3.708 \(\int \frac{(c+d x)^{5/2}}{x^4 \sqrt{a+b x}} \, dx\)

Optimal. Leaf size=157 \[ \frac{5 (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{7/2} \sqrt{c}}-\frac{5 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2}{8 a^3 x}+\frac{5 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)}{12 a^2 x^2}-\frac{\sqrt{a+b x} (c+d x)^{5/2}}{3 a x^3} \]

[Out]

(-5*(b*c - a*d)^2*Sqrt[a + b*x]*Sqrt[c + d*x])/(8*a^3*x) + (5*(b*c - a*d)*Sqrt[a
+ b*x]*(c + d*x)^(3/2))/(12*a^2*x^2) - (Sqrt[a + b*x]*(c + d*x)^(5/2))/(3*a*x^3
) + (5*(b*c - a*d)^3*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(
8*a^(7/2)*Sqrt[c])

_______________________________________________________________________________________

Rubi [A]  time = 0.307965, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{5 (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{7/2} \sqrt{c}}-\frac{5 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2}{8 a^3 x}+\frac{5 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)}{12 a^2 x^2}-\frac{\sqrt{a+b x} (c+d x)^{5/2}}{3 a x^3} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^(5/2)/(x^4*Sqrt[a + b*x]),x]

[Out]

(-5*(b*c - a*d)^2*Sqrt[a + b*x]*Sqrt[c + d*x])/(8*a^3*x) + (5*(b*c - a*d)*Sqrt[a
+ b*x]*(c + d*x)^(3/2))/(12*a^2*x^2) - (Sqrt[a + b*x]*(c + d*x)^(5/2))/(3*a*x^3
) + (5*(b*c - a*d)^3*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(
8*a^(7/2)*Sqrt[c])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 24.1077, size = 143, normalized size = 0.91 \[ - \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{5}{2}}}{3 a x^{3}} - \frac{5 \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )}{12 a^{2} x^{2}} - \frac{5 \sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )^{2}}{8 a^{3} x} - \frac{5 \left (a d - b c\right )^{3} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{8 a^{\frac{7}{2}} \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(5/2)/x**4/(b*x+a)**(1/2),x)

[Out]

-sqrt(a + b*x)*(c + d*x)**(5/2)/(3*a*x**3) - 5*sqrt(a + b*x)*(c + d*x)**(3/2)*(a
*d - b*c)/(12*a**2*x**2) - 5*sqrt(a + b*x)*sqrt(c + d*x)*(a*d - b*c)**2/(8*a**3*
x) - 5*(a*d - b*c)**3*atanh(sqrt(c)*sqrt(a + b*x)/(sqrt(a)*sqrt(c + d*x)))/(8*a*
*(7/2)*sqrt(c))

_______________________________________________________________________________________

Mathematica [A]  time = 0.201565, size = 171, normalized size = 1.09 \[ \frac{-2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} \left (a^2 \left (8 c^2+26 c d x+33 d^2 x^2\right )-10 a b c x (c+4 d x)+15 b^2 c^2 x^2\right )-15 x^3 \log (x) (b c-a d)^3+15 x^3 (b c-a d)^3 \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{48 a^{7/2} \sqrt{c} x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^(5/2)/(x^4*Sqrt[a + b*x]),x]

[Out]

(-2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]*(15*b^2*c^2*x^2 - 10*a*b*c*x*(c
+ 4*d*x) + a^2*(8*c^2 + 26*c*d*x + 33*d^2*x^2)) - 15*(b*c - a*d)^3*x^3*Log[x] +
15*(b*c - a*d)^3*x^3*Log[2*a*c + b*c*x + a*d*x + 2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]
*Sqrt[c + d*x]])/(48*a^(7/2)*Sqrt[c]*x^3)

_______________________________________________________________________________________

Maple [B]  time = 0.036, size = 405, normalized size = 2.6 \[ -{\frac{1}{48\,{a}^{3}{x}^{3}}\sqrt{bx+a}\sqrt{dx+c} \left ( 15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}{a}^{3}{d}^{3}-45\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}{a}^{2}bc{d}^{2}+45\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}a{b}^{2}{c}^{2}d-15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}{b}^{3}{c}^{3}+66\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{d}^{2}{a}^{2}{x}^{2}\sqrt{ac}-80\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }dbca{x}^{2}\sqrt{ac}+30\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{b}^{2}{c}^{2}{x}^{2}\sqrt{ac}+52\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }dc{a}^{2}x\sqrt{ac}-20\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }b{c}^{2}ax\sqrt{ac}+16\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{c}^{2}{a}^{2}\sqrt{ac} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(5/2)/x^4/(b*x+a)^(1/2),x)

[Out]

-1/48*(d*x+c)^(1/2)*(b*x+a)^(1/2)/a^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^3*a^3*d^3-45*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*c*d^2+45*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^2*c^2*d-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^3*b^3*c^3+66*((b*x+a)*(d*x+c))^(1/2)*d^2*a^2*x^
2*(a*c)^(1/2)-80*((b*x+a)*(d*x+c))^(1/2)*d*b*c*a*x^2*(a*c)^(1/2)+30*((b*x+a)*(d*
x+c))^(1/2)*b^2*c^2*x^2*(a*c)^(1/2)+52*((b*x+a)*(d*x+c))^(1/2)*d*c*a^2*x*(a*c)^(
1/2)-20*((b*x+a)*(d*x+c))^(1/2)*b*c^2*a*x*(a*c)^(1/2)+16*((b*x+a)*(d*x+c))^(1/2)
*c^2*a^2*(a*c)^(1/2))/((b*x+a)*(d*x+c))^(1/2)/(a*c)^(1/2)/x^3

_______________________________________________________________________________________

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)^(5/2)/(sqrt(b*x + a)*x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.421454, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{3} \log \left (-\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} -{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right ) + 4 \,{\left (8 \, a^{2} c^{2} +{\left (15 \, b^{2} c^{2} - 40 \, a b c d + 33 \, a^{2} d^{2}\right )} x^{2} - 2 \,{\left (5 \, a b c^{2} - 13 \, a^{2} c d\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c}}{96 \, \sqrt{a c} a^{3} x^{3}}, \frac{15 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{3} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right ) - 2 \,{\left (8 \, a^{2} c^{2} +{\left (15 \, b^{2} c^{2} - 40 \, a b c d + 33 \, a^{2} d^{2}\right )} x^{2} - 2 \,{\left (5 \, a b c^{2} - 13 \, a^{2} c d\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c}}{48 \, \sqrt{-a c} a^{3} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(5/2)/(sqrt(b*x + a)*x^4),x, algorithm="fricas")

[Out]

[-1/96*(15*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*x^3*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*(8*
a^2*c^2 + (15*b^2*c^2 - 40*a*b*c*d + 33*a^2*d^2)*x^2 - 2*(5*a*b*c^2 - 13*a^2*c*d
)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(a*c)*a^3*x^3), 1/48*(15*(b^3*c
^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*x^3*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*(8*a^2*c^2 + (15*b^2*c^2
- 40*a*b*c*d + 33*a^2*d^2)*x^2 - 2*(5*a*b*c^2 - 13*a^2*c*d)*x)*sqrt(-a*c)*sqrt(b
*x + a)*sqrt(d*x + c))/(sqrt(-a*c)*a^3*x^3)]

_______________________________________________________________________________________

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)**(5/2)/x**4/(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)^(5/2)/(sqrt(b*x + a)*x^4),x, algorithm="giac")

[Out]

Exception raised: TypeError