3.677 \(\int \frac{(a+b x)^{5/2}}{x^5 (c+d x)^{3/2}} \, dx\)
Optimal. Leaf size=317 \[ -\frac{\sqrt{a+b x} \left (315 a^2 d^2-322 a b c d+15 b^2 c^2\right ) (b c-a d)}{192 a c^4 x \sqrt{c+d x}}+\frac{5 \left (-63 a^2 d^2+14 a b c d+b^2 c^2\right ) (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{3/2} c^{11/2}}-\frac{d \sqrt{a+b x} \left (-945 a^3 d^3+1785 a^2 b c d^2-839 a b^2 c^2 d+15 b^3 c^3\right )}{192 a c^5 \sqrt{c+d x}}-\frac{\sqrt{a+b x} (59 b c-63 a d) (b c-a d)}{96 c^3 x^2 \sqrt{c+d x}}-\frac{a \sqrt{a+b x} (11 b c-9 a d)}{24 c^2 x^3 \sqrt{c+d x}}-\frac{a (a+b x)^{3/2}}{4 c x^4 \sqrt{c+d x}} \]
[Out]
-(d*(15*b^3*c^3 - 839*a*b^2*c^2*d + 1785*a^2*b*c*d^2 - 945*a^3*d^3)*Sqrt[a + b*x
])/(192*a*c^5*Sqrt[c + d*x]) - (a*(11*b*c - 9*a*d)*Sqrt[a + b*x])/(24*c^2*x^3*Sq
rt[c + d*x]) - ((59*b*c - 63*a*d)*(b*c - a*d)*Sqrt[a + b*x])/(96*c^3*x^2*Sqrt[c
+ d*x]) - ((b*c - a*d)*(15*b^2*c^2 - 322*a*b*c*d + 315*a^2*d^2)*Sqrt[a + b*x])/(
192*a*c^4*x*Sqrt[c + d*x]) - (a*(a + b*x)^(3/2))/(4*c*x^4*Sqrt[c + d*x]) + (5*(b
*c - a*d)^2*(b^2*c^2 + 14*a*b*c*d - 63*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/
(Sqrt[a]*Sqrt[c + d*x])])/(64*a^(3/2)*c^(11/2))
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Rubi [A] time = 1.14008, antiderivative size = 317, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318
\[ -\frac{\sqrt{a+b x} \left (315 a^2 d^2-322 a b c d+15 b^2 c^2\right ) (b c-a d)}{192 a c^4 x \sqrt{c+d x}}+\frac{5 \left (-63 a^2 d^2+14 a b c d+b^2 c^2\right ) (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{3/2} c^{11/2}}-\frac{d \sqrt{a+b x} \left (-945 a^3 d^3+1785 a^2 b c d^2-839 a b^2 c^2 d+15 b^3 c^3\right )}{192 a c^5 \sqrt{c+d x}}-\frac{\sqrt{a+b x} (59 b c-63 a d) (b c-a d)}{96 c^3 x^2 \sqrt{c+d x}}-\frac{a \sqrt{a+b x} (11 b c-9 a d)}{24 c^2 x^3 \sqrt{c+d x}}-\frac{a (a+b x)^{3/2}}{4 c x^4 \sqrt{c+d x}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^(5/2)/(x^5*(c + d*x)^(3/2)),x]
[Out]
-(d*(15*b^3*c^3 - 839*a*b^2*c^2*d + 1785*a^2*b*c*d^2 - 945*a^3*d^3)*Sqrt[a + b*x
])/(192*a*c^5*Sqrt[c + d*x]) - (a*(11*b*c - 9*a*d)*Sqrt[a + b*x])/(24*c^2*x^3*Sq
rt[c + d*x]) - ((59*b*c - 63*a*d)*(b*c - a*d)*Sqrt[a + b*x])/(96*c^3*x^2*Sqrt[c
+ d*x]) - ((b*c - a*d)*(15*b^2*c^2 - 322*a*b*c*d + 315*a^2*d^2)*Sqrt[a + b*x])/(
192*a*c^4*x*Sqrt[c + d*x]) - (a*(a + b*x)^(3/2))/(4*c*x^4*Sqrt[c + d*x]) + (5*(b
*c - a*d)^2*(b^2*c^2 + 14*a*b*c*d - 63*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/
(Sqrt[a]*Sqrt[c + d*x])])/(64*a^(3/2)*c^(11/2))
