3.782 \(\int \frac{x^3 (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx\)

Optimal. Leaf size=377 \[ \frac{\sqrt{a+b x} (c+d x)^{5/2} \left (231 a^2 d^2+2 b d x (59 b c-99 a d)-156 a b c d+5 b^2 c^2\right )}{24 b^4 d (b c-a d)}-\frac{5 (b c-a d) \left (231 a^3 d^3-189 a^2 b c d^2+21 a b^2 c^2 d+b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{13/2} d^{3/2}}-\frac{5 \sqrt{a+b x} \sqrt{c+d x} \left (231 a^3 d^3-189 a^2 b c d^2+21 a b^2 c^2 d+b^3 c^3\right )}{64 b^6 d}-\frac{5 \sqrt{a+b x} (c+d x)^{3/2} \left (231 a^3 d^3-189 a^2 b c d^2+21 a b^2 c^2 d+b^3 c^3\right )}{96 b^5 d (b c-a d)}-\frac{2 x^2 (c+d x)^{5/2} (6 b c-11 a d)}{3 b^2 \sqrt{a+b x} (b c-a d)}-\frac{2 x^3 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}} \]

[Out]

(-5*(b^3*c^3 + 21*a*b^2*c^2*d - 189*a^2*b*c*d^2 + 231*a^3*d^3)*Sqrt[a + b*x]*Sqr
t[c + d*x])/(64*b^6*d) - (5*(b^3*c^3 + 21*a*b^2*c^2*d - 189*a^2*b*c*d^2 + 231*a^
3*d^3)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(96*b^5*d*(b*c - a*d)) - (2*x^3*(c + d*x)^
(5/2))/(3*b*(a + b*x)^(3/2)) - (2*(6*b*c - 11*a*d)*x^2*(c + d*x)^(5/2))/(3*b^2*(
b*c - a*d)*Sqrt[a + b*x]) + (Sqrt[a + b*x]*(c + d*x)^(5/2)*(5*b^2*c^2 - 156*a*b*
c*d + 231*a^2*d^2 + 2*b*d*(59*b*c - 99*a*d)*x))/(24*b^4*d*(b*c - a*d)) - (5*(b*c
- a*d)*(b^3*c^3 + 21*a*b^2*c^2*d - 189*a^2*b*c*d^2 + 231*a^3*d^3)*ArcTanh[(Sqrt
[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(64*b^(13/2)*d^(3/2))

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Rubi [A]  time = 0.882316, antiderivative size = 377, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{\sqrt{a+b x} (c+d x)^{5/2} \left (231 a^2 d^2+2 b d x (59 b c-99 a d)-156 a b c d+5 b^2 c^2\right )}{24 b^4 d (b c-a d)}-\frac{5 (b c-a d) \left (231 a^3 d^3-189 a^2 b c d^2+21 a b^2 c^2 d+b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{13/2} d^{3/2}}-\frac{5 \sqrt{a+b x} \sqrt{c+d x} \left (231 a^3 d^3-189 a^2 b c d^2+21 a b^2 c^2 d+b^3 c^3\right )}{64 b^6 d}-\frac{5 \sqrt{a+b x} (c+d x)^{3/2} \left (231 a^3 d^3-189 a^2 b c d^2+21 a b^2 c^2 d+b^3 c^3\right )}{96 b^5 d (b c-a d)}-\frac{2 x^2 (c+d x)^{5/2} (6 b c-11 a d)}{3 b^2 \sqrt{a+b x} (b c-a d)}-\frac{2 x^3 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-5*(b^3*c^3 + 21*a*b^2*c^2*d - 189*a^2*b*c*d^2 + 231*a^3*d^3)*Sqrt[a + b*x]*Sqr
t[c + d*x])/(64*b^6*d) - (5*(b^3*c^3 + 21*a*b^2*c^2*d - 189*a^2*b*c*d^2 + 231*a^
3*d^3)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(96*b^5*d*(b*c - a*d)) - (2*x^3*(c + d*x)^
(5/2))/(3*b*(a + b*x)^(3/2)) - (2*(6*b*c - 11*a*d)*x^2*(c + d*x)^(5/2))/(3*b^2*(
b*c - a*d)*Sqrt[a + b*x]) + (Sqrt[a + b*x]*(c + d*x)^(5/2)*(5*b^2*c^2 - 156*a*b*
c*d + 231*a^2*d^2 + 2*b*d*(59*b*c - 99*a*d)*x))/(24*b^4*d*(b*c - a*d)) - (5*(b*c
- a*d)*(b^3*c^3 + 21*a*b^2*c^2*d - 189*a^2*b*c*d^2 + 231*a^3*d^3)*ArcTanh[(Sqrt
[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(64*b^(13/2)*d^(3/2))

