3.1765 \(\int \frac{(a+b x)^3 (e+f x)^{5/2}}{c+d x} \, dx\)
Optimal. Leaf size=280 \[ \frac{2 b (e+f x)^{7/2} \left (3 a^2 d^2 f^2-3 a b d f (c f+d e)+b^2 \left (c^2 f^2+c d e f+d^2 e^2\right )\right )}{7 d^3 f^3}-\frac{2 b^2 (e+f x)^{9/2} (-3 a d f+b c f+2 b d e)}{9 d^2 f^3}+\frac{2 (b c-a d)^3 (d e-c f)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{13/2}}-\frac{2 \sqrt{e+f x} (b c-a d)^3 (d e-c f)^2}{d^6}-\frac{2 (e+f x)^{3/2} (b c-a d)^3 (d e-c f)}{3 d^5}-\frac{2 (e+f x)^{5/2} (b c-a d)^3}{5 d^4}+\frac{2 b^3 (e+f x)^{11/2}}{11 d f^3} \]
[Out]
(-2*(b*c - a*d)^3*(d*e - c*f)^2*Sqrt[e + f*x])/d^6 - (2*(b*c - a*d)^3*(d*e - c*f
)*(e + f*x)^(3/2))/(3*d^5) - (2*(b*c - a*d)^3*(e + f*x)^(5/2))/(5*d^4) + (2*b*(3
*a^2*d^2*f^2 - 3*a*b*d*f*(d*e + c*f) + b^2*(d^2*e^2 + c*d*e*f + c^2*f^2))*(e + f
*x)^(7/2))/(7*d^3*f^3) - (2*b^2*(2*b*d*e + b*c*f - 3*a*d*f)*(e + f*x)^(9/2))/(9*
d^2*f^3) + (2*b^3*(e + f*x)^(11/2))/(11*d*f^3) + (2*(b*c - a*d)^3*(d*e - c*f)^(5
/2)*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d*e - c*f]])/d^(13/2)
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Rubi [A] time = 0.666971, antiderivative size = 280, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167
\[ \frac{2 b (e+f x)^{7/2} \left (3 a^2 d^2 f^2-3 a b d f (c f+d e)+b^2 \left (c^2 f^2+c d e f+d^2 e^2\right )\right )}{7 d^3 f^3}-\frac{2 b^2 (e+f x)^{9/2} (-3 a d f+b c f+2 b d e)}{9 d^2 f^3}+\frac{2 (b c-a d)^3 (d e-c f)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{13/2}}-\frac{2 \sqrt{e+f x} (b c-a d)^3 (d e-c f)^2}{d^6}-\frac{2 (e+f x)^{3/2} (b c-a d)^3 (d e-c f)}{3 d^5}-\frac{2 (e+f x)^{5/2} (b c-a d)^3}{5 d^4}+\frac{2 b^3 (e+f x)^{11/2}}{11 d f^3} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)^3*(e + f*x)^(5/2))/(c + d*x),x]
[Out]
(-2*(b*c - a*d)^3*(d*e - c*f)^2*Sqrt[e + f*x])/d^6 - (2*(b*c - a*d)^3*(d*e - c*f
)*(e + f*x)^(3/2))/(3*d^5) - (2*(b*c - a*d)^3*(e + f*x)^(5/2))/(5*d^4) + (2*b*(3
*a^2*d^2*f^2 - 3*a*b*d*f*(d*e + c*f) + b^2*(d^2*e^2 + c*d*e*f + c^2*f^2))*(e + f
*x)^(7/2))/(7*d^3*f^3) - (2*b^2*(2*b*d*e + b*c*f - 3*a*d*f)*(e + f*x)^(9/2))/(9*
d^2*f^3) + (2*b^3*(e + f*x)^(11/2))/(11*d*f^3) + (2*(b*c - a*d)^3*(d*e - c*f)^(5
/2)*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d*e - c*f]])/d^(13/2)
