3.686 \(\int \frac{(a+b x)^{5/2}}{x^5 (c+d x)^{5/2}} \, dx\)
Optimal. Leaf size=388 \[ -\frac{d \sqrt{a+b x} (b c-a d) \left (385 a^2 d^2-238 a b c d+5 b^2 c^2\right )}{64 a c^5 (c+d x)^{3/2}}-\frac{\sqrt{a+b x} (b c-a d) \left (231 a^2 d^2-156 a b c d+5 b^2 c^2\right )}{64 a c^4 x (c+d x)^{3/2}}-\frac{d \sqrt{a+b x} \left (-1155 a^3 d^3+1715 a^2 b c d^2-581 a b^2 c^2 d+5 b^3 c^3\right )}{64 a c^6 \sqrt{c+d x}}+\frac{5 (b c-a d) \left (231 a^3 d^3-189 a^2 b c d^2+21 a b^2 c^2 d+b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{3/2} c^{13/2}}-\frac{\sqrt{a+b x} (59 b c-99 a d) (b c-a d)}{96 c^3 x^2 (c+d x)^{3/2}}-\frac{11 a \sqrt{a+b x} (b c-a d)}{24 c^2 x^3 (c+d x)^{3/2}}-\frac{a (a+b x)^{3/2}}{4 c x^4 (c+d x)^{3/2}} \]
[Out]
-(d*(b*c - a*d)*(5*b^2*c^2 - 238*a*b*c*d + 385*a^2*d^2)*Sqrt[a + b*x])/(64*a*c^5
*(c + d*x)^(3/2)) - (11*a*(b*c - a*d)*Sqrt[a + b*x])/(24*c^2*x^3*(c + d*x)^(3/2)
) - ((59*b*c - 99*a*d)*(b*c - a*d)*Sqrt[a + b*x])/(96*c^3*x^2*(c + d*x)^(3/2)) -
((b*c - a*d)*(5*b^2*c^2 - 156*a*b*c*d + 231*a^2*d^2)*Sqrt[a + b*x])/(64*a*c^4*x
*(c + d*x)^(3/2)) - (a*(a + b*x)^(3/2))/(4*c*x^4*(c + d*x)^(3/2)) - (d*(5*b^3*c^
3 - 581*a*b^2*c^2*d + 1715*a^2*b*c*d^2 - 1155*a^3*d^3)*Sqrt[a + b*x])/(64*a*c^6*
Sqrt[c + d*x]) + (5*(b*c - a*d)*(b^3*c^3 + 21*a*b^2*c^2*d - 189*a^2*b*c*d^2 + 23
1*a^3*d^3)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(64*a^(3/2)
*c^(13/2))
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Rubi [A] time = 1.58613, antiderivative size = 388, normalized size of antiderivative = 1.,
number of steps used = 9, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318
\[ -\frac{d \sqrt{a+b x} (b c-a d) \left (385 a^2 d^2-238 a b c d+5 b^2 c^2\right )}{64 a c^5 (c+d x)^{3/2}}-\frac{\sqrt{a+b x} (b c-a d) \left (231 a^2 d^2-156 a b c d+5 b^2 c^2\right )}{64 a c^4 x (c+d x)^{3/2}}-\frac{d \sqrt{a+b x} \left (-1155 a^3 d^3+1715 a^2 b c d^2-581 a b^2 c^2 d+5 b^3 c^3\right )}{64 a c^6 \sqrt{c+d x}}+\frac{5 (b c-a d) \left (231 a^3 d^3-189 a^2 b c d^2+21 a b^2 c^2 d+b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{3/2} c^{13/2}}-\frac{\sqrt{a+b x} (59 b c-99 a d) (b c-a d)}{96 c^3 x^2 (c+d x)^{3/2}}-\frac{11 a \sqrt{a+b x} (b c-a d)}{24 c^2 x^3 (c+d x)^{3/2}}-\frac{a (a+b x)^{3/2}}{4 c x^4 (c+d x)^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^(5/2)/(x^5*(c + d*x)^(5/2)),x]
[Out]
-(d*(b*c - a*d)*(5*b^2*c^2 - 238*a*b*c*d + 385*a^2*d^2)*Sqrt[a + b*x])/(64*a*c^5
*(c + d*x)^(3/2)) - (11*a*(b*c - a*d)*Sqrt[a + b*x])/(24*c^2*x^3*(c + d*x)^(3/2)
) - ((59*b*c - 99*a*d)*(b*c - a*d)*Sqrt[a + b*x])/(96*c^3*x^2*(c + d*x)^(3/2)) -
((b*c - a*d)*(5*b^2*c^2 - 156*a*b*c*d + 231*a^2*d^2)*Sqrt[a + b*x])/(64*a*c^4*x
*(c + d*x)^(3/2)) - (a*(a + b*x)^(3/2))/(4*c*x^4*(c + d*x)^(3/2)) - (d*(5*b^3*c^
3 - 581*a*b^2*c^2*d + 1715*a^2*b*c*d^2 - 1155*a^3*d^3)*Sqrt[a + b*x])/(64*a*c^6*
Sqrt[c + d*x]) + (5*(b*c - a*d)*(b^3*c^3 + 21*a*b^2*c^2*d - 189*a^2*b*c*d^2 + 23
1*a^3*d^3)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(64*a^(3/2)
*c^(13/2))
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**(5/2)/x**5/(d*x+c)**(5/2),x)
[Out]
Timed out
