3.562 \(\int \frac{\sqrt{a+b x} (c+d x)^{5/2}}{x^6} \, dx\)

Optimal. Leaf size=283 \[ -\frac{(3 a d+7 b c) (b c-a d)^4 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{128 a^{9/2} c^{5/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} (3 a d+7 b c) (b c-a d)^3}{128 a^4 c^2 x}-\frac{\sqrt{a+b x} (c+d x)^{3/2} (3 a d+7 b c) (b c-a d)^2}{192 a^3 c^2 x^2}+\frac{\sqrt{a+b x} (c+d x)^{5/2} (3 a d+7 b c) (b c-a d)}{240 a^2 c^2 x^3}+\frac{\sqrt{a+b x} (c+d x)^{7/2} (3 a d+7 b c)}{40 a c^2 x^4}-\frac{(a+b x)^{3/2} (c+d x)^{7/2}}{5 a c x^5} \]

[Out]

((b*c - a*d)^3*(7*b*c + 3*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(128*a^4*c^2*x) - ((
b*c - a*d)^2*(7*b*c + 3*a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(192*a^3*c^2*x^2) +
((b*c - a*d)*(7*b*c + 3*a*d)*Sqrt[a + b*x]*(c + d*x)^(5/2))/(240*a^2*c^2*x^3) +
((7*b*c + 3*a*d)*Sqrt[a + b*x]*(c + d*x)^(7/2))/(40*a*c^2*x^4) - ((a + b*x)^(3/2
)*(c + d*x)^(7/2))/(5*a*c*x^5) - ((b*c - a*d)^4*(7*b*c + 3*a*d)*ArcTanh[(Sqrt[c]
*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(128*a^(9/2)*c^(5/2))

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Rubi [A]  time = 0.538105, antiderivative size = 283, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{(3 a d+7 b c) (b c-a d)^4 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{128 a^{9/2} c^{5/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} (3 a d+7 b c) (b c-a d)^3}{128 a^4 c^2 x}-\frac{\sqrt{a+b x} (c+d x)^{3/2} (3 a d+7 b c) (b c-a d)^2}{192 a^3 c^2 x^2}+\frac{\sqrt{a+b x} (c+d x)^{5/2} (3 a d+7 b c) (b c-a d)}{240 a^2 c^2 x^3}+\frac{\sqrt{a+b x} (c+d x)^{7/2} (3 a d+7 b c)}{40 a c^2 x^4}-\frac{(a+b x)^{3/2} (c+d x)^{7/2}}{5 a c x^5} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[a + b*x]*(c + d*x)^(5/2))/x^6,x]

[Out]

((b*c - a*d)^3*(7*b*c + 3*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(128*a^4*c^2*x) - ((
b*c - a*d)^2*(7*b*c + 3*a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(192*a^3*c^2*x^2) +
((b*c - a*d)*(7*b*c + 3*a*d)*Sqrt[a + b*x]*(c + d*x)^(5/2))/(240*a^2*c^2*x^3) +
((7*b*c + 3*a*d)*Sqrt[a + b*x]*(c + d*x)^(7/2))/(40*a*c^2*x^4) - ((a + b*x)^(3/2
)*(c + d*x)^(7/2))/(5*a*c*x^5) - ((b*c - a*d)^4*(7*b*c + 3*a*d)*ArcTanh[(Sqrt[c]
*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(128*a^(9/2)*c^(5/2))

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Rubi in Sympy [A]  time = 50.3565, size = 257, normalized size = 0.91 \[ - \frac{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{7}{2}}}{5 a c x^{5}} + \frac{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{5}{2}} \left (3 a d + 7 b c\right )}{40 a^{2} c x^{4}} + \frac{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right ) \left (3 a d + 7 b c\right )}{48 a^{3} c x^{3}} + \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a d - b c\right )^{2} \left (3 a d + 7 b c\right )}{64 a^{4} c x^{2}} + \frac{\sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )^{3} \left (3 a d + 7 b c\right )}{128 a^{4} c^{2} x} - \frac{\left (a d - b c\right )^{4} \left (3 a d + 7 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{128 a^{\frac{9}{2}} c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(5/2)*(b*x+a)**(1/2)/x**6,x)

[Out]

-(a + b*x)**(3/2)*(c + d*x)**(7/2)/(5*a*c*x**5) + (a + b*x)**(3/2)*(c + d*x)**(5
/2)*(3*a*d + 7*b*c)/(40*a**2*c*x**4) + (a + b*x)**(3/2)*(c + d*x)**(3/2)*(a*d -
b*c)*(3*a*d + 7*b*c)/(48*a**3*c*x**3) + (a + b*x)**(3/2)*sqrt(c + d*x)*(a*d - b*
c)**2*(3*a*d + 7*b*c)/(64*a**4*c*x**2) + sqrt(a + b*x)*sqrt(c + d*x)*(a*d - b*c)
**3*(3*a*d + 7*b*c)/(128*a**4*c**2*x) - (a*d - b*c)**4*(3*a*d + 7*b*c)*atanh(sqr
t(c)*sqrt(a + b*x)/(sqrt(a)*sqrt(c + d*x)))/(128*a**(9/2)*c**(5/2))

