3.799 \(\int \frac{1}{x (a+b x)^{5/2} (c+d x)^{5/2}} \, dx\)

Optimal. Leaf size=252 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{5/2} c^{5/2}}+\frac{2 d \sqrt{a+b x} (a d+b c) \left (3 a^2 d^2-14 a b c d+3 b^2 c^2\right )}{3 a^2 c^2 \sqrt{c+d x} (b c-a d)^4}+\frac{2 d \sqrt{a+b x} \left (-a^2 d^2-10 a b c d+3 b^2 c^2\right )}{3 a^2 c (c+d x)^{3/2} (b c-a d)^3}+\frac{2 b (b c-3 a d)}{a^2 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^2}+\frac{2 b}{3 a (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)} \]

[Out]

(2*b)/(3*a*(b*c - a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2)) + (2*b*(b*c - 3*a*d))/(a
^2*(b*c - a*d)^2*Sqrt[a + b*x]*(c + d*x)^(3/2)) + (2*d*(3*b^2*c^2 - 10*a*b*c*d -
a^2*d^2)*Sqrt[a + b*x])/(3*a^2*c*(b*c - a*d)^3*(c + d*x)^(3/2)) + (2*d*(b*c + a
*d)*(3*b^2*c^2 - 14*a*b*c*d + 3*a^2*d^2)*Sqrt[a + b*x])/(3*a^2*c^2*(b*c - a*d)^4
*Sqrt[c + d*x]) - (2*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(
a^(5/2)*c^(5/2))

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Rubi [A]  time = 0.871757, antiderivative size = 252, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{5/2} c^{5/2}}+\frac{2 d \sqrt{a+b x} (a d+b c) \left (3 a^2 d^2-14 a b c d+3 b^2 c^2\right )}{3 a^2 c^2 \sqrt{c+d x} (b c-a d)^4}+\frac{2 d \sqrt{a+b x} \left (-a^2 d^2-10 a b c d+3 b^2 c^2\right )}{3 a^2 c (c+d x)^{3/2} (b c-a d)^3}+\frac{2 b (b c-3 a d)}{a^2 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^2}+\frac{2 b}{3 a (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a + b*x)^(5/2)*(c + d*x)^(5/2)),x]

[Out]

(2*b)/(3*a*(b*c - a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2)) + (2*b*(b*c - 3*a*d))/(a
^2*(b*c - a*d)^2*Sqrt[a + b*x]*(c + d*x)^(3/2)) + (2*d*(3*b^2*c^2 - 10*a*b*c*d -
a^2*d^2)*Sqrt[a + b*x])/(3*a^2*c*(b*c - a*d)^3*(c + d*x)^(3/2)) + (2*d*(b*c + a
*d)*(3*b^2*c^2 - 14*a*b*c*d + 3*a^2*d^2)*Sqrt[a + b*x])/(3*a^2*c^2*(b*c - a*d)^4
*Sqrt[c + d*x]) - (2*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(
a^(5/2)*c^(5/2))

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Rubi in Sympy [A]  time = 100.825, size = 240, normalized size = 0.95 \[ - \frac{2 b}{3 a \left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )} - \frac{2 b \left (3 a d - b c\right )}{a^{2} \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )^{2}} + \frac{2 d \sqrt{a + b x} \left (a^{2} d^{2} + 10 a b c d - 3 b^{2} c^{2}\right )}{3 a^{2} c \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )^{3}} + \frac{2 d \sqrt{a + b x} \left (a d + b c\right ) \left (3 a^{2} d^{2} - 14 a b c d + 3 b^{2} c^{2}\right )}{3 a^{2} c^{2} \sqrt{c + d x} \left (a d - b c\right )^{4}} - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{a^{\frac{5}{2}} c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(b*x+a)**(5/2)/(d*x+c)**(5/2),x)

[Out]

