3.774 \(\int \frac{x^3}{(a+b x)^{3/2} (c+d x)^{5/2}} \, dx\)
Optimal. Leaf size=165 \[ -\frac{2 c \sqrt{a+b x} \left (2 d x \left (-3 a^2 d^2-3 a b c d+2 b^2 c^2\right )+c (b c-3 a d) (a d+3 b c)\right )}{3 b d^2 (c+d x)^{3/2} (b c-a d)^3}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{3/2} d^{5/2}}+\frac{2 a x^2}{b \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)} \]
[Out]
(2*a*x^2)/(b*(b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2)) - (2*c*Sqrt[a + b*x]*(c*
(b*c - 3*a*d)*(3*b*c + a*d) + 2*d*(2*b^2*c^2 - 3*a*b*c*d - 3*a^2*d^2)*x))/(3*b*d
^2*(b*c - a*d)^3*(c + d*x)^(3/2)) + (2*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*
Sqrt[c + d*x])])/(b^(3/2)*d^(5/2))
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Rubi [A] time = 0.336417, antiderivative size = 165, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182
\[ -\frac{2 c \sqrt{a+b x} \left (2 d x \left (-3 a^2 d^2-3 a b c d+2 b^2 c^2\right )+c (b c-3 a d) (a d+3 b c)\right )}{3 b d^2 (c+d x)^{3/2} (b c-a d)^3}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{3/2} d^{5/2}}+\frac{2 a x^2}{b \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[x^3/((a + b*x)^(3/2)*(c + d*x)^(5/2)),x]
[Out]
(2*a*x^2)/(b*(b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2)) - (2*c*Sqrt[a + b*x]*(c*
(b*c - 3*a*d)*(3*b*c + a*d) + 2*d*(2*b^2*c^2 - 3*a*b*c*d - 3*a^2*d^2)*x))/(3*b*d
^2*(b*c - a*d)^3*(c + d*x)^(3/2)) + (2*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*
Sqrt[c + d*x])])/(b^(3/2)*d^(5/2))
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Rubi in Sympy [A] time = 25.9417, size = 158, normalized size = 0.96 \[ - \frac{2 a x^{2}}{b \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )} - \frac{8 c \sqrt{a + b x} \left (\frac{c \left (a d + 3 b c\right ) \left (3 a d - b c\right )}{4} + \frac{d x \left (3 a^{2} d^{2} + 3 a b c d - 2 b^{2} c^{2}\right )}{2}\right )}{3 b d^{2} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )^{3}} + \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{b^{\frac{3}{2}} d^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3/(b*x+a)**(3/2)/(d*x+c)**(5/2),x)
[Out]
-2*a*x**2/(b*sqrt(a + b*x)*(c + d*x)**(3/2)*(a*d - b*c)) - 8*c*sqrt(a + b*x)*(c*
(a*d + 3*b*c)*(3*a*d - b*c)/4 + d*x*(3*a**2*d**2 + 3*a*b*c*d - 2*b**2*c**2)/2)/(
3*b*d**2*(c + d*x)**(3/2)*(a*d - b*c)**3) + 2*atanh(sqrt(d)*sqrt(a + b*x)/(sqrt(
b)*sqrt(c + d*x)))/(b**(3/2)*d**(5/2))
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Mathematica [A] time = 0.674011, size = 160, normalized size = 0.97 \[ \frac{2}{3} \sqrt{a+b x} \sqrt{c+d x} \left (\frac{3 a^3}{b (a+b x) (b c-a d)^3}+\frac{c^3}{d^2 (c+d x)^2 (b c-a d)^2}+\frac{c^2 (4 b c-9 a d)}{d^2 (c+d x) (a d-b c)^3}\right )+\frac{\log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{b^{3/2} d^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[x^3/((a + b*x)^(3/2)*(c + d*x)^(5/2)),x]
[Out]
(2*Sqrt[a + b*x]*Sqrt[c + d*x]*((3*a^3)/(b*(b*c - a*d)^3*(a + b*x)) + c^3/(d^2*(
b*c - a*d)^2*(c + d*x)^2) + (c^2*(4*b*c - 9*a*d))/(d^2*(-(b*c) + a*d)^3*(c + d*x
))))/3 + Log[b*c + a*d + 2*b*d*x + 2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*Sqrt[c + d*x]
]/(b^(3/2)*d^(5/2))
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Maple [B] time = 0.043, size = 1289, normalized size = 7.8 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3/(b*x+a)^(3/2)/(d*x+c)^(5/2),x)
[Out]
1/3*(6*x*b^3*c^4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-6*a^3*c^2*d^2*((b*x+a)*(d*x
+c))^(1/2)*(b*d)^(1/2)+6*a*b^2*c^4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+3*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*d
