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

Optimal. Leaf size=204 \[ \frac{5 (b c-a d)^4 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{7/2} c^{3/2}}-\frac{5 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^3}{64 a^3 c x}+\frac{5 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^2}{96 a^2 c x^2}-\frac{\sqrt{a+b x} (c+d x)^{7/2}}{4 c x^4}-\frac{\sqrt{a+b x} (c+d x)^{5/2} (b c-a d)}{24 a c x^3} \]

[Out]

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

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Rubi [A]  time = 0.398519, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{5 (b c-a d)^4 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{7/2} c^{3/2}}-\frac{5 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^3}{64 a^3 c x}+\frac{5 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^2}{96 a^2 c x^2}-\frac{\sqrt{a+b x} (c+d x)^{7/2}}{4 c x^4}-\frac{\sqrt{a+b x} (c+d x)^{5/2} (b c-a d)}{24 a c x^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 35.2123, size = 178, normalized size = 0.87 \[ - \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{7}{2}}}{4 c x^{4}} + \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{5}{2}} \left (a d - b c\right )}{24 a c x^{3}} + \frac{5 \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )^{2}}{96 a^{2} c x^{2}} + \frac{5 \sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )^{3}}{64 a^{3} c x} + \frac{5 \left (a d - b c\right )^{4} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{64 a^{\frac{7}{2}} c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(a + b*x)*(c + d*x)**(7/2)/(4*c*x**4) + sqrt(a + b*x)*(c + d*x)**(5/2)*(a*d
- b*c)/(24*a*c*x**3) + 5*sqrt(a + b*x)*(c + d*x)**(3/2)*(a*d - b*c)**2/(96*a**2
*c*x**2) + 5*sqrt(a + b*x)*sqrt(c + d*x)*(a*d - b*c)**3/(64*a**3*c*x) + 5*(a*d -
b*c)**4*atanh(sqrt(c)*sqrt(a + b*x)/(sqrt(a)*sqrt(c + d*x)))/(64*a**(7/2)*c**(3
/2))

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Mathematica [A]  time = 0.251114, size = 216, normalized size = 1.06 \[ \frac{-2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} \left (a^3 \left (48 c^3+136 c^2 d x+118 c d^2 x^2+15 d^3 x^3\right )+a^2 b c x \left (8 c^2+36 c d x+73 d^2 x^2\right )-5 a b^2 c^2 x^2 (2 c+11 d x)+15 b^3 c^3 x^3\right )-15 x^4 \log (x) (b c-a d)^4+15 x^4 (b c-a d)^4 \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{384 a^{7/2} c^{3/2} x^4} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]*(15*b^3*c^3*x^3 - 5*a*b^2*c^2*x^
2*(2*c + 11*d*x) + a^2*b*c*x*(8*c^2 + 36*c*d*x + 73*d^2*x^2) + a^3*(48*c^3 + 136
*c^2*d*x + 118*c*d^2*x^2 + 15*d^3*x^3)) - 15*(b*c - a*d)^4*x^4*Log[x] + 15*(b*c
- a*d)^4*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^(3/2)*x^4)

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Maple [B]  time = 0.026, size = 705, normalized size = 3.5 \[{\frac{1}{384\,{a}^{3}c{x}^{4}}\sqrt{bx+a}\sqrt{dx+c} \left ( 15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{4}{a}^{4}{d}^{4}-60\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{4}{a}^{3}bc{d}^{3}+90\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{4}{a}^{2}{b}^{2}{c}^{2}{d}^{2}-60\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{4}a{b}^{3}{c}^{3}d+15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{4}{b}^{4}{c}^{4}-30\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}{a}^{3}{d}^{3}-146\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}{a}^{2}bc{d}^{2}+110\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}a{b}^{2}{c}^{2}d-30\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}{b}^{3}{c}^{3}-236\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{a}^{3}c{d}^{2}-72\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{a}^{2}b{c}^{2}d+20\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}a{b}^{2}{c}^{3}-272\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{3}{c}^{2}d-16\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{2}b{c}^{3}-96\,\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{3}{c}^{3}\sqrt{ac} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/384*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^3/c*(15*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^4*a^4*d^4-60*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^4*a^3*b*c*d^3+90*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^4*a^2*b^2*c^2*d^2-60*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^4*a*b^3*c
^3*d+15*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^4*b^4*c^4-30*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^3*a^3*d^3-146*(a*c)
^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^3*a^2*b*c*d^2+110*(a*c)^(1/2)*(b*d*x^2+
a*d*x+b*c*x+a*c)^(1/2)*x^3*a*b^2*c^2*d-30*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^
(1/2)*x^3*b^3*c^3-236*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^2*a^3*c*d^2-
72*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^2*a^2*b*c^2*d+20*(a*c)^(1/2)*(b
*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^2*a*b^2*c^3-272*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x
+a*c)^(1/2)*x*a^3*c^2*d-16*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^2*b*c
^3-96*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^3*c^3*(a*c)^(1/2))/(b*d*x^2+a*d*x+b*c*x+
a*c)^(1/2)/x^4/(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(sqrt(b*x + a)*(d*x + c)^(5/2)/x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.716462, size = 1, normalized size = 0. \[ \left [\frac{15 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} x^{4} \log \left (\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right ) - 4 \,{\left (48 \, a^{3} c^{3} +{\left (15 \, b^{3} c^{3} - 55 \, a b^{2} c^{2} d + 73 \, a^{2} b c d^{2} + 15 \, a^{3} d^{3}\right )} x^{3} - 2 \,{\left (5 \, a b^{2} c^{3} - 18 \, a^{2} b c^{2} d - 59 \, a^{3} c d^{2}\right )} x^{2} + 8 \,{\left (a^{2} b c^{3} + 17 \, a^{3} c^{2} d\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c}}{768 \, \sqrt{a c} a^{3} c x^{4}}, \frac{15 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} x^{4} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right ) - 2 \,{\left (48 \, a^{3} c^{3} +{\left (15 \, b^{3} c^{3} - 55 \, a b^{2} c^{2} d + 73 \, a^{2} b c d^{2} + 15 \, a^{3} d^{3}\right )} x^{3} - 2 \,{\left (5 \, a b^{2} c^{3} - 18 \, a^{2} b c^{2} d - 59 \, a^{3} c d^{2}\right )} x^{2} + 8 \,{\left (a^{2} b c^{3} + 17 \, a^{3} c^{2} d\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c}}{384 \, \sqrt{-a c} a^{3} c x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/768*(15*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^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 - 55*a*b^2*c^2*d + 73*a^2*b*c*d^2 + 15
*a^3*d^3)*x^3 - 2*(5*a*b^2*c^3 - 18*a^2*b*c^2*d - 59*a^3*c*d^2)*x^2 + 8*(a^2*b*c
^3 + 17*a^3*c^2*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(a*c)*a^3*c*x^
4), 1/384*(15*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^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)*sqrt(d*x
+ c)*a*c)) - 2*(48*a^3*c^3 + (15*b^3*c^3 - 55*a*b^2*c^2*d + 73*a^2*b*c*d^2 + 15*
a^3*d^3)*x^3 - 2*(5*a*b^2*c^3 - 18*a^2*b*c^2*d - 59*a^3*c*d^2)*x^2 + 8*(a^2*b*c^
3 + 17*a^3*c^2*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(-a*c)*a^3*c*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)**(5/2)*(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)^(5/2)/x^5,x, algorithm="giac")

[Out]

Exception raised: TypeError