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Mathematica [A]  time = 0.339603, size = 291, normalized size = 1.03 \[ \frac{-2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} \left (3 a^4 \left (128 c^4+336 c^3 d x+248 c^2 d^2 x^2+10 c d^3 x^3-15 d^4 x^4\right )+2 a^3 b c x \left (24 c^3+88 c^2 d x+109 c d^2 x^2+30 d^3 x^3\right )-2 a^2 b^2 c^2 x^2 \left (28 c^2+111 c d x+173 d^2 x^2\right )+10 a b^3 c^3 x^3 (7 c+34 d x)-105 b^4 c^4 x^4\right )+15 x^5 \log (x) (b c-a d)^4 (3 a d+7 b c)-15 x^5 (b c-a d)^4 (3 a d+7 b c) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{3840 a^{9/2} c^{5/2} x^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[a + b*x]*(c + d*x)^(5/2))/x^6,x]

[Out]

(-2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]*(-105*b^4*c^4*x^4 + 10*a*b^3*c^3
*x^3*(7*c + 34*d*x) - 2*a^2*b^2*c^2*x^2*(28*c^2 + 111*c*d*x + 173*d^2*x^2) + 2*a
^3*b*c*x*(24*c^3 + 88*c^2*d*x + 109*c*d^2*x^2 + 30*d^3*x^3) + 3*a^4*(128*c^4 + 3
36*c^3*d*x + 248*c^2*d^2*x^2 + 10*c*d^3*x^3 - 15*d^4*x^4)) + 15*(b*c - a*d)^4*(7
*b*c + 3*a*d)*x^5*Log[x] - 15*(b*c - a*d)^4*(7*b*c + 3*a*d)*x^5*Log[2*a*c + b*c*
x + a*d*x + 2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(3840*a^(9/2)*c^(5/2
)*x^5)

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Maple [B]  time = 0.029, size = 967, normalized size = 3.4 \[ -{\frac{1}{3840\,{a}^{4}{c}^{2}{x}^{5}}\sqrt{bx+a}\sqrt{dx+c} \left ( 45\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}{a}^{5}{d}^{5}-75\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}{a}^{4}bc{d}^{4}-150\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}{a}^{3}{b}^{2}{c}^{2}{d}^{3}+450\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}{a}^{2}{b}^{3}{c}^{3}{d}^{2}-375\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}a{b}^{4}{c}^{4}d+105\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}{b}^{5}{c}^{5}-90\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}{a}^{4}{d}^{4}+120\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}{a}^{3}bc{d}^{3}-692\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}{a}^{2}{b}^{2}{c}^{2}{d}^{2}+680\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}a{b}^{3}{c}^{3}d-210\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}{b}^{4}{c}^{4}+60\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}{a}^{4}c{d}^{3}+436\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}{a}^{3}b{c}^{2}{d}^{2}-444\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}{a}^{2}{b}^{2}{c}^{3}d+140\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}a{b}^{3}{c}^{4}+1488\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{a}^{4}{c}^{2}{d}^{2}+352\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{a}^{3}b{c}^{3}d-112\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{a}^{2}{b}^{2}{c}^{4}+2016\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{4}{c}^{3}d+96\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{3}b{c}^{4}+768\,\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{4}{c}^{4}\sqrt{ac} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(5/2)*(b*x+a)^(1/2)/x^6,x)

[Out]

