3.778 \(\int \frac{1}{x (a+b x)^{3/2} (c+d x)^{5/2}} \, dx\)
Optimal. Leaf size=180 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{3/2} c^{5/2}}+\frac{2 d \sqrt{a+b x} (3 b c-a d) (3 a d+b c)}{3 a c^2 \sqrt{c+d x} (b c-a d)^3}+\frac{2 b}{a \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)}+\frac{2 d \sqrt{a+b x} (a d+3 b c)}{3 a c (c+d x)^{3/2} (b c-a d)^2} \]
[Out]
(2*b)/(a*(b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2)) + (2*d*(3*b*c + a*d)*Sqrt[a
+ b*x])/(3*a*c*(b*c - a*d)^2*(c + d*x)^(3/2)) + (2*d*(3*b*c - a*d)*(b*c + 3*a*d)
*Sqrt[a + b*x])/(3*a*c^2*(b*c - a*d)^3*Sqrt[c + d*x]) - (2*ArcTanh[(Sqrt[c]*Sqrt
[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(a^(3/2)*c^(5/2))
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Rubi [A] time = 0.553901, antiderivative size = 180, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227
\[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{3/2} c^{5/2}}+\frac{2 d \sqrt{a+b x} (3 b c-a d) (3 a d+b c)}{3 a c^2 \sqrt{c+d x} (b c-a d)^3}+\frac{2 b}{a \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)}+\frac{2 d \sqrt{a+b x} (a d+3 b c)}{3 a c (c+d x)^{3/2} (b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Int[1/(x*(a + b*x)^(3/2)*(c + d*x)^(5/2)),x]
[Out]
(2*b)/(a*(b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2)) + (2*d*(3*b*c + a*d)*Sqrt[a
+ b*x])/(3*a*c*(b*c - a*d)^2*(c + d*x)^(3/2)) + (2*d*(3*b*c - a*d)*(b*c + 3*a*d)
*Sqrt[a + b*x])/(3*a*c^2*(b*c - a*d)^3*Sqrt[c + d*x]) - (2*ArcTanh[(Sqrt[c]*Sqrt
[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(a^(3/2)*c^(5/2))
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Rubi in Sympy [A] time = 60.9026, size = 163, normalized size = 0.91 \[ - \frac{2 b}{a \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )} + \frac{2 d \sqrt{a + b x} \left (a d + 3 b c\right )}{3 a c \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )^{2}} + \frac{2 d \sqrt{a + b x} \left (a d - 3 b c\right ) \left (3 a d + b c\right )}{3 a c^{2} \sqrt{c + d x} \left (a d - b c\right )^{3}} - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{a^{\frac{3}{2}} c^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x/(b*x+a)**(3/2)/(d*x+c)**(5/2),x)
[Out]
-2*b/(a*sqrt(a + b*x)*(c + d*x)**(3/2)*(a*d - b*c)) + 2*d*sqrt(a + b*x)*(a*d + 3
*b*c)/(3*a*c*(c + d*x)**(3/2)*(a*d - b*c)**2) + 2*d*sqrt(a + b*x)*(a*d - 3*b*c)*
(3*a*d + b*c)/(3*a*c**2*sqrt(c + d*x)*(a*d - b*c)**3) - 2*atanh(sqrt(c)*sqrt(a +
b*x)/(sqrt(a)*sqrt(c + d*x)))/(a**(3/2)*c**(5/2))
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Mathematica [A] time = 0.826125, size = 175, normalized size = 0.97 \[ -\frac{\log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{a^{3/2} c^{5/2}}+\frac{\log (x)}{a^{3/2} c^{5/2}}+\frac{2}{3} \sqrt{a+b x} \sqrt{c+d x} \left (-\frac{3 b^3}{a (a+b x) (a d-b c)^3}+\frac{d^2 (8 b c-3 a d)}{c^2 (c+d x) (b c-a d)^3}+\frac{d^2}{c (c+d x)^2 (b c-a d)^2}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[1/(x*(a + b*x)^(3/2)*(c + d*x)^(5/2)),x]
[Out]
(2*Sqrt[a + b*x]*Sqrt[c + d*x]*((-3*b^3)/(a*(-(b*c) + a*d)^3*(a + b*x)) + d^2/(c
*(b*c - a*d)^2*(c + d*x)^2) + (d^2*(8*b*c - 3*a*d))/(c^2*(b*c - a*d)^3*(c + d*x)
)))/3 + Log[x]/(a^(3/2)*c^(5/2)) - Log[2*a*c + b*c*x + a*d*x + 2*Sqrt[a]*Sqrt[c]
*Sqrt[a + b*x]*Sqrt[c + d*x]]/(a^(3/2)*c^(5/2))
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Maple [B] time = 0.06, size = 1253, normalized size = 7. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x/(b*x+a)^(3/2)/(d*x+c)^(5/2),x)
[Out]
-1/3*(-6*x^2*a^2*b*d^4*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-8*a^3*c*d^3*(a*c)^(1/