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Rubi in Sympy [A] time = 158.916, size = 303, normalized size = 0.96 \[ - \frac{a \left (a + b x\right )^{\frac{3}{2}}}{4 c x^{4} \sqrt{c + d x}} + \frac{a \sqrt{a + b x} \left (9 a d - 11 b c\right )}{24 c^{2} x^{3} \sqrt{c + d x}} - \frac{\sqrt{a + b x} \left (a d - b c\right ) \left (63 a d - 59 b c\right )}{96 c^{3} x^{2} \sqrt{c + d x}} + \frac{\sqrt{a + b x} \left (a d - b c\right ) \left (315 a^{2} d^{2} - 322 a b c d + 15 b^{2} c^{2}\right )}{192 a c^{4} x \sqrt{c + d x}} + \frac{d \sqrt{a + b x} \left (945 a^{3} d^{3} - 1785 a^{2} b c d^{2} + 839 a b^{2} c^{2} d - 15 b^{3} c^{3}\right )}{192 a c^{5} \sqrt{c + d x}} - \frac{5 \left (a d - b c\right )^{2} \left (63 a^{2} d^{2} - 14 a b c d - b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{64 a^{\frac{3}{2}} c^{\frac{11}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**(5/2)/x**5/(d*x+c)**(3/2),x)
[Out]
-a*(a + b*x)**(3/2)/(4*c*x**4*sqrt(c + d*x)) + a*sqrt(a + b*x)*(9*a*d - 11*b*c)/
(24*c**2*x**3*sqrt(c + d*x)) - sqrt(a + b*x)*(a*d - b*c)*(63*a*d - 59*b*c)/(96*c
**3*x**2*sqrt(c + d*x)) + sqrt(a + b*x)*(a*d - b*c)*(315*a**2*d**2 - 322*a*b*c*d
+ 15*b**2*c**2)/(192*a*c**4*x*sqrt(c + d*x)) + d*sqrt(a + b*x)*(945*a**3*d**3 -
1785*a**2*b*c*d**2 + 839*a*b**2*c**2*d - 15*b**3*c**3)/(192*a*c**5*sqrt(c + d*x
)) - 5*(a*d - b*c)**2*(63*a**2*d**2 - 14*a*b*c*d - b**2*c**2)*atanh(sqrt(c)*sqrt
(a + b*x)/(sqrt(a)*sqrt(c + d*x)))/(64*a**(3/2)*c**(11/2))
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Mathematica [A] time = 0.419426, size = 292, normalized size = 0.92 \[ \frac{-15 \log (x) \left (-63 a^2 d^2+14 a b c d+b^2 c^2\right ) (b c-a d)^2+15 \left (-63 a^2 d^2+14 a b c d+b^2 c^2\right ) (b c-a d)^2 \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )+\frac{2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \left (a^3 \left (-48 c^4+72 c^3 d x-126 c^2 d^2 x^2+315 c d^3 x^3+945 d^4 x^4\right )-a^2 b c x \left (136 c^3-244 c^2 d x+637 c d^2 x^2+1785 d^3 x^3\right )+a b^2 c^2 x^2 \left (-118 c^2+337 c d x+839 d^2 x^2\right )-15 b^3 c^3 x^3 (c+d x)\right )}{x^4 \sqrt{c+d x}}}{384 a^{3/2} c^{11/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^(5/2)/(x^5*(c + d*x)^(3/2)),x]
[Out]
((2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*(-15*b^3*c^3*x^3*(c + d*x) + a*b^2*c^2*x^2*(-1
18*c^2 + 337*c*d*x + 839*d^2*x^2) - a^2*b*c*x*(136*c^3 - 244*c^2*d*x + 637*c*d^2
*x^2 + 1785*d^3*x^3) + a^3*(-48*c^4 + 72*c^3*d*x - 126*c^2*d^2*x^2 + 315*c*d^3*x
^3 + 945*d^4*x^4)))/(x^4*Sqrt[c + d*x]) - 15*(b*c - a*d)^2*(b^2*c^2 + 14*a*b*c*d
- 63*a^2*d^2)*Log[x] + 15*(b*c - a*d)^2*(b^2*c^2 + 14*a*b*c*d - 63*a^2*d^2)*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^(11/2))
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Maple [B] time = 0.052, size = 982, normalized size = 3.1 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^(5/2)/x^5/(d*x+c)^(3/2),x)
[Out]