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Rubi in Sympy [A] time = 99.3163, size = 282, normalized size = 1.01 \[ \frac{2 b^{3} \left (e + f x\right )^{\frac{11}{2}}}{11 d f^{3}} + \frac{2 b^{2} \left (e + f x\right )^{\frac{9}{2}} \left (3 a d f - b c f - 2 b d e\right )}{9 d^{2} f^{3}} + \frac{2 b \left (e + f x\right )^{\frac{7}{2}} \left (3 a^{2} d^{2} f^{2} - 3 a b c d f^{2} - 3 a b d^{2} e f + b^{2} c^{2} f^{2} + b^{2} c d e f + b^{2} d^{2} e^{2}\right )}{7 d^{3} f^{3}} + \frac{2 \left (e + f x\right )^{\frac{5}{2}} \left (a d - b c\right )^{3}}{5 d^{4}} - \frac{2 \left (e + f x\right )^{\frac{3}{2}} \left (a d - b c\right )^{3} \left (c f - d e\right )}{3 d^{5}} + \frac{2 \sqrt{e + f x} \left (a d - b c\right )^{3} \left (c f - d e\right )^{2}}{d^{6}} - \frac{2 \left (a d - b c\right )^{3} \left (c f - d e\right )^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{d} \sqrt{e + f x}}{\sqrt{c f - d e}} \right )}}{d^{\frac{13}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**3*(f*x+e)**(5/2)/(d*x+c),x)
[Out]
2*b**3*(e + f*x)**(11/2)/(11*d*f**3) + 2*b**2*(e + f*x)**(9/2)*(3*a*d*f - b*c*f
- 2*b*d*e)/(9*d**2*f**3) + 2*b*(e + f*x)**(7/2)*(3*a**2*d**2*f**2 - 3*a*b*c*d*f*
*2 - 3*a*b*d**2*e*f + b**2*c**2*f**2 + b**2*c*d*e*f + b**2*d**2*e**2)/(7*d**3*f*
*3) + 2*(e + f*x)**(5/2)*(a*d - b*c)**3/(5*d**4) - 2*(e + f*x)**(3/2)*(a*d - b*c
)**3*(c*f - d*e)/(3*d**5) + 2*sqrt(e + f*x)*(a*d - b*c)**3*(c*f - d*e)**2/d**6 -
2*(a*d - b*c)**3*(c*f - d*e)**(5/2)*atan(sqrt(d)*sqrt(e + f*x)/sqrt(c*f - d*e))
/d**(13/2)
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Mathematica [A] time = 0.882474, size = 435, normalized size = 1.55 \[ \frac{2 \sqrt{e+f x} \left (231 a^3 d^3 f^3 \left (15 c^2 f^2-5 c d f (7 e+f x)+d^2 \left (23 e^2+11 e f x+3 f^2 x^2\right )\right )+99 a^2 b d^2 f^2 \left (-105 c^3 f^3+35 c^2 d f^2 (7 e+f x)-7 c d^2 f \left (23 e^2+11 e f x+3 f^2 x^2\right )+15 d^3 (e+f x)^3\right )-33 a b^2 d f \left (-315 c^4 f^4+105 c^3 d f^3 (7 e+f x)-21 c^2 d^2 f^2 \left (23 e^2+11 e f x+3 f^2 x^2\right )+45 c d^3 f (e+f x)^3+5 d^4 (2 e-7 f x) (e+f x)^3\right )+b^3 \left (-3465 c^5 f^5+1155 c^4 d f^4 (7 e+f x)-231 c^3 d^2 f^3 \left (23 e^2+11 e f x+3 f^2 x^2\right )+495 c^2 d^3 f^2 (e+f x)^3+55 c d^4 f (2 e-7 f x) (e+f x)^3+5 d^5 (e+f x)^3 \left (8 e^2-28 e f x+63 f^2 x^2\right )\right )\right )}{3465 d^6 f^3}+\frac{2 (b c-a d)^3 (d e-c f)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{13/2}} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)^3*(e + f*x)^(5/2))/(c + d*x),x]
