3.773 \(\int \frac{x^4}{(a+b x)^{3/2} (c+d x)^{5/2}} \, dx\)

Optimal. Leaf size=251 \[ \frac{\sqrt{a+b x} \left (d x (b c-a d) \left (9 a^2 d^2-6 a b c d+5 b^2 c^2\right )+c \left (-9 a^3 d^3+9 a^2 b c d^2-31 a b^2 c^2 d+15 b^3 c^3\right )\right )}{3 b^2 d^3 \sqrt{c+d x} (b c-a d)^3}-\frac{(3 a d+5 b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{5/2} d^{7/2}}+\frac{2 a x^3}{b \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)}-\frac{2 c x^2 \sqrt{a+b x} (3 a d+b c)}{3 b d (c+d x)^{3/2} (b c-a d)^2} \]

[Out]

(2*a*x^3)/(b*(b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2)) - (2*c*(b*c + 3*a*d)*x^2
*Sqrt[a + b*x])/(3*b*d*(b*c - a*d)^2*(c + d*x)^(3/2)) + (Sqrt[a + b*x]*(c*(15*b^
3*c^3 - 31*a*b^2*c^2*d + 9*a^2*b*c*d^2 - 9*a^3*d^3) + d*(b*c - a*d)*(5*b^2*c^2 -
6*a*b*c*d + 9*a^2*d^2)*x))/(3*b^2*d^3*(b*c - a*d)^3*Sqrt[c + d*x]) - ((5*b*c +
3*a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(b^(5/2)*d^(7/2
))

_______________________________________________________________________________________

Rubi [A]  time = 0.590625, antiderivative size = 251, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{\sqrt{a+b x} \left (d x (b c-a d) \left (9 a^2 d^2-6 a b c d+5 b^2 c^2\right )+c \left (-9 a^3 d^3+9 a^2 b c d^2-31 a b^2 c^2 d+15 b^3 c^3\right )\right )}{3 b^2 d^3 \sqrt{c+d x} (b c-a d)^3}-\frac{(3 a d+5 b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{5/2} d^{7/2}}+\frac{2 a x^3}{b \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)}-\frac{2 c x^2 \sqrt{a+b x} (3 a d+b c)}{3 b d (c+d x)^{3/2} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Int[x^4/((a + b*x)^(3/2)*(c + d*x)^(5/2)),x]

[Out]

(2*a*x^3)/(b*(b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2)) - (2*c*(b*c + 3*a*d)*x^2
*Sqrt[a + b*x])/(3*b*d*(b*c - a*d)^2*(c + d*x)^(3/2)) + (Sqrt[a + b*x]*(c*(15*b^
3*c^3 - 31*a*b^2*c^2*d + 9*a^2*b*c*d^2 - 9*a^3*d^3) + d*(b*c - a*d)*(5*b^2*c^2 -
6*a*b*c*d + 9*a^2*d^2)*x))/(3*b^2*d^3*(b*c - a*d)^3*Sqrt[c + d*x]) - ((5*b*c +
3*a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(b^(5/2)*d^(7/2
))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 50.4776, size = 243, normalized size = 0.97 \[ - \frac{2 a x^{3}}{b \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )} - \frac{2 c x^{2} \sqrt{a + b x} \left (3 a d + b c\right )}{3 b d \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )^{2}} + \frac{8 \sqrt{a + b x} \left (\frac{c \left (9 a^{3} d^{3} - 9 a^{2} b c d^{2} + 31 a b^{2} c^{2} d - 15 b^{3} c^{3}\right )}{8} + \frac{d x \left (a d - b c\right ) \left (9 a^{2} d^{2} - 6 a b c d + 5 b^{2} c^{2}\right )}{8}\right )}{3 b^{2} d^{3} \sqrt{c + d x} \left (a d - b c\right )^{3}} - \frac{\left (3 a d + 5 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{b^{\frac{5}{2}} d^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(b*x+a)**(3/2)/(d*x+c)**(5/2),x)

[Out]

-2*a*x**3/(b*sqrt(a + b*x)*(c + d*x)**(3/2)*(a*d - b*c)) - 2*c*x**2*sqrt(a + b*x
)*(3*a*d + b*c)/(3*b*d*(c + d*x)**(3/2)*(a*d - b*c)**2) + 8*sqrt(a + b*x)*(c*(9*
a**3*d**3 - 9*a**2*b*c*d**2 + 31*a*b**2*c**2*d - 15*b**3*c**3)/8 + d*x*(a*d - b*
c)*(9*a**2*d**2 - 6*a*b*c*d + 5*b**2*c**2)/8)/(3*b**2*d**3*sqrt(c + d*x)*(a*d -
b*c)**3) - (3*a*d + 5*b*c)*atanh(sqrt(d)*sqrt(a + b*x)/(sqrt(b)*sqrt(c + d*x)))/
(b**(5/2)*d**(7/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.987292, size = 182, normalized size = 0.73 \[ \sqrt{a+b x} \sqrt{c+d x} \left (-\frac{2 a^4}{b^2 (a+b x) (b c-a d)^3}-\frac{2 c^4}{3 d^3 (c+d x)^2 (a d-b c)^2}-\frac{2 c^3 (7 b c-12 a d)}{3 d^3 (c+d x) (a d-b c)^3}+\frac{1}{b^2 d^3}\right )-\frac{(3 a d+5 b c) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{2 b^{5/2} d^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/((a + b*x)^(3/2)*(c + d*x)^(5/2)),x]