2)*((b*x+a)*(d*x+c))^(1/2)+3*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^3*a^3*b*d^5-3*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^3*b^4*c^3*d^2-6*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^2*b^4*c^4*d+6*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*a^4*c*d^4-9*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/
2)+2*a*c)/x)*a^3*b*c^3*d^2+9*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/
2)+2*a*c)/x)*a^2*b^2*c^4*d+3*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^2*a^4*d^5-3*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*b^4*c^5+3*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a
*c)/x)*a^4*c^2*d^3-3*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c
)/x)*a*b^3*c^5+6*b^3*c^4*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-6*x*a^3*d^4*(a*c)^(
1/2)*((b*x+a)*(d*x+c))^(1/2)-9*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^3*a^2*b^2*c*d^4+9*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^3*a*b^3*c^2*d^3-3*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^2*a^3*b*c*d^4-9*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^2*a^2*b^2*c^2*d^3+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^2*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*a^3*b*c^2*d^3+9*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*a^2*b^2*c^3*d^2+3*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*a*b^3*c^4*d+6*x^2*b^3*c^2*d^2*(a*c)
^(1/2)*((b*x+a)*(d*x+c))^(1/2)+12*x*b^3*c^3*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2
)+18*a^2*b*c^2*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+16*x^2*a*b^2*c*d^3*(a*c)^
(1/2)*((b*x+a)*(d*x+c))^(1/2)+8*x*a^2*b*c*d^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2
)+18*x*a*b^2*c^2*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/a/c^2/(a*c)^(1/2)/(a*d
-b*c)^3/((b*x+a)*(d*x+c))^(1/2)/(d*x+c)^(3/2)/(b*x+a)^(1/2)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{3}{2}}{\left (d x + c\right )}^{\frac{5}{2}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^(3/2)*(d*x + c)^(5/2)*x),x, algorithm="maxima")
[Out]
integrate(1/((b*x + a)^(3/2)*(d*x + c)^(5/2)*x), x)
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Fricas [A] time = 0.454633, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^(3/2)*(d*x + c)^(5/2)*x),x, algorithm="fricas")
[Out]
[1/6*(4*(3*b^3*c^4 + 9*a^2*b*c^2*d^2 - 4*a^3*c*d^3 + (3*b^3*c^2*d^2 + 8*a*b^2*c*
d^3 - 3*a^2*b*d^4)*x^2 + (6*b^3*c^3*d + 9*a*b^2*c^2*d^2 + 4*a^2*b*c*d^3 - 3*a^3*
d^4)*x)*sqrt(a*c)*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*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^2*b^3*c^7 - 3*a^3*b^2*c^6*d + 3*a^4*b*c^5*d^2 -
a^5*c^4*d^3 + (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)*x^3 + (2*a*b^4*c^6*d - 5*a^2*b^3*c^5*d^2 + 3*a^3*b^2*c^4*d^3 + a^4*b*c^3*
d^4 - a^5*c^2*d^5)*x^2 + (a*b^4*c^7 - a^2*b^3*c^6*d - 3*a^3*b^2*c^5*d^2 + 5*a^4*
b*c^4*d^3 - 2*a^5*c^3*d^4)*x)*sqrt(a*c)), 1/3*(2*(3*b^3*c^4 + 9*a^2*b*c^2*d^2 -
4*a^3*c*d^3 + (3*b^3*c^2*d^2 + 8*a*b^2*c*d^3 - 3*a^2*b*d^4)*x^2 + (6*b^3*c^3*d +
9*a*b^2*c^2*d^2 + 4*a^2*b*c*d^3 - 3*a^3*d^4)*x)*sqrt(-a*c)*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*a*
c + (b*c + a*d)*x)*sqrt(-a*c)/(sqrt(b*x + a)*sqrt(d*x + c)*a*c)))/((a^2*b^3*c^7
- 3*a^3*b^2*c^6*d + 3*a^4*b*c^5*d^2 - a^5*c^4*d^3 + (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)*x^3 + (2*a*b^4*c^6*d - 5*a^2*b^3*c^5
*d^2 + 3*a^3*b^2*c^4*d^3 + a^4*b*c^3*d^4 - a^5*c^2*d^5)*x^2 + (a*b^4*c^7 - a^2*b
^3*c^6*d - 3*a^3*b^2*c^5*d^2 + 5*a^4*b*c^4*d^3 - 2*a^5*c^3*d^4)*x)*sqrt(-a*c))]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x/(b*x+a)**(3/2)/(d*x+c)**(5/2),x)
[Out]
Integral(1/(x*(a + b*x)**(3/2)*(c + d*x)**(5/2)), x)
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GIAC/XCAS [A] time = 1.08185, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^(3/2)*(d*x + c)^(5/2)*x),x, algorithm="giac")
[Out]
sage0*x