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

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

[Out]

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

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Rubi [A]  time = 0.971722, 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 \[ -\frac{(a+b x)^{5/2} \sqrt{c+d x} \left (5 a^2 d^2-2 b d x (99 b c-59 a d)-156 a b c d+231 b^2 c^2\right )}{24 b d^4 (b c-a d)}-\frac{5 \sqrt{a+b x} \sqrt{c+d x} \left (a^3 d^3+21 a^2 b c d^2-189 a b^2 c^2 d+231 b^3 c^3\right )}{64 b d^6}+\frac{5 (a+b x)^{3/2} \sqrt{c+d x} \left (a^3 d^3+21 a^2 b c d^2-189 a b^2 c^2 d+231 b^3 c^3\right )}{96 b d^5 (b c-a d)}+\frac{5 (b c-a d) \left (a^3 d^3+21 a^2 b c d^2-189 a b^2 c^2 d+231 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{3/2} d^{13/2}}-\frac{2 x^2 (a+b x)^{5/2} (11 b c-6 a d)}{3 d^2 \sqrt{c+d x} (b c-a d)}-\frac{2 x^3 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 80.6117, size = 326, normalized size = 0.86 \[ - \frac{2 x^{3} \left (a + b x\right )^{\frac{5}{2}}}{3 d \left (c + d x\right )^{\frac{3}{2}}} + \frac{2 x^{3} \left (a + b x\right )^{\frac{3}{2}} \left (6 a d - 11 b c\right )}{3 c d^{2} \sqrt{c + d x}} - \frac{x^{2} \left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (16 a d - 33 b c\right )}{4 c d^{3}} + \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (\frac{45 a^{2} d^{2}}{16} - \frac{1071 a b c d}{8} + \frac{3465 b^{2} c^{2}}{16} + \frac{9 b d x \left (41 a d - 77 b c\right )}{4}\right )}{18 b d^{5}} - \frac{5 \sqrt{a + b x} \sqrt{c + d x} \left (a^{3} d^{3} + 21 a^{2} b c d^{2} - 189 a b^{2} c^{2} d + 231 b^{3} c^{3}\right )}{64 b d^{6}} - \frac{5 \left (a d - b c\right ) \left (a^{3} d^{3} + 21 a^{2} b c d^{2} - 189 a b^{2} c^{2} d + 231 b^{3} c^{3}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{64 b^{\frac{3}{2}} d^{\frac{13}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.470218, size = 288, normalized size = 0.76 \[ \frac{\sqrt{a+b x} \left (15 a^3 d^3 (c+d x)^2+a^2 b d^2 \left (-1743 c^3-2472 c^2 d x-483 c d^2 x^2+118 d^3 x^3\right )+a b^2 d \left (5145 c^4+7014 c^3 d x+1161 c^2 d^2 x^2-316 c d^3 x^3+136 d^4 x^4\right )+b^3 \left (-\left (3465 c^5+4620 c^4 d x+693 c^3 d^2 x^2-198 c^2 d^3 x^3+88 c d^4 x^4-48 d^5 x^5\right )\right )\right )}{192 b d^6 (c+d x)^{3/2}}+\frac{5 (b c-a d) \left (a^3 d^3+21 a^2 b c d^2-189 a b^2 c^2 d+231 b^3 c^3\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{128 b^{3/2} d^{13/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x]*(15*a^3*d^3*(c + d*x)^2 + a^2*b*d^2*(-1743*c^3 - 2472*c^2*d*x - 4
83*c*d^2*x^2 + 118*d^3*x^3) + a*b^2*d*(5145*c^4 + 7014*c^3*d*x + 1161*c^2*d^2*x^
2 - 316*c*d^3*x^3 + 136*d^4*x^4) - b^3*(3465*c^5 + 4620*c^4*d*x + 693*c^3*d^2*x^
2 - 198*c^2*d^3*x^3 + 88*c*d^4*x^4 - 48*d^5*x^5)))/(192*b*d^6*(c + d*x)^(3/2)) +
(5*(b*c - a*d)*(231*b^3*c^3 - 189*a*b^2*c^2*d + 21*a^2*b*c*d^2 + 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^(3/2
)*d^(13/2))

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

[Out]