[Out]
(2*Sqrt[e + f*x]*(231*a^3*d^3*f^3*(15*c^2*f^2 - 5*c*d*f*(7*e + f*x) + d^2*(23*e^
2 + 11*e*f*x + 3*f^2*x^2)) + 99*a^2*b*d^2*f^2*(-105*c^3*f^3 + 15*d^3*(e + f*x)^3
+ 35*c^2*d*f^2*(7*e + f*x) - 7*c*d^2*f*(23*e^2 + 11*e*f*x + 3*f^2*x^2)) - 33*a*
b^2*d*f*(-315*c^4*f^4 + 45*c*d^3*f*(e + f*x)^3 + 5*d^4*(2*e - 7*f*x)*(e + f*x)^3
+ 105*c^3*d*f^3*(7*e + f*x) - 21*c^2*d^2*f^2*(23*e^2 + 11*e*f*x + 3*f^2*x^2)) +
b^3*(-3465*c^5*f^5 + 495*c^2*d^3*f^2*(e + f*x)^3 + 55*c*d^4*f*(2*e - 7*f*x)*(e
+ f*x)^3 + 1155*c^4*d*f^4*(7*e + f*x) - 231*c^3*d^2*f^3*(23*e^2 + 11*e*f*x + 3*f
^2*x^2) + 5*d^5*(e + f*x)^3*(8*e^2 - 28*e*f*x + 63*f^2*x^2))))/(3465*d^6*f^3) +
(2*(b*c - a*d)^3*(d*e - c*f)^(5/2)*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d*e - c*
f]])/d^(13/2)
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Maple [B] time = 0.021, size = 1437, normalized size = 5.1 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^3*(f*x+e)^(5/2)/(d*x+c),x)
[Out]
2*f^2/d^3*a^3*c^2*(f*x+e)^(1/2)+6/7/f/d*(f*x+e)^(7/2)*a^2*b+2/7/f/d^3*(f*x+e)^(7
/2)*b^3*c^2-2/9/f^2/d^2*(f*x+e)^(9/2)*b^3*c-2/3*f/d^2*(f*x+e)^(3/2)*a^3*c-2*f^2/
d^6*b^3*c^5*(f*x+e)^(1/2)+2/3/f^2/d*(f*x+e)^(9/2)*a*b^2+6/5/d^3*(f*x+e)^(5/2)*a*
b^2*c^2-2/d^4*b^3*c^3*e^2*(f*x+e)^(1/2)+2/d*a^3*e^2*(f*x+e)^(1/2)-2/5/d^4*(f*x+e
)^(5/2)*b^3*c^3+2/3/d*(f*x+e)^(3/2)*a^3*e+2/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(
1/2)*d/((c*f-d*e)*d)^(1/2))*a^3*e^3-4/9/f^3/d*(f*x+e)^(9/2)*b^3*e+2/3*f/d^5*(f*x
+e)^(3/2)*b^3*c^4+2/11*b^3*(f*x+e)^(11/2)/d/f^3-18*f^2/d^3/((c*f-d*e)*d)^(1/2)*a
rctan((f*x+e)^(1/2)*d/((c*f-d*e)*d)^(1/2))*a^2*b*c^3*e+18*f/d^2/((c*f-d*e)*d)^(1
/2)*arctan((f*x+e)^(1/2)*d/((c*f-d*e)*d)^(1/2))*a^2*b*c^2*e^2-18*f/d^3/((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^3*e^2+2/5/d*(f*x+
e)^(5/2)*a^3-6*f/d/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)*d/((c*f-d*e)*d)^(1/2
))*a^3*c*e^2+6*f^3/d^4/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)*d/((c*f-d*e)*d)^
(1/2))*a^2*b*c^4-6*f^3/d^5/((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^5-6*f^2/d^5/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)*d/((c*f-
