3.790 \(\int \frac{(c+d x)^{5/2}}{x^5 (a+b x)^{5/2}} \, dx\)
Optimal. Leaf size=388 \[ -\frac{\sqrt{c+d x} (99 b c-59 a d) (b c-a d)}{96 a^3 x^2 (a+b x)^{3/2}}+\frac{11 c \sqrt{c+d x} (b c-a d)}{24 a^2 x^3 (a+b x)^{3/2}}+\frac{b \sqrt{c+d x} (b c-a d) \left (5 a^2 d^2-238 a b c d+385 b^2 c^2\right )}{64 a^5 c (a+b x)^{3/2}}+\frac{\sqrt{c+d x} (b c-a d) \left (5 a^2 d^2-156 a b c d+231 b^2 c^2\right )}{64 a^4 c x (a+b x)^{3/2}}-\frac{5 (b c-a d) \left (a^3 d^3+21 a^2 b c d^2-189 a b^2 c^2 d+231 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{13/2} c^{3/2}}+\frac{b \sqrt{c+d x} \left (-5 a^3 d^3+581 a^2 b c d^2-1715 a b^2 c^2 d+1155 b^3 c^3\right )}{64 a^6 c \sqrt{a+b x}}-\frac{c (c+d x)^{3/2}}{4 a x^4 (a+b x)^{3/2}} \]
[Out]
(b*(b*c - a*d)*(385*b^2*c^2 - 238*a*b*c*d + 5*a^2*d^2)*Sqrt[c + d*x])/(64*a^5*c*
(a + b*x)^(3/2)) + (11*c*(b*c - a*d)*Sqrt[c + d*x])/(24*a^2*x^3*(a + b*x)^(3/2))
- ((99*b*c - 59*a*d)*(b*c - a*d)*Sqrt[c + d*x])/(96*a^3*x^2*(a + b*x)^(3/2)) +
((b*c - a*d)*(231*b^2*c^2 - 156*a*b*c*d + 5*a^2*d^2)*Sqrt[c + d*x])/(64*a^4*c*x*
(a + b*x)^(3/2)) + (b*(1155*b^3*c^3 - 1715*a*b^2*c^2*d + 581*a^2*b*c*d^2 - 5*a^3
*d^3)*Sqrt[c + d*x])/(64*a^6*c*Sqrt[a + b*x]) - (c*(c + d*x)^(3/2))/(4*a*x^4*(a
+ b*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)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(64*a^(13/2)
*c^(3/2))
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Rubi [A] time = 1.61654, 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
\[ -\frac{\sqrt{c+d x} (99 b c-59 a d) (b c-a d)}{96 a^3 x^2 (a+b x)^{3/2}}+\frac{11 c \sqrt{c+d x} (b c-a d)}{24 a^2 x^3 (a+b x)^{3/2}}+\frac{b \sqrt{c+d x} (b c-a d) \left (5 a^2 d^2-238 a b c d+385 b^2 c^2\right )}{64 a^5 c (a+b x)^{3/2}}+\frac{\sqrt{c+d x} (b c-a d) \left (5 a^2 d^2-156 a b c d+231 b^2 c^2\right )}{64 a^4 c x (a+b x)^{3/2}}-\frac{5 (b c-a d) \left (a^3 d^3+21 a^2 b c d^2-189 a b^2 c^2 d+231 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{13/2} c^{3/2}}+\frac{b \sqrt{c+d x} \left (-5 a^3 d^3+581 a^2 b c d^2-1715 a b^2 c^2 d+1155 b^3 c^3\right )}{64 a^6 c \sqrt{a+b x}}-\frac{c (c+d x)^{3/2}}{4 a x^4 (a+b x)^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x)^(5/2)/(x^5*(a + b*x)^(5/2)),x]
[Out]
(b*(b*c - a*d)*(385*b^2*c^2 - 238*a*b*c*d + 5*a^2*d^2)*Sqrt[c + d*x])/(64*a^5*c*
(a + b*x)^(3/2)) + (11*c*(b*c - a*d)*Sqrt[c + d*x])/(24*a^2*x^3*(a + b*x)^(3/2))
- ((99*b*c - 59*a*d)*(b*c - a*d)*Sqrt[c + d*x])/(96*a^3*x^2*(a + b*x)^(3/2)) +
((b*c - a*d)*(231*b^2*c^2 - 156*a*b*c*d + 5*a^2*d^2)*Sqrt[c + d*x])/(64*a^4*c*x*
(a + b*x)^(3/2)) + (b*(1155*b^3*c^3 - 1715*a*b^2*c^2*d + 581*a^2*b*c*d^2 - 5*a^3
*d^3)*Sqrt[c + d*x])/(64*a^6*c*Sqrt[a + b*x]) - (c*(c + d*x)^(3/2))/(4*a*x^4*(a
+ b*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)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(64*a^(13/2)
*c^(3/2))
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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((d*x+c)**(5/2)/x**5/(b*x+a)**(5/2),x)
[Out]
Timed out