-2*b/(3*a*(a + b*x)**(3/2)*(c + d*x)**(3/2)*(a*d - b*c)) - 2*b*(3*a*d - b*c)/(a*
*2*sqrt(a + b*x)*(c + d*x)**(3/2)*(a*d - b*c)**2) + 2*d*sqrt(a + b*x)*(a**2*d**2
+ 10*a*b*c*d - 3*b**2*c**2)/(3*a**2*c*(c + d*x)**(3/2)*(a*d - b*c)**3) + 2*d*sq
rt(a + b*x)*(a*d + b*c)*(3*a**2*d**2 - 14*a*b*c*d + 3*b**2*c**2)/(3*a**2*c**2*sq
rt(c + d*x)*(a*d - b*c)**4) - 2*atanh(sqrt(c)*sqrt(a + b*x)/(sqrt(a)*sqrt(c + d*
x)))/(a**(5/2)*c**(5/2))

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Mathematica [A]  time = 1.392, size = 209, normalized size = 0.83 \[ -\frac{\log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{a^{5/2} c^{5/2}}+\frac{\log (x)}{a^{5/2} c^{5/2}}+\frac{2}{3} \sqrt{a+b x} \sqrt{c+d x} \left (\frac{b^3 (3 b c-11 a d)}{a^2 (a+b x) (b c-a d)^4}-\frac{b^3}{a (a+b x)^2 (a d-b c)^3}+\frac{d^3 (3 a d-11 b c)}{c^2 (c+d x) (b c-a d)^4}-\frac{d^3}{c (c+d x)^2 (b c-a d)^3}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(a + b*x)^(5/2)*(c + d*x)^(5/2)),x]

[Out]

(2*Sqrt[a + b*x]*Sqrt[c + d*x]*(-(b^3/(a*(-(b*c) + a*d)^3*(a + b*x)^2)) + (b^3*(
3*b*c - 11*a*d))/(a^2*(b*c - a*d)^4*(a + b*x)) - d^3/(c*(b*c - a*d)^3*(c + d*x)^
2) + (d^3*(-11*b*c + 3*a*d))/(c^2*(b*c - a*d)^4*(c + d*x))))/3 + Log[x]/(a^(5/2)
*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^(5/2)*c^(5/2))

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Maple [B]  time = 0.069, size = 2033, normalized size = 8.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(b*x+a)^(5/2)/(d*x+c)^(5/2),x)

[Out]

-1/3/c^2/a^2*(-12*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*x^2*b^5*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^6*d^6+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*b^6*c^6+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^6*c^2*d^4+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^2*b^4*c^6-6*((b*x+a)*(d*x+c))^
(1/2)*(a*c)^(1/2)*x*a^5*d^5+24*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^4*b*c^2*d^3
+24*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^2*b^3*c^4*d-6*((b*x+a)*(d*x+c))^(1/2)*
(a*c)^(1/2)*x^3*a^3*b^2*d^5-6*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*x^3*b^5*c^3*d^
2-12*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*x^2*a^4*b*d^5-12*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*d^5+18*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^2*d^4-12*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^5*c^3*d^3-18*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^4*b^2*c*d^5+12*
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^3*c^2*
d^4+12*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
^4*c^3*d^3-18*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^5*c^4*d^2-27*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*b^2*c^2*d^4+48*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^3*c^3*d^3-27*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^4*c^4*d^2-18*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^5*b*c^2*d^4+12*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*b^2*c^3*d^3+12*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^3*c^4*d^2-18*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^4*c^5*d+22*((b*x+a)*(d*x+c))^(1/2)*
(a*c)^(1/2)*x^3*a*b^4*c^2*d^3+36*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*x^2*a^3*b^2
*c*d^4+48*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*x^2*a^2*b^3*c^2*d^3+6*((b*x+a)*(d*
x+c))^(1/2)*(a*c)^(1/2)*x*a*b^4*c^4*d+22*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*x^3
*a^2*b^3*c*d^4+48*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*x*a^3*b^2*c^2*d^3+48*((b*x
+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*x*a^2*b^3*c^3*d^2+36*((b*x+a)*(d*x+c))^(1/2)*(a*c
)^(1/2)*x^2*a*b^4*c^3*d^2+6*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*x*a^4*b*c*d^4-6*
((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*x*b^5*c^5-8*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1
/2)*a^5*c*d^4-8*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a*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)*x^4*a^4*b^2*d^6+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^4*b^6*c^4*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^3*a^5*b*d^6+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^3*b^6*c^5*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^6*c*d^5+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*b^5*c^6-12*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^5*b*c^3*d^3+18*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*b^2*c^4*d^2-12*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^3*c^5*d)/((b*x+a)*(d*
x+c))^(1/2)/(a*d-b*c)^4/(a*c)^(1/2)/(b*x+a)^(3/2)/(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(1/((b*x + a)^(5/2)*(d*x + c)^(5/2)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.912071, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(5/2)*(d*x + c)^(5/2)*x),x, algorithm="fricas")