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Rubi in Sympy [A]  time = 81.0518, size = 326, normalized size = 0.86 \[ - \frac{2 x^{3} \left (c + d x\right )^{\frac{5}{2}}}{3 b \left (a + b x\right )^{\frac{3}{2}}} + \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (\frac{3465 a^{2} d^{2}}{16} - \frac{1071 a b c d}{8} + \frac{45 b^{2} c^{2}}{16} - \frac{9 b d x \left (77 a d - 41 b c\right )}{4}\right )}{18 b^{5} d} - \frac{5 \sqrt{a + b x} \sqrt{c + d x} \left (231 a^{3} d^{3} - 189 a^{2} b c d^{2} + 21 a b^{2} c^{2} d + b^{3} c^{3}\right )}{64 b^{6} d} + \frac{5 \left (a d - b c\right ) \left (231 a^{3} d^{3} - 189 a^{2} b c d^{2} + 21 a b^{2} c^{2} d + b^{3} c^{3}\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{64 b^{\frac{13}{2}} d^{\frac{3}{2}}} - \frac{2 x^{3} \left (c + d x\right )^{\frac{3}{2}} \left (11 a d - 6 b c\right )}{3 a b^{2} \sqrt{a + b x}} + \frac{x^{2} \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (33 a d - 16 b c\right )}{4 a b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(d*x+c)**(5/2)/(b*x+a)**(5/2),x)

[Out]

-2*x**3*(c + d*x)**(5/2)/(3*b*(a + b*x)**(3/2)) + sqrt(a + b*x)*(c + d*x)**(3/2)
*(3465*a**2*d**2/16 - 1071*a*b*c*d/8 + 45*b**2*c**2/16 - 9*b*d*x*(77*a*d - 41*b*
c)/4)/(18*b**5*d) - 5*sqrt(a + b*x)*sqrt(c + d*x)*(231*a**3*d**3 - 189*a**2*b*c*
d**2 + 21*a*b**2*c**2*d + b**3*c**3)/(64*b**6*d) + 5*(a*d - b*c)*(231*a**3*d**3
- 189*a**2*b*c*d**2 + 21*a*b**2*c**2*d + b**3*c**3)*atanh(sqrt(b)*sqrt(c + d*x)/
(sqrt(d)*sqrt(a + b*x)))/(64*b**(13/2)*d**(3/2)) - 2*x**3*(c + d*x)**(3/2)*(11*a
*d - 6*b*c)/(3*a*b**2*sqrt(a + b*x)) + x**2*sqrt(a + b*x)*(c + d*x)**(3/2)*(33*a
*d - 16*b*c)/(4*a*b**3)