^5-3*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^4*c^3*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^2*b^4*c^4*d+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^4*c*d^4-9*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^3*b*c^3*d^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))*a^2*b^2*c^4*d-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^3*b*c^2*d^3+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*b^3*c^2*d^3-3*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^4-9*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^2*b^2*c^2*d^3+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*b^3*c^3*d^2+3*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^5-3*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
+3*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^3-3*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^5-6*x^2*a^3*d^4*((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*a^2*b^2*c^3
*d^2+3*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+8*x^2*b^3*c^3*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-12*x*a^3*c*
d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-16*a^2*b*c^3*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^3*a^2*b^2*c*d^4-18*x^2*a*b^2*c^2*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1
/2)-18*x*a^2*b*c^2*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-8*x*a*b^2*c^3*d*((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/d^2/(d*x+c)^(3/2)/(b*x+a)^(1/2)
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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(x^3/((b*x + a)^(3/2)*(d*x + c)^(5/2)),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.530217, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/((b*x + a)^(3/2)*(d*x + c)^(5/2)),x, algorithm="fricas")
[Out]
[-1/6*(4*(3*a*b^2*c^4 - 8*a^2*b*c^3*d - 3*a^3*c^2*d^2 + (4*b^3*c^3*d - 9*a*b^2*c
^2*d^2 - 3*a^3*d^4)*x^2 + (3*b^3*c^4 - 4*a*b^2*c^3*d - 9*a^2*b*c^2*d^2 - 6*a^3*c
*d^3)*x)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) - 3*(a*b^3*c^5 - 3*a^2*b^2*c^4*d
+ 3*a^3*b*c^3*d^2 - a^4*c^2*d^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 + (2*b^4*c^4*d - 5*a*b^3*c^3*d^2 + 3*a^2*b^2*c^2*d^3 + a^3*b
*c*d^4 - a^4*d^5)*x^2 + (b^4*c^5 - a*b^3*c^4*d - 3*a^2*b^2*c^3*d^2 + 5*a^3*b*c^2
*d^3 - 2*a^4*c*d^4)*x)*log(4*(2*b^2*d^2*x + b^2*c*d + a*b*d^2)*sqrt(b*x + a)*sqr
t(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^4*c^5*d^2 - 3*a^2*b^3*c^4*d^3 + 3*a^3*b^2*c^3*d^4 - a^4
*b*c^2*d^5 + (b^5*c^3*d^4 - 3*a*b^4*c^2*d^5 + 3*a^2*b^3*c*d^6 - a^3*b^2*d^7)*x^3
+ (2*b^5*c^4*d^3 - 5*a*b^4*c^3*d^4 + 3*a^2*b^3*c^2*d^5 + a^3*b^2*c*d^6 - a^4*b*
d^7)*x^2 + (b^5*c^5*d^2 - a*b^4*c^4*d^3 - 3*a^2*b^3*c^3*d^4 + 5*a^3*b^2*c^2*d^5
- 2*a^4*b*c*d^6)*x)*sqrt(b*d)), -1/3*(2*(3*a*b^2*c^4 - 8*a^2*b*c^3*d - 3*a^3*c^2
*d^2 + (4*b^3*c^3*d - 9*a*b^2*c^2*d^2 - 3*a^3*d^4)*x^2 + (3*b^3*c^4 - 4*a*b^2*c^
3*d - 9*a^2*b*c^2*d^2 - 6*a^3*c*d^3)*x)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c) -
3*(a*b^3*c^5 - 3*a^2*b^2*c^4*d + 3*a^3*b*c^3*d^2 - a^4*c^2*d^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 + (2*b^4*c^4*d - 5*a*b^3*c^3
*d^2 + 3*a^2*b^2*c^2*d^3 + a^3*b*c*d^4 - a^4*d^5)*x^2 + (b^4*c^5 - a*b^3*c^4*d -
3*a^2*b^2*c^3*d^2 + 5*a^3*b*c^2*d^3 - 2*a^4*c*d^4)*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^4*c^5*d^2 - 3*a^2*b
^3*c^4*d^3 + 3*a^3*b^2*c^3*d^4 - a^4*b*c^2*d^5 + (b^5*c^3*d^4 - 3*a*b^4*c^2*d^5
+ 3*a^2*b^3*c*d^6 - a^3*b^2*d^7)*x^3 + (2*b^5*c^4*d^3 - 5*a*b^4*c^3*d^4 + 3*a^2*
b^3*c^2*d^5 + a^3*b^2*c*d^6 - a^4*b*d^7)*x^2 + (b^5*c^5*d^2 - a*b^4*c^4*d^3 - 3*
a^2*b^3*c^3*d^4 + 5*a^3*b^2*c^2*d^5 - 2*a^4*b*c*d^6)*x)*sqrt(-b*d))]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3/(b*x+a)**(3/2)/(d*x+c)**(5/2),x)
[Out]
Integral(x**3/((a + b*x)**(3/2)*(c + d*x)**(5/2)), x)
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GIAC/XCAS [A] time = 0.595411, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/((b*x + a)^(3/2)*(d*x + c)^(5/2)),x, algorithm="giac")
[Out]
sage0*x