3.620 \(\int \frac{(a+b x)^{3/2}}{x^4 \sqrt{c+d x}} \, dx\)

Optimal. Leaf size=180 \[ \frac{(b c-a d)^2 (5 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{3/2} c^{7/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} (b c-a d) (5 a d+b c)}{8 a c^3 x}+\frac{(a+b x)^{3/2} \sqrt{c+d x} (5 a d+b c)}{12 a c^2 x^2}-\frac{(a+b x)^{5/2} \sqrt{c+d x}}{3 a c x^3} \]

[Out]

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

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Rubi [A]  time = 0.312252, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{(b c-a d)^2 (5 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{3/2} c^{7/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} (b c-a d) (5 a d+b c)}{8 a c^3 x}+\frac{(a+b x)^{3/2} \sqrt{c+d x} (5 a d+b c)}{12 a c^2 x^2}-\frac{(a+b x)^{5/2} \sqrt{c+d x}}{3 a c x^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(3/2)/(x^4*Sqrt[c + d*x]),x]

[Out]

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

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Rubi in Sympy [A]  time = 25.5569, size = 158, normalized size = 0.88 \[ - \frac{\left (a + b x\right )^{\frac{5}{2}} \sqrt{c + d x}}{3 a c x^{3}} + \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (5 a d + b c\right )}{12 a c^{2} x^{2}} - \frac{\sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right ) \left (5 a d + b c\right )}{8 a c^{3} x} + \frac{\left (a d - b c\right )^{2} \left (5 a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{8 a^{\frac{3}{2}} c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(3/2)/x**4/(d*x+c)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.193683, size = 189, normalized size = 1.05 \[ \frac{-2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} \left (a^2 \left (8 c^2-10 c d x+15 d^2 x^2\right )+2 a b c x (7 c-11 d x)+3 b^2 c^2 x^2\right )-3 x^3 \log (x) (b c-a d)^2 (5 a d+b c)+3 x^3 (b c-a d)^2 (5 a d+b c) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{48 a^{3/2} c^{7/2} x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(3/2)/(x^4*Sqrt[c + d*x]),x]

[Out]

(-2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]*(3*b^2*c^2*x^2 + 2*a*b*c*x*(7*c
- 11*d*x) + a^2*(8*c^2 - 10*c*d*x + 15*d^2*x^2)) - 3*(b*c - a*d)^2*(b*c + 5*a*d)
*x^3*Log[x] + 3*(b*c - a*d)^2*(b*c + 5*a*d)*x^3*Log[2*a*c + b*c*x + a*d*x + 2*Sq
rt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(48*a^(3/2)*c^(7/2)*x^3)

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Maple [B]  time = 0.035, size = 408, normalized size = 2.3 \[{\frac{1}{48\,a{c}^{3}{x}^{3}}\sqrt{bx+a}\sqrt{dx+c} \left ( 15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}{a}^{3}{d}^{3}-27\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}{a}^{2}bc{d}^{2}+9\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}a{b}^{2}{c}^{2}d+3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}{b}^{3}{c}^{3}-30\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{d}^{2}{a}^{2}{x}^{2}\sqrt{ac}+44\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }dbca{x}^{2}\sqrt{ac}-6\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{b}^{2}{c}^{2}{x}^{2}\sqrt{ac}+20\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }dc{a}^{2}x\sqrt{ac}-28\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }b{c}^{2}ax\sqrt{ac}-16\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{c}^{2}{a}^{2}\sqrt{ac} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(3/2)/x^4/(d*x+c)^(1/2),x)

[Out]

1/48*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a/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^3-27*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+9*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+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^3*b^3*c^3-30*((b*x+a)*(d*x+c))^(1/2)*d^2*a^2*x^2
*(a*c)^(1/2)+44*((b*x+a)*(d*x+c))^(1/2)*d*b*c*a*x^2*(a*c)^(1/2)-6*((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)-28*((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)^(3/2)/(sqrt(d*x + c)*x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.423885, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (b^{3} c^{3} + 3 \, a b^{2} c^{2} d - 9 \, a^{2} b c d^{2} + 5 \, a^{3} d^{3}\right )} x^{3} \log \left (\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right ) - 4 \,{\left (8 \, a^{2} c^{2} +{\left (3 \, b^{2} c^{2} - 22 \, a b c d + 15 \, a^{2} d^{2}\right )} x^{2} + 2 \,{\left (7 \, a b c^{2} - 5 \, a^{2} c d\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c}}{96 \, \sqrt{a c} a c^{3} x^{3}}, \frac{3 \,{\left (b^{3} c^{3} + 3 \, a b^{2} c^{2} d - 9 \, a^{2} b c d^{2} + 5 \, a^{3} d^{3}\right )} x^{3} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right ) - 2 \,{\left (8 \, a^{2} c^{2} +{\left (3 \, b^{2} c^{2} - 22 \, a b c d + 15 \, a^{2} d^{2}\right )} x^{2} + 2 \,{\left (7 \, a b c^{2} - 5 \, a^{2} c d\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c}}{48 \, \sqrt{-a c} a c^{3} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(3/2)/(sqrt(d*x + c)*x^4),x, algorithm="fricas")

[Out]

[1/96*(3*(b^3*c^3 + 3*a*b^2*c^2*d - 9*a^2*b*c*d^2 + 5*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 + (3*b^2*c^2 - 22*a*b*c*d + 15*a^2*d^2)*x^2 + 2*(7*a*b*c^2 - 5*a^2*c*d)*x
)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(a*c)*a*c^3*x^3), 1/48*(3*(b^3*c^3
+ 3*a*b^2*c^2*d - 9*a^2*b*c*d^2 + 5*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 + (3*b^2*c^2 -
22*a*b*c*d + 15*a^2*d^2)*x^2 + 2*(7*a*b*c^2 - 5*a^2*c*d)*x)*sqrt(-a*c)*sqrt(b*x
+ a)*sqrt(d*x + c))/(sqrt(-a*c)*a*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)**(3/2)/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)^(3/2)/(sqrt(d*x + c)*x^4),x, algorithm="giac")

[Out]

Exception raised: TypeError