d*e)*d)^(1/2))*b^3*c^5*e+6*f/d^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^4*e^2-12*f/d^4*a*b^2*c^3*e*(f*x+e)^(1/2)+18*f^2/d^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^4*e-2/3/d^4*(
f*x+e)^(3/2)*b^3*c^3*e+4*f/d^5*b^3*c^4*e*(f*x+e)^(1/2)+2*f/d^3*(f*x+e)^(3/2)*a^2
*b*c^2+2*f^3/d^6/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)*d/((c*f-d*e)*d)^(1/2))
*b^3*c^6-6/5/d^2*(f*x+e)^(5/2)*a^2*b*c+2/7/f^3/d*(f*x+e)^(7/2)*b^3*e^2-6/7/f/d^2
*(f*x+e)^(7/2)*a*b^2*c-6/7/f^2/d*(f*x+e)^(7/2)*a*b^2*e-2*f/d^4*(f*x+e)^(3/2)*a*b
^2*c^3-6*f^2/d^4*a^2*b*c^3*(f*x+e)^(1/2)+2/7/f^2/d^2*(f*x+e)^(7/2)*b^3*c*e-4*f/d
^2*a^3*c*e*(f*x+e)^(1/2)-2*f^3/d^3/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)*d/((
c*f-d*e)*d)^(1/2))*a^3*c^3-2/d^3/((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*e^3+6/d^3*a*b^2*c^2*e^2*(f*x+e)^(1/2)+2/d^3*(f*x+e)^(3/
2)*a*b^2*c^2*e-2/d^2*(f*x+e)^(3/2)*a^2*b*c*e-6/d^2*a^2*b*c*e^2*(f*x+e)^(1/2)+6*f
^2/d^5*a*b^2*c^4*(f*x+e)^(1/2)+12*f/d^3*a^2*b*c^2*e*(f*x+e)^(1/2)+6*f^2/d^2/((c*
f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)*d/((c*f-d*e)*d)^(1/2))*a^3*c^2*e+6/d^2/((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*e^3-6/d/((
c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)*d/((c*f-d*e)*d)^(1/2))*a^2*b*c*e^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*(f*x + e)^(5/2)/(d*x + c),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.23622, 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*(f*x + e)^(5/2)/(d*x + c),x, algorithm="fricas")
[Out]
[-1/3465*(3465*((b^3*c^3*d^2 - 3*a*b^2*c^2*d^3 + 3*a^2*b*c*d^4 - a^3*d^5)*e^2*f^
3 - 2*(b^3*c^4*d - 3*a*b^2*c^3*d^2 + 3*a^2*b*c^2*d^3 - a^3*c*d^4)*e*f^4 + (b^3*c
^5 - 3*a*b^2*c^4*d + 3*a^2*b*c^3*d^2 - a^3*c^2*d^3)*f^5)*sqrt((d*e - c*f)/d)*log
((d*f*x + 2*d*e - c*f - 2*sqrt(f*x + e)*d*sqrt((d*e - c*f)/d))/(d*x + c)) - 2*(3
15*b^3*d^5*f^5*x^5 + 40*b^3*d^5*e^5 + 110*(b^3*c*d^4 - 3*a*b^2*d^5)*e^4*f + 495*
(b^3*c^2*d^3 - 3*a*b^2*c*d^4 + 3*a^2*b*d^5)*e^3*f^2 - 5313*(b^3*c^3*d^2 - 3*a*b^
2*c^2*d^3 + 3*a^2*b*c*d^4 - a^3*d^5)*e^2*f^3 + 8085*(b^3*c^4*d - 3*a*b^2*c^3*d^2
+ 3*a^2*b*c^2*d^3 - a^3*c*d^4)*e*f^4 - 3465*(b^3*c^5 - 3*a*b^2*c^4*d + 3*a^2*b*
c^3*d^2 - a^3*c^2*d^3)*f^5 + 35*(23*b^3*d^5*e*f^4 - 11*(b^3*c*d^4 - 3*a*b^2*d^5)