[Out]

Sqrt[a + b*x]*Sqrt[c + d*x]*(1/(b^2*d^3) - (2*a^4)/(b^2*(b*c - a*d)^3*(a + b*x))
- (2*c^4)/(3*d^3*(-(b*c) + a*d)^2*(c + d*x)^2) - (2*c^3*(7*b*c - 12*a*d))/(3*d^
3*(-(b*c) + a*d)^3*(c + d*x))) - ((5*b*c + 3*a*d)*Log[b*c + a*d + 2*b*d*x + 2*Sq
rt[b]*Sqrt[d]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(2*b^(5/2)*d^(7/2))

_______________________________________________________________________________________

Maple [B]  time = 0.046, size = 1714, normalized size = 6.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(b*x+a)^(3/2)/(d*x+c)^(5/2),x)

[Out]

-1/6*(-6*x^3*a^3*b*d^5*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+18*x^3*a^2*b^2*c*d^4*
((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-18*x^3*a*b^3*c^2*d^3*((b*x+a)*(d*x+c))^(1/2)
*(b*d)^(1/2)+6*x^2*a^3*b*c*d^4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+18*x^2*a^2*b^
2*c^2*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-78*x^2*a*b^3*c^3*d^2*((b*x+a)*(d*x
+c))^(1/2)*(b*d)^(1/2)+30*x*a^3*b*c^2*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-66
*x*a^2*b^2*c^3*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-22*x*a*b^3*c^4*d*((b*x+a)
*(d*x+c))^(1/2)*(b*d)^(1/2)+9*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^5*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))*x*b^5*c^6+9*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*c^2*d^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*b^4*c^6-18*x^2*a^4*d^5
*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+30*x*b^4*c^5*((b*x+a)*(d*x+c))^(1/2)*(b*d)^
(1/2)-18*a^4*c^2*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+30*a*b^3*c^5*((b*x+a)*(
d*x+c))^(1/2)*(b*d)^(1/2)+9*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^3*a^4*b*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))*x^3*b^5*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^2*b^5*c^5*d+18*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*c*d^5-12*
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*c^3*d^3-18*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^2*c^4*d^2+36*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^3*c^5*d-62*a^2*b^2*c^4*d*((b*x+a)*(d*x+c))^(1/2)*(
b*d)^(1/2)-12*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^3*a^3*b^2*c*d^5-18*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^3*a^2*b^3*c^2*d^4+36*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^3*a*b^4*c^3*d^3+6*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*c*d^5-42
*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^2*c^2*d^4+57*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^4*c^4*d^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*a^4*b*c^2*d^4-48*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^2*c^3*d^3+54*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^3*c^4*d
^2+6*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^4*c^5*d+6*x^3*b^4*c^3*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+40*x^2*b^4*
c^4*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-36*x*a^4*c*d^4*((b*x+a)*(d*x+c))^(1/2)
*(b*d)^(1/2)+18*a^3*b*c^3*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2))/(b*d)^(1/2)/(
a*d-b*c)^3/((b*x+a)*(d*x+c))^(1/2)/b^2/d^3/(d*x+c)^(3/2)/(b*x+a)^(1/2)

_______________________________________________________________________________________

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(x^4/((b*x + a)^(3/2)*(d*x + c)^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.749091, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/((b*x + a)^(3/2)*(d*x + c)^(5/2)),x, algorithm="fricas")

[Out]