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Mathematica [A]  time = 0.449525, size = 296, normalized size = 0.79 \[ \frac{\sqrt{c+d x} \left (-3465 a^5 d^3+105 a^4 b d^2 (49 c-44 d x)-21 a^3 b^2 d \left (83 c^2-334 c d x+33 d^2 x^2\right )+3 a^2 b^3 \left (5 c^3-824 c^2 d x+387 c d^2 x^2+66 d^3 x^3\right )-a b^4 x \left (-30 c^3+483 c^2 d x+316 c d^2 x^2+88 d^3 x^3\right )+b^5 x^2 \left (15 c^3+118 c^2 d x+136 c d^2 x^2+48 d^3 x^3\right )\right )}{192 b^6 d (a+b x)^{3/2}}-\frac{5 (b c-a d) \left (231 a^3 d^3-189 a^2 b c d^2+21 a b^2 c^2 d+b^3 c^3\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{128 b^{13/2} d^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[c + d*x]*(-3465*a^5*d^3 + 105*a^4*b*d^2*(49*c - 44*d*x) - 21*a^3*b^2*d*(83
*c^2 - 334*c*d*x + 33*d^2*x^2) + b^5*x^2*(15*c^3 + 118*c^2*d*x + 136*c*d^2*x^2 +
48*d^3*x^3) + 3*a^2*b^3*(5*c^3 - 824*c^2*d*x + 387*c*d^2*x^2 + 66*d^3*x^3) - a*
b^4*x*(-30*c^3 + 483*c^2*d*x + 316*c*d^2*x^2 + 88*d^3*x^3)))/(192*b^6*d*(a + b*x
)^(3/2)) - (5*(b*c - a*d)*(b^3*c^3 + 21*a*b^2*c^2*d - 189*a^2*b*c*d^2 + 231*a^3*
d^3)*Log[b*c + a*d + 2*b*d*x + 2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(
128*b^(13/2)*d^(3/2))

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Maple [B]  time = 0.048, size = 1366, normalized size = 3.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(d*x+c)^(5/2)/(b*x+a)^(5/2),x)

[Out]

1/384*(d*x+c)^(1/2)*(-632*x^3*a*b^4*c*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+23
22*x^2*a^2*b^3*c*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-966*x^2*a*b^4*c^2*d*((b
*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+14028*x*a^3*b^2*c*d^2*((b*x+a)*(d*x+c))^(1/2)*(
b*d)^(1/2)-4944*x*a^2*b^3*c^2*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+30*x^2*b^5*c
^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+30*a^2*b^3*c^3*((b*x+a)*(d*x+c))^(1/2)*(b
*d)^(1/2)-176*x^4*a*b^4*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-15*ln(1/2*(2*b*d
*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x^2*b^6*c^4-15*ln
(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^2*b^
4*c^4-6930*a^5*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+3465*ln(1/2*(2*b*d*x+2*((
b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x^2*a^4*b^2*d^4+6930*ln(
1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x*a^5*b
*d^4-30*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/
2))*x*a*b^5*c^4-6300*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b
*c)/(b*d)^(1/2))*a^5*b*c*d^3+3150*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d
)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^4*b^2*c^2*d^2-300*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*
x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^3*b^3*c^3*d+96*x^5*b^5*d^3*((b*x
+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-12600*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(
b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x*a^4*b^2*c*d^3+6300*ln(1/2*(2*b*d*x+2*((b*x+a)
*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x*a^3*b^3*c^2*d^2-600*ln(1/2*(
2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x*a^2*b^4*c^
3*d+272*x^4*b^5*c*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+396*x^3*a^2*b^3*d^3*((
b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+236*x^3*b^5*c^2*d*((b*x+a)*(d*x+c))^(1/2)*(b*d
)^(1/2)-1386*x^2*a^3*b^2*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-9240*x*a^4*b*d^
3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+60*x*a*b^4*c^3*((b*x+a)*(d*x+c))^(1/2)*(b*
d)^(1/2)+10290*a^4*b*c*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-3486*a^3*b^2*c^2*
d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-6300*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(
1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x^2*a^3*b^3*c*d^3+3150*ln(1/2*(2*b*d*x+2*
((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x^2*a^2*b^4*c^2*d^2-30
0*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x^
2*a*b^5*c^3*d+3465*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c
)/(b*d)^(1/2))*a^6*d^4)/((b*x+a)*(d*x+c))^(1/2)/(b*d)^(1/2)/d/(b*x+a)^(3/2)/b^6