-1/384*(b*x+a)^(1/2)*(945*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*d^5-2100*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*d^4+1350*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^2*d^3-180*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^3*d^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^5*b^4*c^4*d+945*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*d^4-2100*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^2*d^3+1350*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^3*d^2-180*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^4*d-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^5-1890*x^4*a^3*d^4
*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+3570*x^4*a^2*b*c*d^3*(a*c)^(1/2)*((b*x+a)*(
d*x+c))^(1/2)-1678*x^4*a*b^2*c^2*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+30*x^4*
b^3*c^3*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-630*x^3*a^3*c*d^3*(a*c)^(1/2)*((b*
x+a)*(d*x+c))^(1/2)+1274*x^3*a^2*b*c^2*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-6
74*x^3*a*b^2*c^3*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+30*x^3*b^3*c^4*(a*c)^(1/2
)*((b*x+a)*(d*x+c))^(1/2)+252*x^2*a^3*c^2*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2
)-488*x^2*a^2*b*c^3*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+236*x^2*a*b^2*c^4*(a*c
)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-144*x*a^3*c^3*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1
/2)+272*x*a^2*b*c^4*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+96*a^3*c^4*(a*c)^(1/2)*(
(b*x+a)*(d*x+c))^(1/2))/c^5/a/((b*x+a)*(d*x+c))^(1/2)/x^4/(a*c)^(1/2)/(d*x+c)^(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((b*x + a)^(5/2)/((d*x + c)^(3/2)*x^5),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 2.12195, 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)^(3/2)*x^5),x, algorithm="fricas")
[Out]
[-1/768*(4*(48*a^3*c^4 + (15*b^3*c^3*d - 839*a*b^2*c^2*d^2 + 1785*a^2*b*c*d^3 -
945*a^3*d^4)*x^4 + (15*b^3*c^4 - 337*a*b^2*c^3*d + 637*a^2*b*c^2*d^2 - 315*a^3*c
*d^3)*x^3 + 2*(59*a*b^2*c^4 - 122*a^2*b*c^3*d + 63*a^3*c^2*d^2)*x^2 + 8*(17*a^2*
b*c^4 - 9*a^3*c^3*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c) + 15*((b^4*c^4*d +
12*a*b^3*c^3*d^2 - 90*a^2*b^2*c^2*d^3 + 140*a^3*b*c*d^4 - 63*a^4*d^5)*x^5 + (b^
4*c^5 + 12*a*b^3*c^4*d - 90*a^2*b^2*c^3*d^2 + 140*a^3*b*c^2*d^3 - 63*a^4*c*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^5*d*x^5 + a*c^6*x^4)*sqrt(a*c)), -1/384*(2*(48*a^3*c^4 + (15*b
^3*c^3*d - 839*a*b^2*c^2*d^2 + 1785*a^2*b*c*d^3 - 945*a^3*d^4)*x^4 + (15*b^3*c^4
- 337*a*b^2*c^3*d + 637*a^2*b*c^2*d^2 - 315*a^3*c*d^3)*x^3 + 2*(59*a*b^2*c^4 -
122*a^2*b*c^3*d + 63*a^3*c^2*d^2)*x^2 + 8*(17*a^2*b*c^4 - 9*a^3*c^3*d)*x)*sqrt(-
a*c)*sqrt(b*x + a)*sqrt(d*x + c) - 15*((b^4*c^4*d + 12*a*b^3*c^3*d^2 - 90*a^2*b^
2*c^2*d^3 + 140*a^3*b*c*d^4 - 63*a^4*d^5)*x^5 + (b^4*c^5 + 12*a*b^3*c^4*d - 90*a
^2*b^2*c^3*d^2 + 140*a^3*b*c^2*d^3 - 63*a^4*c*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^5*d*x^5 + a*c^6*
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)**(3/2),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)^(3/2)*x^5),x, algorithm="giac")
[Out]
Exception raised: TypeError