[1/12*(4*(15*a*b^3*c^5 - 31*a^2*b^2*c^4*d + 9*a^3*b*c^3*d^2 - 9*a^4*c^2*d^3 + 3*
(b^4*c^3*d^2 - 3*a*b^3*c^2*d^3 + 3*a^2*b^2*c*d^4 - a^3*b*d^5)*x^3 + (20*b^4*c^4*
d - 39*a*b^3*c^3*d^2 + 9*a^2*b^2*c^2*d^3 + 3*a^3*b*c*d^4 - 9*a^4*d^5)*x^2 + (15*
b^4*c^5 - 11*a*b^3*c^4*d - 33*a^2*b^2*c^3*d^2 + 15*a^3*b*c^2*d^3 - 18*a^4*c*d^4)
*x)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 3*(5*a*b^4*c^6 - 12*a^2*b^3*c^5*d +
6*a^3*b^2*c^4*d^2 + 4*a^4*b*c^3*d^3 - 3*a^5*c^2*d^4 + (5*b^5*c^4*d^2 - 12*a*b^4*
c^3*d^3 + 6*a^2*b^3*c^2*d^4 + 4*a^3*b^2*c*d^5 - 3*a^4*b*d^6)*x^3 + (10*b^5*c^5*d
- 19*a*b^4*c^4*d^2 + 14*a^3*b^2*c^2*d^4 - 2*a^4*b*c*d^5 - 3*a^5*d^6)*x^2 + (5*b
^5*c^6 - 2*a*b^4*c^5*d - 18*a^2*b^3*c^4*d^2 + 16*a^3*b^2*c^3*d^3 + 5*a^4*b*c^2*d
^4 - 6*a^5*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)))/((a*b^5*c^5*d^3 - 3*a^2*b^4*c^4*d^4 + 3*a^3*b^3*c^3*d^5 - a^4*
b^2*c^2*d^6 + (b^6*c^3*d^5 - 3*a*b^5*c^2*d^6 + 3*a^2*b^4*c*d^7 - a^3*b^3*d^8)*x^
3 + (2*b^6*c^4*d^4 - 5*a*b^5*c^3*d^5 + 3*a^2*b^4*c^2*d^6 + a^3*b^3*c*d^7 - a^4*b
^2*d^8)*x^2 + (b^6*c^5*d^3 - a*b^5*c^4*d^4 - 3*a^2*b^4*c^3*d^5 + 5*a^3*b^3*c^2*d
^6 - 2*a^4*b^2*c*d^7)*x)*sqrt(b*d)), 1/6*(2*(15*a*b^3*c^5 - 31*a^2*b^2*c^4*d + 9
*a^3*b*c^3*d^2 - 9*a^4*c^2*d^3 + 3*(b^4*c^3*d^2 - 3*a*b^3*c^2*d^3 + 3*a^2*b^2*c*
d^4 - a^3*b*d^5)*x^3 + (20*b^4*c^4*d - 39*a*b^3*c^3*d^2 + 9*a^2*b^2*c^2*d^3 + 3*
a^3*b*c*d^4 - 9*a^4*d^5)*x^2 + (15*b^4*c^5 - 11*a*b^3*c^4*d - 33*a^2*b^2*c^3*d^2
+ 15*a^3*b*c^2*d^3 - 18*a^4*c*d^4)*x)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c) -
3*(5*a*b^4*c^6 - 12*a^2*b^3*c^5*d + 6*a^3*b^2*c^4*d^2 + 4*a^4*b*c^3*d^3 - 3*a^5*
c^2*d^4 + (5*b^5*c^4*d^2 - 12*a*b^4*c^3*d^3 + 6*a^2*b^3*c^2*d^4 + 4*a^3*b^2*c*d^
5 - 3*a^4*b*d^6)*x^3 + (10*b^5*c^5*d - 19*a*b^4*c^4*d^2 + 14*a^3*b^2*c^2*d^4 - 2
*a^4*b*c*d^5 - 3*a^5*d^6)*x^2 + (5*b^5*c^6 - 2*a*b^4*c^5*d - 18*a^2*b^3*c^4*d^2
+ 16*a^3*b^2*c^3*d^3 + 5*a^4*b*c^2*d^4 - 6*a^5*c*d^5)*x)*arctan(1/2*(2*b*d*x + b
*c + a*d)*sqrt(-b*d)/(sqrt(b*x + a)*sqrt(d*x + c)*b*d)))/((a*b^5*c^5*d^3 - 3*a^2
*b^4*c^4*d^4 + 3*a^3*b^3*c^3*d^5 - a^4*b^2*c^2*d^6 + (b^6*c^3*d^5 - 3*a*b^5*c^2*
d^6 + 3*a^2*b^4*c*d^7 - a^3*b^3*d^8)*x^3 + (2*b^6*c^4*d^4 - 5*a*b^5*c^3*d^5 + 3*
a^2*b^4*c^2*d^6 + a^3*b^3*c*d^7 - a^4*b^2*d^8)*x^2 + (b^6*c^5*d^3 - a*b^5*c^4*d^
4 - 3*a^2*b^4*c^3*d^5 + 5*a^3*b^3*c^2*d^6 - 2*a^4*b^2*c*d^7)*x)*sqrt(-b*d))]

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(b*x+a)**(3/2)/(d*x+c)**(5/2),x)

[Out]

Integral(x**4/((a + b*x)**(3/2)*(c + d*x)**(5/2)), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.581727, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/((b*x + a)^(3/2)*(d*x + c)^(5/2)),x, algorithm="giac")

[Out]

sage0*x