3.552 \(\int \frac{\sqrt{a+b x} (c+d x)^{3/2}}{x^5} \, dx\)

Optimal. Leaf size=233 \[ \frac{(3 a d+5 b c) (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{7/2} c^{5/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} (3 a d+5 b c) (b c-a d)^2}{64 a^3 c^2 x}+\frac{\sqrt{a+b x} (c+d x)^{3/2} (3 a d+5 b c) (b c-a d)}{96 a^2 c^2 x^2}+\frac{\sqrt{a+b x} (c+d x)^{5/2} (3 a d+5 b c)}{24 a c^2 x^3}-\frac{(a+b x)^{3/2} (c+d x)^{5/2}}{4 a c x^4} \]

[Out]

-((b*c - a*d)^2*(5*b*c + 3*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*a^3*c^2*x) + ((
b*c - a*d)*(5*b*c + 3*a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(96*a^2*c^2*x^2) + ((5
*b*c + 3*a*d)*Sqrt[a + b*x]*(c + d*x)^(5/2))/(24*a*c^2*x^3) - ((a + b*x)^(3/2)*(
c + d*x)^(5/2))/(4*a*c*x^4) + ((b*c - a*d)^3*(5*b*c + 3*a*d)*ArcTanh[(Sqrt[c]*Sq
rt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(64*a^(7/2)*c^(5/2))

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Rubi [A]  time = 0.421311, antiderivative size = 233, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{(3 a d+5 b c) (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{7/2} c^{5/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} (3 a d+5 b c) (b c-a d)^2}{64 a^3 c^2 x}+\frac{\sqrt{a+b x} (c+d x)^{3/2} (3 a d+5 b c) (b c-a d)}{96 a^2 c^2 x^2}+\frac{\sqrt{a+b x} (c+d x)^{5/2} (3 a d+5 b c)}{24 a c^2 x^3}-\frac{(a+b x)^{3/2} (c+d x)^{5/2}}{4 a c x^4} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[a + b*x]*(c + d*x)^(3/2))/x^5,x]

[Out]

-((b*c - a*d)^2*(5*b*c + 3*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(64*a^3*c^2*x) + ((
b*c - a*d)*(5*b*c + 3*a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(96*a^2*c^2*x^2) + ((5
*b*c + 3*a*d)*Sqrt[a + b*x]*(c + d*x)^(5/2))/(24*a*c^2*x^3) - ((a + b*x)^(3/2)*(
c + d*x)^(5/2))/(4*a*c*x^4) + ((b*c - a*d)^3*(5*b*c + 3*a*d)*ArcTanh[(Sqrt[c]*Sq
rt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(64*a^(7/2)*c^(5/2))

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Rubi in Sympy [A]  time = 37.181, size = 211, normalized size = 0.91 \[ - \frac{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{5}{2}}}{4 a c x^{4}} + \frac{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (3 a d + 5 b c\right )}{24 a^{2} c x^{3}} + \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a d - b c\right ) \left (3 a d + 5 b c\right )}{32 a^{3} c x^{2}} + \frac{\sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )^{2} \left (3 a d + 5 b c\right )}{64 a^{3} c^{2} x} - \frac{\left (a d - b c\right )^{3} \left (3 a d + 5 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{64 a^{\frac{7}{2}} c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(3/2)*(b*x+a)**(1/2)/x**5,x)

[Out]

-(a + b*x)**(3/2)*(c + d*x)**(5/2)/(4*a*c*x**4) + (a + b*x)**(3/2)*(c + d*x)**(3
/2)*(3*a*d + 5*b*c)/(24*a**2*c*x**3) + (a + b*x)**(3/2)*sqrt(c + d*x)*(a*d - b*c
)*(3*a*d + 5*b*c)/(32*a**3*c*x**2) + sqrt(a + b*x)*sqrt(c + d*x)*(a*d - b*c)**2*
(3*a*d + 5*b*c)/(64*a**3*c**2*x) - (a*d - b*c)**3*(3*a*d + 5*b*c)*atanh(sqrt(c)*
sqrt(a + b*x)/(sqrt(a)*sqrt(c + d*x)))/(64*a**(7/2)*c**(5/2))