[Out]

[1/6*(4*(4*a*b^4*c^5 - 12*a^2*b^3*c^4*d - 12*a^4*b*c^2*d^3 + 4*a^5*c*d^4 + (3*b^
5*c^3*d^2 - 11*a*b^4*c^2*d^3 - 11*a^2*b^3*c*d^4 + 3*a^3*b^2*d^5)*x^3 + 6*(b^5*c^
4*d - 3*a*b^4*c^3*d^2 - 4*a^2*b^3*c^2*d^3 - 3*a^3*b^2*c*d^4 + a^4*b*d^5)*x^2 + 3
*(b^5*c^5 - a*b^4*c^4*d - 8*a^2*b^3*c^3*d^2 - 8*a^3*b^2*c^2*d^3 - a^4*b*c*d^4 +
a^5*d^5)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c) + 3*(a^2*b^4*c^6 - 4*a^3*b^3*c
^5*d + 6*a^4*b^2*c^4*d^2 - 4*a^5*b*c^3*d^3 + a^6*c^2*d^4 + (b^6*c^4*d^2 - 4*a*b^
5*c^3*d^3 + 6*a^2*b^4*c^2*d^4 - 4*a^3*b^3*c*d^5 + a^4*b^2*d^6)*x^4 + 2*(b^6*c^5*
d - 3*a*b^5*c^4*d^2 + 2*a^2*b^4*c^3*d^3 + 2*a^3*b^3*c^2*d^4 - 3*a^4*b^2*c*d^5 +
a^5*b*d^6)*x^3 + (b^6*c^6 - 9*a^2*b^4*c^4*d^2 + 16*a^3*b^3*c^3*d^3 - 9*a^4*b^2*c
^2*d^4 + a^6*d^6)*x^2 + 2*(a*b^5*c^6 - 3*a^2*b^4*c^5*d + 2*a^3*b^3*c^4*d^2 + 2*a
^4*b^2*c^3*d^3 - 3*a^5*b*c^2*d^4 + a^6*c*d^5)*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^4*b^4*c^8 - 4*a^5*b^
3*c^7*d + 6*a^6*b^2*c^6*d^2 - 4*a^7*b*c^5*d^3 + a^8*c^4*d^4 + (a^2*b^6*c^6*d^2 -
4*a^3*b^5*c^5*d^3 + 6*a^4*b^4*c^4*d^4 - 4*a^5*b^3*c^3*d^5 + a^6*b^2*c^2*d^6)*x^
4 + 2*(a^2*b^6*c^7*d - 3*a^3*b^5*c^6*d^2 + 2*a^4*b^4*c^5*d^3 + 2*a^5*b^3*c^4*d^4
- 3*a^6*b^2*c^3*d^5 + a^7*b*c^2*d^6)*x^3 + (a^2*b^6*c^8 - 9*a^4*b^4*c^6*d^2 + 1
6*a^5*b^3*c^5*d^3 - 9*a^6*b^2*c^4*d^4 + a^8*c^2*d^6)*x^2 + 2*(a^3*b^5*c^8 - 3*a^
4*b^4*c^7*d + 2*a^5*b^3*c^6*d^2 + 2*a^6*b^2*c^5*d^3 - 3*a^7*b*c^4*d^4 + a^8*c^3*
d^5)*x)*sqrt(a*c)), 1/3*(2*(4*a*b^4*c^5 - 12*a^2*b^3*c^4*d - 12*a^4*b*c^2*d^3 +
4*a^5*c*d^4 + (3*b^5*c^3*d^2 - 11*a*b^4*c^2*d^3 - 11*a^2*b^3*c*d^4 + 3*a^3*b^2*d
