3.699 \(\int \frac{(c+d x)^{3/2}}{x^3 \sqrt{a+b x}} \, dx\)

Optimal. Leaf size=119 \[ -\frac{3 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{5/2} \sqrt{c}}+\frac{3 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)}{4 a^2 x}-\frac{\sqrt{a+b x} (c+d x)^{3/2}}{2 a x^2} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.212592, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ -\frac{3 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{5/2} \sqrt{c}}+\frac{3 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)}{4 a^2 x}-\frac{\sqrt{a+b x} (c+d x)^{3/2}}{2 a x^2} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^(3/2)/(x^3*Sqrt[a + b*x]),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 16.3142, size = 107, normalized size = 0.9 \[ - \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}}}{2 a x^{2}} - \frac{3 \sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )}{4 a^{2} x} - \frac{3 \left (a d - b c\right )^{2} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{4 a^{\frac{5}{2}} \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(3/2)/x**3/(b*x+a)**(1/2),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.131573, size = 139, normalized size = 1.17 \[ \frac{3 x^2 \log (x) (b c-a d)^2-3 x^2 (b c-a d)^2 \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )-2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} (2 a c+5 a d x-3 b c x)}{8 a^{5/2} \sqrt{c} x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^(3/2)/(x^3*Sqrt[a + b*x]),x]

[Out]

(-2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]*(2*a*c - 3*b*c*x + 5*a*d*x) + 3*
(b*c - a*d)^2*x^2*Log[x] - 3*(b*c - a*d)^2*x^2*Log[2*a*c + b*c*x + a*d*x + 2*Sqr
t[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(8*a^(5/2)*Sqrt[c]*x^2)

_______________________________________________________________________________________

Maple [B]  time = 0.033, size = 255, normalized size = 2.1 \[ -{\frac{1}{8\,{a}^{2}{x}^{2}}\sqrt{bx+a}\sqrt{dx+c} \left ( 3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}{a}^{2}{d}^{2}-6\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}abcd+3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}{b}^{2}{c}^{2}+10\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }dax\sqrt{ac}-6\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }bcx\sqrt{ac}+4\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }ca\sqrt{ac} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(3/2)/x^3/(b*x+a)^(1/2),x)

[Out]

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

_______________________________________________________________________________________

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((d*x + c)^(3/2)/(sqrt(b*x + a)*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.2964, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{2} \log \left (-\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} -{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right ) - 4 \,{\left (2 \, a c -{\left (3 \, b c - 5 \, a d\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c}}{16 \, \sqrt{a c} a^{2} x^{2}}, -\frac{3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{2} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right ) + 2 \,{\left (2 \, a c -{\left (3 \, b c - 5 \, a d\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c}}{8 \, \sqrt{-a c} a^{2} x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(3/2)/(sqrt(b*x + a)*x^3),x, algorithm="fricas")

[Out]

[1/16*(3*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x^2*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) - 4*(2*a*c - (3*b*c - 5*a*d)*
x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(a*c)*a^2*x^2), -1/8*(3*(b^2*c^2
- 2*a*b*c*d + a^2*d^2)*x^2*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)/(sqrt(b
*x + a)*sqrt(d*x + c)*a*c)) + 2*(2*a*c - (3*b*c - 5*a*d)*x)*sqrt(-a*c)*sqrt(b*x
+ a)*sqrt(d*x + c))/(sqrt(-a*c)*a^2*x^2)]

_______________________________________________________________________________________

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((d*x+c)**(3/2)/x**3/(b*x+a)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

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((d*x + c)^(3/2)/(sqrt(b*x + a)*x^3),x, algorithm="giac")

[Out]

Exception raised: TypeError