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Mathematica [A]  time = 0.256089, size = 234, normalized size = 1. \[ \frac{-2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} \left (a^3 \left (48 c^3+72 c^2 d x+6 c d^2 x^2-9 d^3 x^3\right )+a^2 b c x \left (8 c^2+20 c d x+9 d^2 x^2\right )-a b^2 c^2 x^2 (10 c+31 d x)+15 b^3 c^3 x^3\right )-3 x^4 \log (x) (b c-a d)^3 (3 a d+5 b c)+3 x^4 (b c-a d)^3 (3 a d+5 b c) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{384 a^{7/2} c^{5/2} x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[a + b*x]*(c + d*x)^(3/2))/x^5,x]

[Out]

(-2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]*(15*b^3*c^3*x^3 - a*b^2*c^2*x^2*
(10*c + 31*d*x) + a^2*b*c*x*(8*c^2 + 20*c*d*x + 9*d^2*x^2) + a^3*(48*c^3 + 72*c^
2*d*x + 6*c*d^2*x^2 - 9*d^3*x^3)) - 3*(b*c - a*d)^3*(5*b*c + 3*a*d)*x^4*Log[x] +
3*(b*c - a*d)^3*(5*b*c + 3*a*d)*x^4*Log[2*a*c + b*c*x + a*d*x + 2*Sqrt[a]*Sqrt[
c]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(384*a^(7/2)*c^(5/2)*x^4)

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Maple [F]  time = 180., size = 0, normalized size = 0. \[ \text{hanged} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(3/2)*(b*x+a)^(1/2)/x^5,x)

[Out]

int((d*x+c)^(3/2)*(b*x+a)^(1/2)/x^5,x)

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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(sqrt(b*x + a)*(d*x + c)^(3/2)/x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.743114, size = 1, normalized size = 0. \[ \left [-\frac{3 \,{\left (5 \, b^{4} c^{4} - 12 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} - 3 \, a^{4} d^{4}\right )} x^{4} \log \left (-\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} -{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right ) + 4 \,{\left (48 \, a^{3} c^{3} +{\left (15 \, b^{3} c^{3} - 31 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - 9 \, a^{3} d^{3}\right )} x^{3} - 2 \,{\left (5 \, a b^{2} c^{3} - 10 \, a^{2} b c^{2} d - 3 \, a^{3} c d^{2}\right )} x^{2} + 8 \,{\left (a^{2} b c^{3} + 9 \, a^{3} c^{2} d\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c}}{768 \, \sqrt{a c} a^{3} c^{2} x^{4}}, \frac{3 \,{\left (5 \, b^{4} c^{4} - 12 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} - 3 \, a^{4} d^{4}\right )} x^{4} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right ) - 2 \,{\left (48 \, a^{3} c^{3} +{\left (15 \, b^{3} c^{3} - 31 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - 9 \, a^{3} d^{3}\right )} x^{3} - 2 \,{\left (5 \, a b^{2} c^{3} - 10 \, a^{2} b c^{2} d - 3 \, a^{3} c d^{2}\right )} x^{2} + 8 \,{\left (a^{2} b c^{3} + 9 \, a^{3} c^{2} d\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c}}{384 \, \sqrt{-a c} a^{3} c^{2} x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x + a)*(d*x + c)^(3/2)/x^5,x, algorithm="fricas")

[Out]

[-1/768*(3*(5*b^4*c^4 - 12*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 + 4*a^3*b*c*d^3 - 3*a
^4*d^4)*x^4*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*(48*a^3*c^3 + (15*b^3*c^3 - 31*a*b^2*c^2*d + 9*a^2*b*c*d^2
- 9*a^3*d^3)*x^3 - 2*(5*a*b^2*c^3 - 10*a^2*b*c^2*d - 3*a^3*c*d^2)*x^2 + 8*(a^2*
b*c^3 + 9*a^3*c^2*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(a*c)*a^3*c^
2*x^4), 1/384*(3*(5*b^4*c^4 - 12*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 + 4*a^3*b*c*d^3
- 3*a^4*d^4)*x^4*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)/(sqrt(b*x + a)*s
qrt(d*x + c)*a*c)) - 2*(48*a^3*c^3 + (15*b^3*c^3 - 31*a*b^2*c^2*d + 9*a^2*b*c*d^
2 - 9*a^3*d^3)*x^3 - 2*(5*a*b^2*c^3 - 10*a^2*b*c^2*d - 3*a^3*c*d^2)*x^2 + 8*(a^2
*b*c^3 + 9*a^3*c^2*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(-a*c)*a^3
*c^2*x^4)]

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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((d*x+c)**(3/2)*(b*x+a)**(1/2)/x**5,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(sqrt(b*x + a)*(d*x + c)^(3/2)/x^5,x, algorithm="giac")

[Out]

Exception raised: TypeError