-1/3840*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^4/c^2*(45*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^5*a^5*d^5-75*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^5*a^4*b*c*d^4-150*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^5*a^3*b^2*c^2*d^3+4
50*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^5*a
^2*b^3*c^3*d^2-375*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^5*a*b^4*c^4*d+105*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^5*b^5*c^5-90*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2
)*x^4*a^4*d^4+120*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^4*a^3*b*c*d^3-69
2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^4*a^2*b^2*c^2*d^2+680*(a*c)^(1/2
)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^4*a*b^3*c^3*d-210*(a*c)^(1/2)*(b*d*x^2+a*d*x
+b*c*x+a*c)^(1/2)*x^4*b^4*c^4+60*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^3
*a^4*c*d^3+436*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^3*a^3*b*c^2*d^2-444
*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^3*a^2*b^2*c^3*d+140*(a*c)^(1/2)*(
b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^3*a*b^3*c^4+1488*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c
*x+a*c)^(1/2)*x^2*a^4*c^2*d^2+352*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^
2*a^3*b*c^3*d-112*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^2*a^2*b^2*c^4+20
16*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^4*c^3*d+96*(a*c)^(1/2)*(b*d*x
^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^3*b*c^4+768*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^4*c^
4*(a*c)^(1/2))/(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)/x^5/(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^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.81009, size = 1, normalized size = 0. \[ \left [\frac{15 \,{\left (7 \, b^{5} c^{5} - 25 \, a b^{4} c^{4} d + 30 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} + 3 \, a^{5} d^{5}\right )} x^{5} \log \left (-\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} -{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right ) - 4 \,{\left (384 \, a^{4} c^{4} -{\left (105 \, b^{4} c^{4} - 340 \, a b^{3} c^{3} d + 346 \, a^{2} b^{2} c^{2} d^{2} - 60 \, a^{3} b c d^{3} + 45 \, a^{4} d^{4}\right )} x^{4} + 2 \,{\left (35 \, a b^{3} c^{4} - 111 \, a^{2} b^{2} c^{3} d + 109 \, a^{3} b c^{2} d^{2} + 15 \, a^{4} c d^{3}\right )} x^{3} - 8 \,{\left (7 \, a^{2} b^{2} c^{4} - 22 \, a^{3} b c^{3} d - 93 \, a^{4} c^{2} d^{2}\right )} x^{2} + 48 \,{\left (a^{3} b c^{4} + 21 \, a^{4} c^{3} d\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c}}{7680 \, \sqrt{a c} a^{4} c^{2} x^{5}}, -\frac{15 \,{\left (7 \, b^{5} c^{5} - 25 \, a b^{4} c^{4} d + 30 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} + 3 \, a^{5} d^{5}\right )} x^{5} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right ) + 2 \,{\left (384 \, a^{4} c^{4} -{\left (105 \, b^{4} c^{4} - 340 \, a b^{3} c^{3} d + 346 \, a^{2} b^{2} c^{2} d^{2} - 60 \, a^{3} b c d^{3} + 45 \, a^{4} d^{4}\right )} x^{4} + 2 \,{\left (35 \, a b^{3} c^{4} - 111 \, a^{2} b^{2} c^{3} d + 109 \, a^{3} b c^{2} d^{2} + 15 \, a^{4} c d^{3}\right )} x^{3} - 8 \,{\left (7 \, a^{2} b^{2} c^{4} - 22 \, a^{3} b c^{3} d - 93 \, a^{4} c^{2} d^{2}\right )} x^{2} + 48 \,{\left (a^{3} b c^{4} + 21 \, a^{4} c^{3} d\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c}}{3840 \, \sqrt{-a c} a^{4} c^{2} x^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x + a)*(d*x + c)^(5/2)/x^6,x, algorithm="fricas")

[Out]

[1/7680*(15*(7*b^5*c^5 - 25*a*b^4*c^4*d + 30*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^
3 - 5*a^4*b*c*d^4 + 3*a^5*d^5)*x^5*log(-(4*(2*a^2*c^2 + (a*b*c^2 + a^2*c*d)*x)*s
qrt(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*(384*a^4*c^4 - (105*b^4*c^4 - 340*a
*b^3*c^3*d + 346*a^2*b^2*c^2*d^2 - 60*a^3*b*c*d^3 + 45*a^4*d^4)*x^4 + 2*(35*a*b^
3*c^4 - 111*a^2*b^2*c^3*d + 109*a^3*b*c^2*d^2 + 15*a^4*c*d^3)*x^3 - 8*(7*a^2*b^2
*c^4 - 22*a^3*b*c^3*d - 93*a^4*c^2*d^2)*x^2 + 48*(a^3*b*c^4 + 21*a^4*c^3*d)*x)*s
qrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(a*c)*a^4*c^2*x^5), -1/3840*(15*(7*b^
5*c^5 - 25*a*b^4*c^4*d + 30*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 - 5*a^4*b*c*d^4
+ 3*a^5*d^5)*x^5*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*(384*a^4*c^4 - (105*b^4*c^4 - 340*a*b^3*c^3*d + 346*a^2*b
^2*c^2*d^2 - 60*a^3*b*c*d^3 + 45*a^4*d^4)*x^4 + 2*(35*a*b^3*c^4 - 111*a^2*b^2*c^
3*d + 109*a^3*b*c^2*d^2 + 15*a^4*c*d^3)*x^3 - 8*(7*a^2*b^2*c^4 - 22*a^3*b*c^3*d
- 93*a^4*c^2*d^2)*x^2 + 48*(a^3*b*c^4 + 21*a^4*c^3*d)*x)*sqrt(-a*c)*sqrt(b*x + a
)*sqrt(d*x + c))/(sqrt(-a*c)*a^4*c^2*x^5)]

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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**6,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^6,x, algorithm="giac")

[Out]

Exception raised: TypeError