-1/384*(b*x+a)^(1/2)*(-272*x^4*a*b^2*d^5*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-96*
x^5*b^3*d^5*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-30*x^2*a^3*d^5*((b*x+a)*(d*x+c))
^(1/2)*(b*d)^(1/2)-30*a^3*c^2*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*b^4*c
^4*d^2+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^4*c*d^5-6930*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*b^4*c^5*d+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*c^3*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^2*b^2*c^4*d^2+6930*b^3*c^5*((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*a^4*d^6+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^4*c^2*d^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*b^3*c^5*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))*b^4*c^6-6300*
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^
2*b^2*c^3*d^3+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*b^3*c^4*d^2+176*x^4*b^3*c*d^4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^
(1/2)-236*x^3*a^2*b*d^5*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-396*x^3*b^3*c^2*d^3*
((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+1386*x^2*b^3*c^3*d^2*((b*x+a)*(d*x+c))^(1/2)
*(b*d)^(1/2)-60*x*a^3*c*d^4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+9240*x*b^3*c^4*d
*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+3486*a^2*b*c^3*d^2*((b*x+a)*(d*x+c))^(1/2)*
(b*d)^(1/2)-10290*a*b^2*c^4*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+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))*x^2*a^3*b*c*d^
5-3150*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^2*c^2*d^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))*x^2*a*b^3*c^3*d^3+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^3*b*c^2*d^4-2322*x^2*a*b^2*c^2*d^3
*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+4944*x*a^2*b*c^2*d^3*((b*x+a)*(d*x+c))^(1/2
)*(b*d)^(1/2)-14028*x*a*b^2*c^3*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+632*x^3*
a*b^2*c*d^4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+966*x^2*a^2*b*c*d^4*((b*x+a)*(d*
x+c))^(1/2)*(b*d)^(1/2))/((b*x+a)*(d*x+c))^(1/2)/(b*d)^(1/2)/b/(d*x+c)^(3/2)/d^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((b*x + a)^(5/2)*x^3/(d*x + c)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.62514, 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)^(5/2)*x^3/(d*x + c)^(5/2),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.299332, size = 926, normalized size = 2.46 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/192*(((2*(4*(b*x + a)*(6*(b^5*c*d^10*abs(b) - a*b^4*d^11*abs(b))*(b*x + a)/(b^
6*c*d^11 - a*b^5*d^12) - (11*b^6*c^2*d^9*abs(b) + 2*a*b^5*c*d^10*abs(b) - 13*a^2
*b^4*d^11*abs(b))/(b^6*c*d^11 - a*b^5*d^12)) + 9*(11*b^7*c^3*d^8*abs(b) - 9*a*b^
6*c^2*d^9*abs(b) + a^2*b^5*c*d^10*abs(b) - 3*a^3*b^4*d^11*abs(b))/(b^6*c*d^11 -
a*b^5*d^12))*(b*x + a) - 3*(231*b^8*c^4*d^7*abs(b) - 420*a*b^7*c^3*d^8*abs(b) +
210*a^2*b^6*c^2*d^9*abs(b) - 20*a^3*b^5*c*d^10*abs(b) - a^4*b^4*d^11*abs(b))/(b^
6*c*d^11 - a*b^5*d^12))*(b*x + a) - 20*(231*b^9*c^5*d^6*abs(b) - 651*a*b^8*c^4*d
^7*abs(b) + 630*a^2*b^7*c^3*d^8*abs(b) - 230*a^3*b^6*c^2*d^9*abs(b) + 19*a^4*b^5
*c*d^10*abs(b) + a^5*b^4*d^11*abs(b))/(b^6*c*d^11 - a*b^5*d^12))*(b*x + a) - 15*
(231*b^10*c^6*d^5*abs(b) - 882*a*b^9*c^5*d^6*abs(b) + 1281*a^2*b^8*c^4*d^7*abs(b
) - 860*a^3*b^7*c^3*d^8*abs(b) + 249*a^4*b^6*c^2*d^9*abs(b) - 18*a^5*b^5*c*d^10*
abs(b) - a^6*b^4*d^11*abs(b))/(b^6*c*d^11 - a*b^5*d^12))*sqrt(b*x + a)/(b^2*c +
(b*x + a)*b*d - a*b*d)^(3/2) - 5/64*(231*b^4*c^4*abs(b) - 420*a*b^3*c^3*d*abs(b)
+ 210*a^2*b^2*c^2*d^2*abs(b) - 20*a^3*b*c*d^3*abs(b) - a^4*d^4*abs(b))*ln(abs(-
sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b^2*d
^6)