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Mathematica [A] time = 0.604326, size = 351, normalized size = 0.9 \[ \frac{15 \log (x) (a d-b c) \left (231 a^3 d^3-189 a^2 b c d^2+21 a b^2 c^2 d+b^3 c^3\right )+15 (b c-a d) \left (231 a^3 d^3-189 a^2 b c d^2+21 a b^2 c^2 d+b^3 c^3\right ) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )+\frac{2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \left (a^3 \left (-48 c^5+88 c^4 d x-198 c^3 d^2 x^2+693 c^2 d^3 x^3+4620 c d^4 x^4+3465 d^5 x^5\right )-a^2 b c x \left (136 c^4-316 c^3 d x+1161 c^2 d^2 x^2+7014 c d^3 x^3+5145 d^4 x^4\right )+a b^2 c^2 x^2 \left (-118 c^3+483 c^2 d x+2472 c d^2 x^2+1743 d^3 x^3\right )-15 b^3 c^3 x^3 (c+d x)^2\right )}{x^4 (c+d x)^{3/2}}}{384 a^{3/2} c^{13/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^(5/2)/(x^5*(c + d*x)^(5/2)),x]
[Out]
((2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*(-15*b^3*c^3*x^3*(c + d*x)^2 + a*b^2*c^2*x^2*(
-118*c^3 + 483*c^2*d*x + 2472*c*d^2*x^2 + 1743*d^3*x^3) - a^2*b*c*x*(136*c^4 - 3
16*c^3*d*x + 1161*c^2*d^2*x^2 + 7014*c*d^3*x^3 + 5145*d^4*x^4) + a^3*(-48*c^5 +
88*c^4*d*x - 198*c^3*d^2*x^2 + 693*c^2*d^3*x^3 + 4620*c*d^4*x^4 + 3465*d^5*x^5))
)/(x^4*(c + d*x)^(3/2)) + 15*(-(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[x] + 15*(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[2*a*c + b*c*x + a*d*x + 2*Sqrt[a]*Sqrt[c]*Sqrt[a +
b*x]*Sqrt[c + d*x]])/(384*a^(3/2)*c^(13/2))
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Maple [B] time = 0.053, size = 1377, normalized size = 3.6 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^(5/2)/x^5/(d*x+c)^(5/2),x)
[Out]
-1/384*(b*x+a)^(1/2)*(30*x^5*b^3*c^3*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+102
90*x^5*a^2*b*c*d^4*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-3486*x^5*a*b^2*c^2*d^3*(a
*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+14028*x^4*a^2*b*c^2*d^3*(a*c)^(1/2)*((b*x+a)*(
d*x+c))^(1/2)-4944*x^4*a*b^2*c^3*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2322*x^
3*a^2*b*c^3*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-966*x^3*a*b^2*c^4*d*(a*c)^(1
/2)*((b*x+a)*(d*x+c))^(1/2)-632*x^2*a^2*b*c^4*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1
/2)+30*x^3*b^3*c^5*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-15*ln((a*d*x+b*c*x+2*(a*c
)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^6*b^4*c^4*d^2+6930*ln((a*d*x+b*c*x+2
*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^5*a^4*c*d^5-30*ln((a*d*x+b*c*x+
2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^5*b^4*c^5*d+3465*ln((a*d*x+b*c
*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^4*a^4*c^2*d^4+96*a^3*c^5*(a
*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+3465*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d
*x+c))^(1/2)+2*a*c)/x)*x^6*a^4*d^6-15*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*
x+c))^(1/2)+2*a*c)/x)*x^4*b^4*c^6-6930*x^5*a^3*d^5*(a*c)^(1/2)*((b*x+a)*(d*x+c))
^(1/2)-9240*x^4*a^3*c*d^4*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+60*x^4*b^3*c^4*d*(
a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-1386*x^3*a^3*c^2*d^3*(a*c)^(1/2)*((b*x+a)*(d*
x+c))^(1/2)+396*x^2*a^3*c^3*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+236*x^2*a*b^
2*c^5*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-176*x*a^3*c^4*d*(a*c)^(1/2)*((b*x+a)*(
d*x+c))^(1/2)+272*x*a^2*b*c^5*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-6300*ln((a*d*x
+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^6*a^3*b*c*d^5+3150*ln((