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Mathematica [A] time = 1.88524, size = 344, normalized size = 0.89 \[ \frac{2 \sqrt{a} \sqrt{a+b x} \sqrt{c+d x} \left (-\frac{48 a^3 c^2}{x^4}-\frac{2 a \left (59 a^2 d^2-294 a b c d+259 b^2 c^2\right )}{x^2}+\frac{128 b^2 \left (8 a^2 d^2-23 a b c d+15 b^2 c^2\right )}{a+b x}-\frac{8 a^2 c (17 a d-23 b c)}{x^3}+\frac{-15 a^3 d^3+719 a^2 b c d^2-2201 a b^2 c^2 d+1545 b^3 c^3}{c x}+\frac{128 a b^2 (b c-a d)^2}{(a+b x)^2}\right )+\frac{15 \log (x) (b c-a d) \left (a^3 d^3+21 a^2 b c d^2-189 a b^2 c^2 d+231 b^3 c^3\right )}{c^{3/2}}+\frac{15 (a d-b c) \left (a^3 d^3+21 a^2 b c d^2-189 a b^2 c^2 d+231 b^3 c^3\right ) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{c^{3/2}}}{384 a^{13/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x)^(5/2)/(x^5*(a + b*x)^(5/2)),x]
[Out]
(2*Sqrt[a]*Sqrt[a + b*x]*Sqrt[c + d*x]*((-48*a^3*c^2)/x^4 - (8*a^2*c*(-23*b*c +
17*a*d))/x^3 - (2*a*(259*b^2*c^2 - 294*a*b*c*d + 59*a^2*d^2))/x^2 + (1545*b^3*c^
3 - 2201*a*b^2*c^2*d + 719*a^2*b*c*d^2 - 15*a^3*d^3)/(c*x) + (128*a*b^2*(b*c - a
*d)^2)/(a + b*x)^2 + (128*b^2*(15*b^2*c^2 - 23*a*b*c*d + 8*a^2*d^2))/(a + b*x))
+ (15*(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
[x])/c^(3/2) + (15*(-(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[2*a*c + b*c*x + a*d*x + 2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c
+ d*x]])/c^(3/2))/(384*a^(13/2))
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Maple [B] time = 0.056, size = 1377, normalized size = 3.6 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)^(5/2)/x^5/(b*x+a)^(5/2),x)
[Out]
1/384*(d*x+c)^(1/2)*(-30*x^5*a^3*b^2*d^3*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-346
5*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^6*c^4+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*a^6*d^4-96*
a^5*c^3*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+6930*x^5*b^5*c^3*((b*x+a)*(d*x+c))^(
1/2)*(a*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^4*a^3*b^3*c^3*d-60*x^4*a^4*b*d^3*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+924
0*x^4*a*b^4*c^3*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+1386*x^3*a^2*b^3*c^3*((b*x+a
)*(d*x+c))^(1/2)*(a*c)^(1/2)-236*x^2*a^5*c*d^2*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/
2)-396*x^2*a^3*b^2*c^3*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-272*x*a^5*c^2*d*((b*x
+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+176*x*a^4*b*c^3*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/
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^6*a^3*b
^3*c*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^
6*a^2*b^4*c^2*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^6*a*b^5*c^3*d+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^4*b^2*c*d^3-6300*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^3*b^3*c^2*d^2+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^2*b^4*c^3*d+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^5*b*c*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^4*b^2*c^2*d^2+3486*x^5*a^2*b^3*c*d^
