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

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

[Out]

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

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Rubi [A]  time = 0.277589, 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 \[ -\frac{5 (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 \sqrt{a} c^{7/2}}-\frac{5 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2}{8 c^3 x}-\frac{5 (a+b x)^{3/2} \sqrt{c+d x} (b c-a d)}{12 c^2 x^2}-\frac{(a+b x)^{5/2} \sqrt{c+d x}}{3 c x^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 24.8534, size = 141, normalized size = 0.9 \[ - \frac{\left (a + b x\right )^{\frac{5}{2}} \sqrt{c + d x}}{3 c x^{3}} + \frac{5 \left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a d - b c\right )}{12 c^{2} x^{2}} - \frac{5 \sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )^{2}}{8 c^{3} x} + \frac{5 \left (a d - b c\right )^{3} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{8 \sqrt{a} c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.194094, size = 173, normalized size = 1.1 \[ \frac{-2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} \left (a^2 \left (8 c^2-10 c d x+15 d^2 x^2\right )+2 a b c x (13 c-20 d x)+33 b^2 c^2 x^2\right )+15 x^3 \log (x) (b c-a d)^3-15 x^3 (b c-a d)^3 \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{48 \sqrt{a} c^{7/2} x^3} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]*(33*b^2*c^2*x^2 + 2*a*b*c*x*(13*
c - 20*d*x) + a^2*(8*c^2 - 10*c*d*x + 15*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*Sqrt[a]*c^(7/2)*x^3)

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Maple [B]  time = 0.034, size = 405, normalized size = 2.6 \[{\frac{1}{48\,{c}^{3}{x}^{3}}\sqrt{bx+a}\sqrt{dx+c} \left ( 15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}{a}^{3}{d}^{3}-45\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}{a}^{2}bc{d}^{2}+45\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}a{b}^{2}{c}^{2}d-15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}{b}^{3}{c}^{3}-30\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{d}^{2}{a}^{2}{x}^{2}\sqrt{ac}+80\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }dbca{x}^{2}\sqrt{ac}-66\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{b}^{2}{c}^{2}{x}^{2}\sqrt{ac}+20\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }dc{a}^{2}x\sqrt{ac}-52\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }b{c}^{2}ax\sqrt{ac}-16\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{c}^{2}{a}^{2}\sqrt{ac} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.519818, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{3} \log \left (\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right ) + 4 \,{\left (8 \, a^{2} c^{2} +{\left (33 \, b^{2} c^{2} - 40 \, a b c d + 15 \, a^{2} d^{2}\right )} x^{2} + 2 \,{\left (13 \, a b c^{2} - 5 \, a^{2} c d\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c}}{96 \, \sqrt{a c} c^{3} x^{3}}, -\frac{15 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{3} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right ) + 2 \,{\left (8 \, a^{2} c^{2} +{\left (33 \, b^{2} c^{2} - 40 \, a b c d + 15 \, a^{2} d^{2}\right )} x^{2} + 2 \,{\left (13 \, a b c^{2} - 5 \, a^{2} c d\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c}}{48 \, \sqrt{-a c} c^{3} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/2)/(sqrt(d*x + c)*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 + (33*b^2*c^2 - 40*a*b*c*d + 15*a^2*d^2)*x^2 + 2*(13*a*b*c^2 - 5*a^2*c*d)
*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(a*c)*c^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 + (33*b^2*c^2
- 40*a*b*c*d + 15*a^2*d^2)*x^2 + 2*(13*a*b*c^2 - 5*a^2*c*d)*x)*sqrt(-a*c)*sqrt(b
*x + a)*sqrt(d*x + c))/(sqrt(-a*c)*c^3*x^3)]

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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)**(5/2)/x**4/(d*x+c)**(1/2),x)

[Out]

Timed out

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

[Out]

Exception raised: TypeError