a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^6*a^2*b^2*c^2*d^4-
300*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^6*a*b^3*c^
3*d^3-12600*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^5*
a^3*b*c^2*d^4+6300*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/
x)*x^5*a^2*b^2*c^3*d^3-600*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)
+2*a*c)/x)*x^5*a*b^3*c^4*d^2-6300*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c)
)^(1/2)+2*a*c)/x)*x^4*a^3*b*c^3*d^3+3150*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*
(d*x+c))^(1/2)+2*a*c)/x)*x^4*a^2*b^2*c^4*d^2-300*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*(
(b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^4*a*b^3*c^5*d)/c^6/a/((b*x+a)*(d*x+c))^(1/2)/
(a*c)^(1/2)/x^4/(d*x+c)^(3/2)
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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((b*x + a)^(5/2)/((d*x + c)^(5/2)*x^5),x, algorithm="maxima")
[Out]
Exception raised: ValueError
_______________________________________________________________________________________
Fricas [A] time = 6.12641, 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)/((d*x + c)^(5/2)*x^5),x, algorithm="fricas")
[Out]
[-1/768*(4*(48*a^3*c^5 + 3*(5*b^3*c^3*d^2 - 581*a*b^2*c^2*d^3 + 1715*a^2*b*c*d^4
- 1155*a^3*d^5)*x^5 + 6*(5*b^3*c^4*d - 412*a*b^2*c^3*d^2 + 1169*a^2*b*c^2*d^3 -
770*a^3*c*d^4)*x^4 + 3*(5*b^3*c^5 - 161*a*b^2*c^4*d + 387*a^2*b*c^3*d^2 - 231*a
^3*c^2*d^3)*x^3 + 2*(59*a*b^2*c^5 - 158*a^2*b*c^4*d + 99*a^3*c^3*d^2)*x^2 + 8*(1
7*a^2*b*c^5 - 11*a^3*c^4*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c) + 15*((b^4*
c^4*d^2 + 20*a*b^3*c^3*d^3 - 210*a^2*b^2*c^2*d^4 + 420*a^3*b*c*d^5 - 231*a^4*d^6
)*x^6 + 2*(b^4*c^5*d + 20*a*b^3*c^4*d^2 - 210*a^2*b^2*c^3*d^3 + 420*a^3*b*c^2*d^
4 - 231*a^4*c*d^5)*x^5 + (b^4*c^6 + 20*a*b^3*c^5*d - 210*a^2*b^2*c^4*d^2 + 420*a
^3*b*c^3*d^3 - 231*a^4*c^2*d^4)*x^4)*log(-(4*(2*a^2*c^2 + (a*b*c^2 + a^2*c*d)*x)
*sqrt(b*x + a)*sqrt(d*x + c) - (8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2
+ 8*(a*b*c^2 + a^2*c*d)*x)*sqrt(a*c))/x^2))/((a*c^6*d^2*x^6 + 2*a*c^7*d*x^5 + a*
c^8*x^4)*sqrt(a*c)), -1/384*(2*(48*a^3*c^5 + 3*(5*b^3*c^3*d^2 - 581*a*b^2*c^2*d^
3 + 1715*a^2*b*c*d^4 - 1155*a^3*d^5)*x^5 + 6*(5*b^3*c^4*d - 412*a*b^2*c^3*d^2 +
1169*a^2*b*c^2*d^3 - 770*a^3*c*d^4)*x^4 + 3*(5*b^3*c^5 - 161*a*b^2*c^4*d + 387*a
^2*b*c^3*d^2 - 231*a^3*c^2*d^3)*x^3 + 2*(59*a*b^2*c^5 - 158*a^2*b*c^4*d + 99*a^3
*c^3*d^2)*x^2 + 8*(17*a^2*b*c^5 - 11*a^3*c^4*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt
(d*x + c) - 15*((b^4*c^4*d^2 + 20*a*b^3*c^3*d^3 - 210*a^2*b^2*c^2*d^4 + 420*a^3*
b*c*d^5 - 231*a^4*d^6)*x^6 + 2*(b^4*c^5*d + 20*a*b^3*c^4*d^2 - 210*a^2*b^2*c^3*d
^3 + 420*a^3*b*c^2*d^4 - 231*a^4*c*d^5)*x^5 + (b^4*c^6 + 20*a*b^3*c^5*d - 210*a^
2*b^2*c^4*d^2 + 420*a^3*b*c^3*d^3 - 231*a^4*c^2*d^4)*x^4)*arctan(1/2*(2*a*c + (b
*c + a*d)*x)*sqrt(-a*c)/(sqrt(b*x + a)*sqrt(d*x + c)*a*c)))/((a*c^6*d^2*x^6 + 2*
a*c^7*d*x^5 + a*c^8*x^4)*sqrt(-a*c))]
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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)**(5/2)/x**5/(d*x+c)**(5/2),x)
[Out]
Timed out
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(5/2)/((d*x + c)^(5/2)*x^5),x, algorithm="giac")
[Out]
Exception raised: TypeError