2*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-10290*x^5*a*b^4*c^2*d*((b*x+a)*(d*x+c))^(1
/2)*(a*c)^(1/2)+4944*x^4*a^3*b^2*c*d^2*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-14028
*x^4*a^2*b^3*c^2*d*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+966*x^3*a^4*b*c*d^2*((b*x
+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-2322*x^3*a^3*b^2*c^2*d*((b*x+a)*(d*x+c))^(1/2)*(a
*c)^(1/2)+632*x^2*a^4*b*c^2*d*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-30*x^3*a^5*d^3
*((b*x+a)*(d*x+c))^(1/2)*(a*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*a^4*b^2*d^4+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*a^5*b*d^4-6930*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*b^5*c^4-3465*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^2*b^4*c^4)/c/a^6/((b*x+a)*(d*x+c))^(1/2)/
(a*c)^(1/2)/x^4/(b*x+a)^(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((d*x + c)^(5/2)/((b*x + a)^(5/2)*x^5),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 7.05391, 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)/((b*x + a)^(5/2)*x^5),x, algorithm="fricas")
[Out]
[-1/768*(4*(48*a^5*c^3 - 3*(1155*b^5*c^3 - 1715*a*b^4*c^2*d + 581*a^2*b^3*c*d^2
- 5*a^3*b^2*d^3)*x^5 - 6*(770*a*b^4*c^3 - 1169*a^2*b^3*c^2*d + 412*a^3*b^2*c*d^2
- 5*a^4*b*d^3)*x^4 - 3*(231*a^2*b^3*c^3 - 387*a^3*b^2*c^2*d + 161*a^4*b*c*d^2 -
5*a^5*d^3)*x^3 + 2*(99*a^3*b^2*c^3 - 158*a^4*b*c^2*d + 59*a^5*c*d^2)*x^2 - 8*(1
1*a^4*b*c^3 - 17*a^5*c^2*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c) + 15*((231*
b^6*c^4 - 420*a*b^5*c^3*d + 210*a^2*b^4*c^2*d^2 - 20*a^3*b^3*c*d^3 - a^4*b^2*d^4
)*x^6 + 2*(231*a*b^5*c^4 - 420*a^2*b^4*c^3*d + 210*a^3*b^3*c^2*d^2 - 20*a^4*b^2*
c*d^3 - a^5*b*d^4)*x^5 + (231*a^2*b^4*c^4 - 420*a^3*b^3*c^3*d + 210*a^4*b^2*c^2*
d^2 - 20*a^5*b*c*d^3 - a^6*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^6*b^2*c*x^6 + 2*a^7*b*c*x^5 + a^8
*c*x^4)*sqrt(a*c)), -1/384*(2*(48*a^5*c^3 - 3*(1155*b^5*c^3 - 1715*a*b^4*c^2*d +
581*a^2*b^3*c*d^2 - 5*a^3*b^2*d^3)*x^5 - 6*(770*a*b^4*c^3 - 1169*a^2*b^3*c^2*d
+ 412*a^3*b^2*c*d^2 - 5*a^4*b*d^3)*x^4 - 3*(231*a^2*b^3*c^3 - 387*a^3*b^2*c^2*d
+ 161*a^4*b*c*d^2 - 5*a^5*d^3)*x^3 + 2*(99*a^3*b^2*c^3 - 158*a^4*b*c^2*d + 59*a^
5*c*d^2)*x^2 - 8*(11*a^4*b*c^3 - 17*a^5*c^2*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(
d*x + c) + 15*((231*b^6*c^4 - 420*a*b^5*c^3*d + 210*a^2*b^4*c^2*d^2 - 20*a^3*b^3
*c*d^3 - a^4*b^2*d^4)*x^6 + 2*(231*a*b^5*c^4 - 420*a^2*b^4*c^3*d + 210*a^3*b^3*c
^2*d^2 - 20*a^4*b^2*c*d^3 - a^5*b*d^4)*x^5 + (231*a^2*b^4*c^4 - 420*a^3*b^3*c^3*
d + 210*a^4*b^2*c^2*d^2 - 20*a^5*b*c*d^3 - a^6*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^6*b^2*c*x^6 + 2*a
^7*b*c*x^5 + a^8*c*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((d*x+c)**(5/2)/x**5/(b*x+a)**(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((d*x + c)^(5/2)/((b*x + a)^(5/2)*x^5),x, algorithm="giac")
[Out]
Exception raised: TypeError