^5)*x^3 + 6*(b^5*c^4*d - 3*a*b^4*c^3*d^2 - 4*a^2*b^3*c^2*d^3 - 3*a^3*b^2*c*d^4 +
a^4*b*d^5)*x^2 + 3*(b^5*c^5 - a*b^4*c^4*d - 8*a^2*b^3*c^3*d^2 - 8*a^3*b^2*c^2*d
^3 - a^4*b*c*d^4 + a^5*d^5)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c) - 3*(a^2*b
^4*c^6 - 4*a^3*b^3*c^5*d + 6*a^4*b^2*c^4*d^2 - 4*a^5*b*c^3*d^3 + a^6*c^2*d^4 + (
b^6*c^4*d^2 - 4*a*b^5*c^3*d^3 + 6*a^2*b^4*c^2*d^4 - 4*a^3*b^3*c*d^5 + a^4*b^2*d^
6)*x^4 + 2*(b^6*c^5*d - 3*a*b^5*c^4*d^2 + 2*a^2*b^4*c^3*d^3 + 2*a^3*b^3*c^2*d^4
- 3*a^4*b^2*c*d^5 + a^5*b*d^6)*x^3 + (b^6*c^6 - 9*a^2*b^4*c^4*d^2 + 16*a^3*b^3*c
^3*d^3 - 9*a^4*b^2*c^2*d^4 + a^6*d^6)*x^2 + 2*(a*b^5*c^6 - 3*a^2*b^4*c^5*d + 2*a
^3*b^3*c^4*d^2 + 2*a^4*b^2*c^3*d^3 - 3*a^5*b*c^2*d^4 + a^6*c*d^5)*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^4*b^4
*c^8 - 4*a^5*b^3*c^7*d + 6*a^6*b^2*c^6*d^2 - 4*a^7*b*c^5*d^3 + a^8*c^4*d^4 + (a^
2*b^6*c^6*d^2 - 4*a^3*b^5*c^5*d^3 + 6*a^4*b^4*c^4*d^4 - 4*a^5*b^3*c^3*d^5 + a^6*
b^2*c^2*d^6)*x^4 + 2*(a^2*b^6*c^7*d - 3*a^3*b^5*c^6*d^2 + 2*a^4*b^4*c^5*d^3 + 2*
a^5*b^3*c^4*d^4 - 3*a^6*b^2*c^3*d^5 + a^7*b*c^2*d^6)*x^3 + (a^2*b^6*c^8 - 9*a^4*
b^4*c^6*d^2 + 16*a^5*b^3*c^5*d^3 - 9*a^6*b^2*c^4*d^4 + a^8*c^2*d^6)*x^2 + 2*(a^3
*b^5*c^8 - 3*a^4*b^4*c^7*d + 2*a^5*b^3*c^6*d^2 + 2*a^6*b^2*c^5*d^3 - 3*a^7*b*c^4
*d^4 + a^8*c^3*d^5)*x)*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(1/x/(b*x+a)**(5/2)/(d*x+c)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 1.43369, 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)^(5/2)*(d*x + c)^(5/2)*x),x, algorithm="giac")

[Out]

sage0*x