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

Optimal. Leaf size=160 \[ \frac{(b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{3/2} c^{5/2}}-\frac{\sqrt{a+b x} (c+d x)^{3/2} (b c-a d)}{4 c^2 x^2}-\frac{\sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2}{8 a c^2 x}-\frac{(a+b x)^{3/2} (c+d x)^{3/2}}{3 c x^3} \]

[Out]

-((b*c - a*d)^2*Sqrt[a + b*x]*Sqrt[c + d*x])/(8*a*c^2*x) - ((b*c - a*d)*Sqrt[a +
b*x]*(c + d*x)^(3/2))/(4*c^2*x^2) - ((a + b*x)^(3/2)*(c + d*x)^(3/2))/(3*c*x^3)
+ ((b*c - a*d)^3*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(8*a
^(3/2)*c^(5/2))

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Rubi [A]  time = 0.280793, antiderivative size = 160, 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 \[ \frac{(b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{3/2} c^{5/2}}-\frac{\sqrt{a+b x} (c+d x)^{3/2} (b c-a d)}{4 c^2 x^2}-\frac{\sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2}{8 a c^2 x}-\frac{(a+b x)^{3/2} (c+d x)^{3/2}}{3 c x^3} \]

Antiderivative was successfully verified.

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

[Out]

-((b*c - a*d)^2*Sqrt[a + b*x]*Sqrt[c + d*x])/(8*a*c^2*x) - ((b*c - a*d)*Sqrt[a +
b*x]*(c + d*x)^(3/2))/(4*c^2*x^2) - ((a + b*x)^(3/2)*(c + d*x)^(3/2))/(3*c*x^3)
+ ((b*c - a*d)^3*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(8*a
^(3/2)*c^(5/2))

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Rubi in Sympy [A]  time = 25.2888, size = 138, normalized size = 0.86 \[ - \frac{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}}}{3 c x^{3}} + \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a d - b c\right )}{4 a c x^{2}} + \frac{\sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )^{2}}{8 a c^{2} x} - \frac{\left (a d - b c\right )^{3} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{8 a^{\frac{3}{2}} c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a + b*x)**(3/2)*(c + d*x)**(3/2)/(3*c*x**3) + (a + b*x)**(3/2)*sqrt(c + d*x)*(
a*d - b*c)/(4*a*c*x**2) + sqrt(a + b*x)*sqrt(c + d*x)*(a*d - b*c)**2/(8*a*c**2*x
) - (a*d - b*c)**3*atanh(sqrt(c)*sqrt(a + b*x)/(sqrt(a)*sqrt(c + d*x)))/(8*a**(3
/2)*c**(5/2))

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Mathematica [A]  time = 0.174493, size = 173, normalized size = 1.08 \[ \frac{-2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} \left (a^2 \left (8 c^2+2 c d x-3 d^2 x^2\right )+2 a b c x (7 c+4 d x)+3 b^2 c^2 x^2\right )-3 x^3 \log (x) (b c-a d)^3+3 x^3 (b c-a d)^3 \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{48 a^{3/2} c^{5/2} x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^(3/2)*Sqrt[c + d*x])/x^4,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
+ 4*d*x) + a^2*(8*c^2 + 2*c*d*x - 3*d^2*x^2)) - 3*(b*c - a*d)^3*x^3*Log[x] + 3*(
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]*Sqr
t[c + d*x]])/(48*a^(3/2)*c^(5/2)*x^3)

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Maple [B]  time = 0.021, size = 485, normalized size = 3. \[ -{\frac{1}{48\,a{c}^{2}{x}^{3}}\sqrt{bx+a}\sqrt{dx+c} \left ( 3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{3}{a}^{3}{d}^{3}-9\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{3}{a}^{2}bc{d}^{2}+9\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{3}a{b}^{2}{c}^{2}d-3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{3}{b}^{3}{c}^{3}-6\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{a}^{2}{d}^{2}+16\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}abcd+6\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{b}^{2}{c}^{2}+4\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{2}cd+28\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}xab{c}^{2}+16\,\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{2}{c}^{2}\sqrt{ac} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/48*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a/c^2*(3*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*(b*d*x^
2+a*d*x+b*c*x+a*c)^(1/2)+2*a*c)/x)*x^3*a^3*d^3-9*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*(
b*d*x^2+a*d*x+b*c*x+a*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*d*x^2+a*d*x+b*c*x+a*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*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+2*a*c)/x)*x^3*b^3*c^3-6*(a*c)
^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^2*a^2*d^2+16*(a*c)^(1/2)*(b*d*x^2+a*d*x
+b*c*x+a*c)^(1/2)*x^2*a*b*c*d+6*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^2*
b^2*c^2+4*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^2*c*d+28*(a*c)^(1/2)*(
b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a*b*c^2+16*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^2*
c^2*(a*c)^(1/2))/(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)/x^3/(a*c)^(1/2)

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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((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 = 0.5065, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{3} \log \left (-\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} -{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right ) + 4 \,{\left (8 \, a^{2} c^{2} +{\left (3 \, b^{2} c^{2} + 8 \, a b c d - 3 \, a^{2} d^{2}\right )} x^{2} + 2 \,{\left (7 \, a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c}}{96 \, \sqrt{a c} a c^{2} x^{3}}, \frac{3 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{3} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right ) - 2 \,{\left (8 \, a^{2} c^{2} +{\left (3 \, b^{2} c^{2} + 8 \, a b c d - 3 \, a^{2} d^{2}\right )} x^{2} + 2 \,{\left (7 \, a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c}}{48 \, \sqrt{-a c} a c^{2} x^{3}}\right ] \]

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 + 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 + (3*b^2*c^2 + 8*a*b*c*d - 3*a^2*d^2)*x^2 + 2*(7*a*b*c^2 + a^2*c*d)*x)*sq
rt(a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(a*c)*a*c^2*x^3), 1/48*(3*(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)*s
qrt(-a*c)/(sqrt(b*x + a)*sqrt(d*x + c)*a*c)) - 2*(8*a^2*c^2 + (3*b^2*c^2 + 8*a*b
*c*d - 3*a^2*d^2)*x^2 + 2*(7*a*b*c^2 + a^2*c*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt
(d*x + c))/(sqrt(-a*c)*a*c^2*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)*(d*x+c)**(1/2)/x**4,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