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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((d*x + c)^(5/2)*x^3/(b*x + a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.34333, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(5/2)*x^3/(b*x + a)^(5/2),x, algorithm="fricas")

[Out]

[1/768*(4*(48*b^5*d^3*x^5 + 15*a^2*b^3*c^3 - 1743*a^3*b^2*c^2*d + 5145*a^4*b*c*d
^2 - 3465*a^5*d^3 + 8*(17*b^5*c*d^2 - 11*a*b^4*d^3)*x^4 + 2*(59*b^5*c^2*d - 158*
a*b^4*c*d^2 + 99*a^2*b^3*d^3)*x^3 + 3*(5*b^5*c^3 - 161*a*b^4*c^2*d + 387*a^2*b^3
*c*d^2 - 231*a^3*b^2*d^3)*x^2 + 6*(5*a*b^4*c^3 - 412*a^2*b^3*c^2*d + 1169*a^3*b^
2*c*d^2 - 770*a^4*b*d^3)*x)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) - 15*(a^2*b^4*
c^4 + 20*a^3*b^3*c^3*d - 210*a^4*b^2*c^2*d^2 + 420*a^5*b*c*d^3 - 231*a^6*d^4 + (
b^6*c^4 + 20*a*b^5*c^3*d - 210*a^2*b^4*c^2*d^2 + 420*a^3*b^3*c*d^3 - 231*a^4*b^2
*d^4)*x^2 + 2*(a*b^5*c^4 + 20*a^2*b^4*c^3*d - 210*a^3*b^3*c^2*d^2 + 420*a^4*b^2*
c*d^3 - 231*a^5*b*d^4)*x)*log(4*(2*b^2*d^2*x + b^2*c*d + a*b*d^2)*sqrt(b*x + a)*
sqrt(d*x + c) + (8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 8*(b^2*c*d + a*
b*d^2)*x)*sqrt(b*d)))/((b^8*d*x^2 + 2*a*b^7*d*x + a^2*b^6*d)*sqrt(b*d)), 1/384*(
2*(48*b^5*d^3*x^5 + 15*a^2*b^3*c^3 - 1743*a^3*b^2*c^2*d + 5145*a^4*b*c*d^2 - 346
5*a^5*d^3 + 8*(17*b^5*c*d^2 - 11*a*b^4*d^3)*x^4 + 2*(59*b^5*c^2*d - 158*a*b^4*c*
d^2 + 99*a^2*b^3*d^3)*x^3 + 3*(5*b^5*c^3 - 161*a*b^4*c^2*d + 387*a^2*b^3*c*d^2 -
231*a^3*b^2*d^3)*x^2 + 6*(5*a*b^4*c^3 - 412*a^2*b^3*c^2*d + 1169*a^3*b^2*c*d^2
- 770*a^4*b*d^3)*x)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c) - 15*(a^2*b^4*c^4 + 2
0*a^3*b^3*c^3*d - 210*a^4*b^2*c^2*d^2 + 420*a^5*b*c*d^3 - 231*a^6*d^4 + (b^6*c^4
+ 20*a*b^5*c^3*d - 210*a^2*b^4*c^2*d^2 + 420*a^3*b^3*c*d^3 - 231*a^4*b^2*d^4)*x
^2 + 2*(a*b^5*c^4 + 20*a^2*b^4*c^3*d - 210*a^3*b^3*c^2*d^2 + 420*a^4*b^2*c*d^3 -
231*a^5*b*d^4)*x)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)/(sqrt(b*x + a)*sq
rt(d*x + c)*b*d)))/((b^8*d*x^2 + 2*a*b^7*d*x + a^2*b^6*d)*sqrt(-b*d))]

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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(x**3*(d*x+c)**(5/2)/(b*x+a)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.691274, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x