d^3)*e^3*f^3 + 4*(12*a*b^2*c^3 - 48*a^2*b*c^2*d + 61*a^3
*c*d^2)*e^2*f^4 + 12*(3*a^2*b*c^3 - 11*a^3*c^2*d)*e*f^5 + 70*(b^3*d^3*e^4*f^2 -
4*b^3*c*d^2*e^3*f^3 + 6*b^3*c^2*d*e^2*f^4 + 2*(3*b^3*c^3 - 15*a*b^2*c^2*d + 15*a
^2*b*c*d^2 - 5*a^3*d^3)*e*f^5 + (3*a*b^2*c^3 - 3*a^2*b*c^2*d + a^3*c*d^2)*f^6)*x
^2 + 105*((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*f^6*x^3 + 3*(b^3*c
^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*e*f^5*x^2 + 3*(b^3*c^3 - 3*a*b^2*c
^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*e^2*f^4*x + (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c
*d^2 - a^3*d^3)*e^3*f^3)*sqrt(f*x + e)*sqrt(d/(d*e - c*f))*log((d*f*x + 2*d*e -
c*f - 2*(d*e - c*f)*sqrt(f*x + e)*sqrt(d/(d*e - c*f)))/(d*x + c)) + 14*(4*b^3*d^
3*e^5*f - (19*b^3*c*d^2 - 9*a*b^2*d^3)*e^4*f^2 + 36*(b^3*c^2*d - a*b^2*c*d^2)*e^
3*f^3 + 2*(12*b^3*c^3 - 60*a*b^2*c^2*d + 87*a^2*b*c*d^2 - 29*a^3*d^3)*e^2*f^4 +
4*(3*a*b^2*c^3 - 12*a^2*b*c^2*d + 4*a^3*c*d^2)*e*f^5 + 3*(3*a^2*b*c^3 - a^3*c^2*
d)*f^6)*x)/((d^4*e^7*f^3 - 4*c*d^3*e^6*f^4 + 6*c^2*d^2*e^5*f^5 - 4*c^3*d*e^4*f^6
+ c^4*e^3*f^7 + (d^4*e^4*f^6 - 4*c*d^3*e^3*f^7 + 6*c^2*d^2*e^2*f^8 - 4*c^3*d*e*
f^9 + c^4*f^10)*x^3 + 3*(d^4*e^5*f^5 - 4*c*d^3*e^4*f^6 + 6*c^2*d^2*e^3*f^7 - 4*c
^3*d*e^2*f^8 + c^4*e*f^9)*x^2 + 3*(d^4*e^6*f^4 - 4*c*d^3*e^5*f^5 + 6*c^2*d^2*e^4
*f^6 - 4*c^3*d*e^3*f^7 + c^4*e^2*f^8)*x)*sqrt(f*x + e)), -2/105*(8*b^3*d^3*e^6 +
15*a^3*c^3*f^6 + 105*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*f^6*x^
3 - 2*(19*b^3*c*d^2 - 9*a*b^2*d^3)*e^5*f + 3*(29*b^3*c^2*d - 39*a*b^2*c*d^2 + 15
*a^2*b*d^3)*e^4*f^2 + 4*(12*b^3*c^3 - 60*a*b^2*c^2*d + 87*a^2*b*c*d^2 - 44*a^3*d
^3)*e^3*f^3 + 2*(12*a*b^2*c^3 - 48*a^2*b*c^2*d + 61*a^3*c*d^2)*e^2*f^4 + 6*(3*a^
2*b*c^3 - 11*a^3*c^2*d)*e*f^5 + 35*(b^3*d^3*e^4*f^2 - 4*b^3*c*d^2*e^3*f^3 + 6*b^
3*c^2*d*e^2*f^4 + 2*(3*b^3*c^3 - 15*a*b^2*c^2*d + 15*a^2*b*c*d^2 - 5*a^3*d^3)*e*
f^5 + (3*a*b^2*c^3 - 3*a^2*b*c^2*d + a^3*c*d^2)*f^6)*x^2 - 105*((b^3*c^3 - 3*a*b
^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*f^6*x^3 + 3*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2
*b*c*d^2 - a^3*d^3)*e*f^5*x^2 + 3*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3
*d^3)*e^2*f^4*x + (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*e^3*f^3)*s
qrt(f*x + e)*sqrt(-d/(d*e - c*f))*arctan(-(d*e - c*f)*sqrt(-d/(d*e - c*f))/(sqrt
(f*x + e)*d)) + 7*(4*b^3*d^3*e^5*f - (19*b^3*c*d^2 - 9*a*b^2*d^3)*e^4*f^2 + 36*(
b^3*c^2*d - a*b^2*c*d^2)*e^3*f^3 + 2*(12*b^3*c^3 - 60*a*b^2*c^2*d + 87*a^2*b*c*d
^2 - 29*a^3*d^3)*e^2*f^4 + 4*(3*a*b^2*c^3 - 12*a^2*b*c^2*d + 4*a^3*c*d^2)*e*f^5
+ 3*(3*a^2*b*c^3 - a^3*c^2*d)*f^6)*x)/((d^4*e^7*f^3 - 4*c*d^3*e^6*f^4 + 6*c^2*d^
2*e^5*f^5 - 4*c^3*d*e^4*f^6 + c^4*e^3*f^7 + (d^4*e^4*f^6 - 4*c*d^3*e^3*f^7 + 6*c
^2*d^2*e^2*f^8 - 4*c^3*d*e*f^9 + c^4*f^10)*x^3 + 3*(d^4*e^5*f^5 - 4*c*d^3*e^4*f^
6 + 6*c^2*d^2*e^3*f^7 - 4*c^3*d*e^2*f^8 + c^4*e*f^9)*x^2 + 3*(d^4*e^6*f^4 - 4*c*
d^3*e^5*f^5 + 6*c^2*d^2*e^4*f^6 - 4*c^3*d*e^3*f^7 + c^4*e^2*f^8)*x)*sqrt(f*x + e
))]
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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)**3/(d*x+c)/(f*x+e)**(9/2),x)
[Out]
Timed out
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GIAC/XCAS [A] time = 0.231277, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^3/((d*x + c)*(f*x + e)^(9/2)),x, algorithm="giac")
[Out]
Done