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

Optimal. Leaf size=102 \[ -\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{c^{5/2}}+\frac{2 \sqrt{a+b x}}{c^2 \sqrt{c+d x}}-\frac{2 d (a+b x)^{3/2}}{3 c (c+d x)^{3/2} (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.196291, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{c^{5/2}}+\frac{2 \sqrt{a+b x}}{c^2 \sqrt{c+d x}}-\frac{2 d (a+b x)^{3/2}}{3 c (c+d x)^{3/2} (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.8331, size = 92, normalized size = 0.9 \[ - \frac{2 \sqrt{a} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{c^{\frac{5}{2}}} + \frac{2 d \left (a + b x\right )^{\frac{3}{2}}}{3 c \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )} + \frac{2 \sqrt{a + b x}}{c^{2} \sqrt{c + d x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.367501, size = 129, normalized size = 1.26 \[ \frac{\frac{2 \sqrt{c} \sqrt{a+b x} (b c (3 c+2 d x)-a d (4 c+3 d x))}{(c+d x)^{3/2} (b c-a d)}-3 \sqrt{a} \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )+3 \sqrt{a} \log (x)}{3 c^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.036, size = 430, normalized size = 4.2 \[ -{\frac{1}{3\,{c}^{2} \left ( ad-bc \right ) } \left ( 3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}{a}^{2}{d}^{3}-3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}abc{d}^{2}+6\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ) x{a}^{2}c{d}^{2}-6\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ) xab{c}^{2}d+3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){a}^{2}{c}^{2}d-3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ) ab{c}^{3}-6\,xa{d}^{2}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+4\,xbcd\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }-8\,acd\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+6\,b{c}^{2}\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) } \right ) \sqrt{bx+a}{\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(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^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*b*c*
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^
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*b*c^2*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-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*c^3-6*x*a*d^2*
(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+4*x*b*c*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2
)-8*a*c*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+6*b*c^2*(a*c)^(1/2)*((b*x+a)*(d*x+
c))^(1/2))/c^2*(b*x+a)^(1/2)/(a*d-b*c)/(a*c)^(1/2)/((b*x+a)*(d*x+c))^(1/2)/(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(sqrt(b*x + a)/((d*x + c)^(5/2)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.355929, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (b c^{3} - a c^{2} d +{\left (b c d^{2} - a d^{3}\right )} x^{2} + 2 \,{\left (b c^{2} d - a c d^{2}\right )} x\right )} \sqrt{\frac{a}{c}} \log \left (\frac{8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \,{\left (2 \, a c^{2} +{\left (b c^{2} + a c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{a}{c}} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \,{\left (3 \, b c^{2} - 4 \, a c d +{\left (2 \, b c d - 3 \, a d^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{6 \,{\left (b c^{5} - a c^{4} d +{\left (b c^{3} d^{2} - a c^{2} d^{3}\right )} x^{2} + 2 \,{\left (b c^{4} d - a c^{3} d^{2}\right )} x\right )}}, -\frac{3 \,{\left (b c^{3} - a c^{2} d +{\left (b c d^{2} - a d^{3}\right )} x^{2} + 2 \,{\left (b c^{2} d - a c d^{2}\right )} x\right )} \sqrt{-\frac{a}{c}} \arctan \left (\frac{2 \, a c +{\left (b c + a d\right )} x}{2 \, \sqrt{b x + a} \sqrt{d x + c} c \sqrt{-\frac{a}{c}}}\right ) - 2 \,{\left (3 \, b c^{2} - 4 \, a c d +{\left (2 \, b c d - 3 \, a d^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{3 \,{\left (b c^{5} - a c^{4} d +{\left (b c^{3} d^{2} - a c^{2} d^{3}\right )} x^{2} + 2 \,{\left (b c^{4} d - a c^{3} d^{2}\right )} x\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/6*(3*(b*c^3 - a*c^2*d + (b*c*d^2 - a*d^3)*x^2 + 2*(b*c^2*d - a*c*d^2)*x)*sqrt
(a/c)*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 - 4*(2*a*c^2 + (b*c^2
+ a*c*d)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(a/c) + 8*(a*b*c^2 + a^2*c*d)*x)/x^
2) + 4*(3*b*c^2 - 4*a*c*d + (2*b*c*d - 3*a*d^2)*x)*sqrt(b*x + a)*sqrt(d*x + c))/
(b*c^5 - a*c^4*d + (b*c^3*d^2 - a*c^2*d^3)*x^2 + 2*(b*c^4*d - a*c^3*d^2)*x), -1/
3*(3*(b*c^3 - a*c^2*d + (b*c*d^2 - a*d^3)*x^2 + 2*(b*c^2*d - a*c*d^2)*x)*sqrt(-a
/c)*arctan(1/2*(2*a*c + (b*c + a*d)*x)/(sqrt(b*x + a)*sqrt(d*x + c)*c*sqrt(-a/c)
)) - 2*(3*b*c^2 - 4*a*c*d + (2*b*c*d - 3*a*d^2)*x)*sqrt(b*x + a)*sqrt(d*x + c))/
(b*c^5 - a*c^4*d + (b*c^3*d^2 - a*c^2*d^3)*x^2 + 2*(b*c^4*d - a*c^3*d^2)*x)]

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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)**(1/2)/x/(d*x+c)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.260392, size = 355, normalized size = 3.48 \[ -\frac{\sqrt{b x + a}{\left (\frac{{\left (2 \, b^{4} c^{3} d^{2}{\left | b \right |} - 3 \, a b^{3} c^{2} d^{3}{\left | b \right |}\right )}{\left (b x + a\right )}}{b^{8} c^{2} d^{4} - 2 \, a b^{7} c d^{5} + a^{2} b^{6} d^{6}} + \frac{3 \,{\left (b^{5} c^{4} d{\left | b \right |} - 2 \, a b^{4} c^{3} d^{2}{\left | b \right |} + a^{2} b^{3} c^{2} d^{3}{\left | b \right |}\right )}}{b^{8} c^{2} d^{4} - 2 \, a b^{7} c d^{5} + a^{2} b^{6} d^{6}}\right )}}{12 \,{\left (b^{2} c +{\left (b x + a\right )} b d - a b d\right )}^{\frac{3}{2}}} - \frac{2 \, \sqrt{b d} a b \arctan \left (-\frac{b^{2} c + a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt{-a b c d} b}\right )}{\sqrt{-a b c d} c^{2}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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