*f^5)*x^4 + 5*(113*b^3*d^5*e^2*f^3 - 209*(b^3*c*d^4 - 3*a*b^2*d^5)*e*f^4 + 99*(b
^3*c^2*d^3 - 3*a*b^2*c*d^4 + 3*a^2*b*d^5)*f^5)*x^3 + 3*(5*b^3*d^5*e^3*f^2 - 275*
(b^3*c*d^4 - 3*a*b^2*d^5)*e^2*f^3 + 495*(b^3*c^2*d^3 - 3*a*b^2*c*d^4 + 3*a^2*b*d
^5)*e*f^4 - 231*(b^3*c^3*d^2 - 3*a*b^2*c^2*d^3 + 3*a^2*b*c*d^4 - a^3*d^5)*f^5)*x
^2 - (20*b^3*d^5*e^4*f + 55*(b^3*c*d^4 - 3*a*b^2*d^5)*e^3*f^2 - 1485*(b^3*c^2*d^
3 - 3*a*b^2*c*d^4 + 3*a^2*b*d^5)*e^2*f^3 + 2541*(b^3*c^3*d^2 - 3*a*b^2*c^2*d^3 +
3*a^2*b*c*d^4 - a^3*d^5)*e*f^4 - 1155*(b^3*c^4*d - 3*a*b^2*c^3*d^2 + 3*a^2*b*c^
2*d^3 - a^3*c*d^4)*f^5)*x)*sqrt(f*x + e))/(d^6*f^3), 2/3465*(3465*((b^3*c^3*d^2
- 3*a*b^2*c^2*d^3 + 3*a^2*b*c*d^4 - a^3*d^5)*e^2*f^3 - 2*(b^3*c^4*d - 3*a*b^2*c^
3*d^2 + 3*a^2*b*c^2*d^3 - a^3*c*d^4)*e*f^4 + (b^3*c^5 - 3*a*b^2*c^4*d + 3*a^2*b*
c^3*d^2 - a^3*c^2*d^3)*f^5)*sqrt(-(d*e - c*f)/d)*arctan(sqrt(f*x + e)/sqrt(-(d*e
- c*f)/d)) + (315*b^3*d^5*f^5*x^5 + 40*b^3*d^5*e^5 + 110*(b^3*c*d^4 - 3*a*b^2*d
^5)*e^4*f + 495*(b^3*c^2*d^3 - 3*a*b^2*c*d^4 + 3*a^2*b*d^5)*e^3*f^2 - 5313*(b^3*
c^3*d^2 - 3*a*b^2*c^2*d^3 + 3*a^2*b*c*d^4 - a^3*d^5)*e^2*f^3 + 8085*(b^3*c^4*d -
3*a*b^2*c^3*d^2 + 3*a^2*b*c^2*d^3 - a^3*c*d^4)*e*f^4 - 3465*(b^3*c^5 - 3*a*b^2*
c^4*d + 3*a^2*b*c^3*d^2 - a^3*c^2*d^3)*f^5 + 35*(23*b^3*d^5*e*f^4 - 11*(b^3*c*d^
4 - 3*a*b^2*d^5)*f^5)*x^4 + 5*(113*b^3*d^5*e^2*f^3 - 209*(b^3*c*d^4 - 3*a*b^2*d^
5)*e*f^4 + 99*(b^3*c^2*d^3 - 3*a*b^2*c*d^4 + 3*a^2*b*d^5)*f^5)*x^3 + 3*(5*b^3*d^
5*e^3*f^2 - 275*(b^3*c*d^4 - 3*a*b^2*d^5)*e^2*f^3 + 495*(b^3*c^2*d^3 - 3*a*b^2*c
*d^4 + 3*a^2*b*d^5)*e*f^4 - 231*(b^3*c^3*d^2 - 3*a*b^2*c^2*d^3 + 3*a^2*b*c*d^4 -
a^3*d^5)*f^5)*x^2 - (20*b^3*d^5*e^4*f + 55*(b^3*c*d^4 - 3*a*b^2*d^5)*e^3*f^2 -
1485*(b^3*c^2*d^3 - 3*a*b^2*c*d^4 + 3*a^2*b*d^5)*e^2*f^3 + 2541*(b^3*c^3*d^2 - 3
*a*b^2*c^2*d^3 + 3*a^2*b*c*d^4 - a^3*d^5)*e*f^4 - 1155*(b^3*c^4*d - 3*a*b^2*c^3*
d^2 + 3*a^2*b*c^2*d^3 - a^3*c*d^4)*f^5)*x)*sqrt(f*x + e))/(d^6*f^3)]
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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*(f*x+e)**(5/2)/(d*x+c),x)
[Out]
Timed out
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GIAC/XCAS [A] time = 0.237423, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^3*(f*x + e)^(5/2)/(d*x + c),x, algorithm="giac")
[Out]
Done