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

Optimal. Leaf size=124 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{\sqrt{a} c^{5/2}}-\frac{2 d \sqrt{a+b x} (5 b c-3 a d)}{3 c^2 \sqrt{c+d x} (b c-a d)^2}-\frac{2 d \sqrt{a+b x}}{3 c (c+d x)^{3/2} (b c-a d)} \]

[Out]

(-2*d*Sqrt[a + b*x])/(3*c*(b*c - a*d)*(c + d*x)^(3/2)) - (2*d*(5*b*c - 3*a*d)*Sq
rt[a + b*x])/(3*c^2*(b*c - a*d)^2*Sqrt[c + d*x]) - (2*ArcTanh[(Sqrt[c]*Sqrt[a +
b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(Sqrt[a]*c^(5/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.314089, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{\sqrt{a} c^{5/2}}-\frac{2 d \sqrt{a+b x} (5 b c-3 a d)}{3 c^2 \sqrt{c+d x} (b c-a d)^2}-\frac{2 d \sqrt{a+b x}}{3 c (c+d x)^{3/2} (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

(-2*d*Sqrt[a + b*x])/(3*c*(b*c - a*d)*(c + d*x)^(3/2)) - (2*d*(5*b*c - 3*a*d)*Sq
rt[a + b*x])/(3*c^2*(b*c - a*d)^2*Sqrt[c + d*x]) - (2*ArcTanh[(Sqrt[c]*Sqrt[a +
b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(Sqrt[a]*c^(5/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 33.8376, size = 114, normalized size = 0.92 \[ \frac{2 d \sqrt{a + b x}}{3 c \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )} + \frac{2 d \sqrt{a + b x} \left (3 a d - 5 b c\right )}{3 c^{2} \sqrt{c + d x} \left (a d - b c\right )^{2}} - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{\sqrt{a} c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*d*sqrt(a + b*x)/(3*c*(c + d*x)**(3/2)*(a*d - b*c)) + 2*d*sqrt(a + b*x)*(3*a*d
- 5*b*c)/(3*c**2*sqrt(c + d*x)*(a*d - b*c)**2) - 2*atanh(sqrt(c)*sqrt(a + b*x)/(
sqrt(a)*sqrt(c + d*x)))/(sqrt(a)*c**(5/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.334796, size = 130, normalized size = 1.05 \[ \frac{\frac{2 \sqrt{c} d \sqrt{a+b x} (a d (4 c+3 d x)-b c (6 c+5 d x))}{(c+d x)^{3/2} (b c-a d)^2}-\frac{3 \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{\sqrt{a}}+\frac{3 \log (x)}{\sqrt{a}}}{3 c^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

((2*Sqrt[c]*d*Sqrt[a + b*x]*(a*d*(4*c + 3*d*x) - b*c*(6*c + 5*d*x)))/((b*c - a*d
)^2*(c + d*x)^(3/2)) + (3*Log[x])/Sqrt[a] - (3*Log[2*a*c + b*c*x + a*d*x + 2*Sqr
t[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]])/Sqrt[a])/(3*c^(5/2))

_______________________________________________________________________________________

Maple [B]  time = 0.045, size = 586, normalized size = 4.7 \[ -{\frac{1}{3\,{c}^{2} \left ( ad-bc \right ) ^{2}}\sqrt{bx+a} \left ( 3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}{a}^{2}{d}^{4}-6\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}abc{d}^{3}+3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}{b}^{2}{c}^{2}{d}^{2}+6\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ) x{a}^{2}c{d}^{3}-12\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ) xab{c}^{2}{d}^{2}+6\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ) x{b}^{2}{c}^{3}d+3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){a}^{2}{c}^{2}{d}^{2}-6\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ) ab{c}^{3}d+3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){b}^{2}{c}^{4}-6\,xa{d}^{3}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+10\,xbc{d}^{2}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }-8\,ac{d}^{2}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+12\,b{c}^{2}d\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) } \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(b*x+a)^(1/2)/c^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^2*a^2*d^4-6*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*a*b*c*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*b^2*c^2*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*a^2*c*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*b*c^2*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*b^2*c^3*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)*a^2*c^2*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)*a*b*c^3*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)
*b^2*c^4-6*x*a*d^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+10*x*b*c*d^2*(a*c)^(1/2)*
((b*x+a)*(d*x+c))^(1/2)-8*a*c*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+12*b*c^2*d
*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/(a*d-b*c)^2/(a*c)^(1/2)/((b*x+a)*(d*x+c))^
(1/2)/(d*x+c)^(3/2)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x + a}{\left (d x + c\right )}^{\frac{5}{2}} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(b*x + a)*(d*x + c)^(5/2)*x), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.382234, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \,{\left (6 \, b c^{2} d - 4 \, a c d^{2} +{\left (5 \, b c d^{2} - 3 \, a d^{3}\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c} - 3 \,{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} +{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2} + 2 \,{\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x\right )} \log \left (-\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} -{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right )}{6 \,{\left (b^{2} c^{6} - 2 \, a b c^{5} d + a^{2} c^{4} d^{2} +{\left (b^{2} c^{4} d^{2} - 2 \, a b c^{3} d^{3} + a^{2} c^{2} d^{4}\right )} x^{2} + 2 \,{\left (b^{2} c^{5} d - 2 \, a b c^{4} d^{2} + a^{2} c^{3} d^{3}\right )} x\right )} \sqrt{a c}}, -\frac{2 \,{\left (6 \, b c^{2} d - 4 \, a c d^{2} +{\left (5 \, b c d^{2} - 3 \, a d^{3}\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c} + 3 \,{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} +{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2} + 2 \,{\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x\right )} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right )}{3 \,{\left (b^{2} c^{6} - 2 \, a b c^{5} d + a^{2} c^{4} d^{2} +{\left (b^{2} c^{4} d^{2} - 2 \, a b c^{3} d^{3} + a^{2} c^{2} d^{4}\right )} x^{2} + 2 \,{\left (b^{2} c^{5} d - 2 \, a b c^{4} d^{2} + a^{2} c^{3} d^{3}\right )} x\right )} \sqrt{-a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/6*(4*(6*b*c^2*d - 4*a*c*d^2 + (5*b*c*d^2 - 3*a*d^3)*x)*sqrt(a*c)*sqrt(b*x +
a)*sqrt(d*x + c) - 3*(b^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2 + (b^2*c^2*d^2 - 2*a*b
*c*d^3 + a^2*d^4)*x^2 + 2*(b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3)*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))/((b^
2*c^6 - 2*a*b*c^5*d + a^2*c^4*d^2 + (b^2*c^4*d^2 - 2*a*b*c^3*d^3 + a^2*c^2*d^4)*
x^2 + 2*(b^2*c^5*d - 2*a*b*c^4*d^2 + a^2*c^3*d^3)*x)*sqrt(a*c)), -1/3*(2*(6*b*c^
2*d - 4*a*c*d^2 + (5*b*c*d^2 - 3*a*d^3)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c
) + 3*(b^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2 + (b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^
4)*x^2 + 2*(b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3)*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)))/((b^2*c^6 - 2*a*b*c^5*d
+ a^2*c^4*d^2 + (b^2*c^4*d^2 - 2*a*b*c^3*d^3 + a^2*c^2*d^4)*x^2 + 2*(b^2*c^5*d -
2*a*b*c^4*d^2 + a^2*c^3*d^3)*x)*sqrt(-a*c))]

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \sqrt{a + b x} \left (c + d x\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x*sqrt(a + b*x)*(c + d*x)**(5/2)), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.261117, size = 358, normalized size = 2.89 \[ \frac{\sqrt{b x + a}{\left (\frac{{\left (5 \, b^{4} c^{3} d^{3}{\left | b \right |} - 3 \, a b^{3} c^{2} d^{4}{\left | b \right |}\right )}{\left (b x + a\right )}}{b^{8} c^{2} d^{4} - 2 \, a b^{7} c d^{5} + a^{2} b^{6} d^{6}} + \frac{3 \,{\left (2 \, b^{5} c^{4} d^{2}{\left | b \right |} - 3 \, a b^{4} c^{3} d^{3}{\left | b \right |} + a^{2} b^{3} c^{2} d^{4}{\left | b \right |}\right )}}{b^{8} c^{2} d^{4} - 2 \, a b^{7} c d^{5} + a^{2} b^{6} d^{6}}\right )}}{12 \,{\left (b^{2} c +{\left (b x + a\right )} b d - a b d\right )}^{\frac{3}{2}}} - \frac{2 \, \sqrt{b d} b \arctan \left (-\frac{b^{2} c + a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt{-a b c d} b}\right )}{\sqrt{-a b c d} c^{2}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*sqrt(b*x + a)*((5*b^4*c^3*d^3*abs(b) - 3*a*b^3*c^2*d^4*abs(b))*(b*x + a)/(b
^8*c^2*d^4 - 2*a*b^7*c*d^5 + a^2*b^6*d^6) + 3*(2*b^5*c^4*d^2*abs(b) - 3*a*b^4*c^
3*d^3*abs(b) + a^2*b^3*c^2*d^4*abs(b))/(b^8*c^2*d^4 - 2*a*b^7*c*d^5 + a^2*b^6*d^
6))/(b^2*c + (b*x + a)*b*d - a*b*d)^(3/2) - 2*sqrt(b*d)*b*arctan(-1/2*(b^2*c + a
*b*d - (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/(sqrt(
-a*b*c*d)*b))/(sqrt(-a*b*c*d)*c^2*abs(b))