3.594 \(\int \frac{(a+b x)^{3/2} \sqrt{c+d x}}{x^6} \, dx\)

Optimal. Leaf size=340 \[ \frac{\left (7 a^2 d^2+6 a b c d+3 b^2 c^2\right ) (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{128 a^{7/2} c^{9/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (-35 a^3 d^3+61 a^2 b c d^2-9 a b^2 c^2 d+15 b^3 c^3\right )}{960 a^2 c^3 x^2}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (-105 a^4 d^4+190 a^3 b c d^3-36 a^2 b^2 c^2 d^2-30 a b^3 c^3 d+45 b^4 c^4\right )}{1920 a^3 c^4 x}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (\frac{3 b^2 c}{a}-\frac{7 a d^2}{c}+12 b d\right )}{240 c x^3}-\frac{(a+b x)^{3/2} \sqrt{c+d x}}{5 x^5}-\frac{\sqrt{a+b x} \sqrt{c+d x} (a d+3 b c)}{40 c x^4} \]

[Out]

-((3*b*c + a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(40*c*x^4) - (((3*b^2*c)/a + 12*b*d
- (7*a*d^2)/c)*Sqrt[a + b*x]*Sqrt[c + d*x])/(240*c*x^3) + ((15*b^3*c^3 - 9*a*b^
2*c^2*d + 61*a^2*b*c*d^2 - 35*a^3*d^3)*Sqrt[a + b*x]*Sqrt[c + d*x])/(960*a^2*c^3
*x^2) - ((45*b^4*c^4 - 30*a*b^3*c^3*d - 36*a^2*b^2*c^2*d^2 + 190*a^3*b*c*d^3 - 1
05*a^4*d^4)*Sqrt[a + b*x]*Sqrt[c + d*x])/(1920*a^3*c^4*x) - ((a + b*x)^(3/2)*Sqr
t[c + d*x])/(5*x^5) + ((b*c - a*d)^3*(3*b^2*c^2 + 6*a*b*c*d + 7*a^2*d^2)*ArcTanh
[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(128*a^(7/2)*c^(9/2))

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Rubi [A]  time = 1.04439, antiderivative size = 340, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{\left (7 a^2 d^2+6 a b c d+3 b^2 c^2\right ) (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{128 a^{7/2} c^{9/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (-35 a^3 d^3+61 a^2 b c d^2-9 a b^2 c^2 d+15 b^3 c^3\right )}{960 a^2 c^3 x^2}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (-105 a^4 d^4+190 a^3 b c d^3-36 a^2 b^2 c^2 d^2-30 a b^3 c^3 d+45 b^4 c^4\right )}{1920 a^3 c^4 x}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (\frac{3 b^2 c}{a}-\frac{7 a d^2}{c}+12 b d\right )}{240 c x^3}-\frac{(a+b x)^{3/2} \sqrt{c+d x}}{5 x^5}-\frac{\sqrt{a+b x} \sqrt{c+d x} (a d+3 b c)}{40 c x^4} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^(3/2)*Sqrt[c + d*x])/x^6,x]

[Out]

-((3*b*c + a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(40*c*x^4) - (((3*b^2*c)/a + 12*b*d
- (7*a*d^2)/c)*Sqrt[a + b*x]*Sqrt[c + d*x])/(240*c*x^3) + ((15*b^3*c^3 - 9*a*b^
2*c^2*d + 61*a^2*b*c*d^2 - 35*a^3*d^3)*Sqrt[a + b*x]*Sqrt[c + d*x])/(960*a^2*c^3
*x^2) - ((45*b^4*c^4 - 30*a*b^3*c^3*d - 36*a^2*b^2*c^2*d^2 + 190*a^3*b*c*d^3 - 1
05*a^4*d^4)*Sqrt[a + b*x]*Sqrt[c + d*x])/(1920*a^3*c^4*x) - ((a + b*x)^(3/2)*Sqr
t[c + d*x])/(5*x^5) + ((b*c - a*d)^3*(3*b^2*c^2 + 6*a*b*c*d + 7*a^2*d^2)*ArcTanh
[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(128*a^(7/2)*c^(9/2))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(3/2)*(d*x+c)**(1/2)/x**6,x)

[Out]

Timed out

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Mathematica [A]  time = 0.349407, size = 315, normalized size = 0.93 \[ \frac{-15 x^5 \log (x) (b c-a d)^3 \left (7 a^2 d^2+6 a b c d+3 b^2 c^2\right )+15 x^5 (b c-a d)^3 \left (7 a^2 d^2+6 a b c d+3 b^2 c^2\right ) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )-2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} \left (a^4 \left (384 c^4+48 c^3 d x-56 c^2 d^2 x^2+70 c d^3 x^3-105 d^4 x^4\right )+2 a^3 b c x \left (264 c^3+48 c^2 d x-61 c d^2 x^2+95 d^3 x^3\right )+6 a^2 b^2 c^2 x^2 \left (4 c^2+3 c d x-6 d^2 x^2\right )-30 a b^3 c^3 x^3 (c+d x)+45 b^4 c^4 x^4\right )}{3840 a^{7/2} c^{9/2} x^5} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^(3/2)*Sqrt[c + d*x])/x^6,x]

[Out]

(-2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]*(45*b^4*c^4*x^4 - 30*a*b^3*c^3*x
^3*(c + d*x) + 6*a^2*b^2*c^2*x^2*(4*c^2 + 3*c*d*x - 6*d^2*x^2) + 2*a^3*b*c*x*(26
4*c^3 + 48*c^2*d*x - 61*c*d^2*x^2 + 95*d^3*x^3) + a^4*(384*c^4 + 48*c^3*d*x - 56
*c^2*d^2*x^2 + 70*c*d^3*x^3 - 105*d^4*x^4)) - 15*(b*c - a*d)^3*(3*b^2*c^2 + 6*a*
b*c*d + 7*a^2*d^2)*x^5*Log[x] + 15*(b*c - a*d)^3*(3*b^2*c^2 + 6*a*b*c*d + 7*a^2*
d^2)*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^(7/2)*c^(9/2)*x^5)

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Maple [B]  time = 0.028, size = 967, normalized size = 2.8 \[ -{\frac{1}{3840\,{a}^{3}{c}^{4}{x}^{5}}\sqrt{bx+a}\sqrt{dx+c} \left ( 105\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}{a}^{5}{d}^{5}-225\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}{a}^{4}bc{d}^{4}+90\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}{a}^{3}{b}^{2}{c}^{2}{d}^{3}+30\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}{a}^{2}{b}^{3}{c}^{3}{d}^{2}+45\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}a{b}^{4}{c}^{4}d-45\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}{b}^{5}{c}^{5}-210\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}{a}^{4}{d}^{4}+380\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}{a}^{3}bc{d}^{3}-72\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}{a}^{2}{b}^{2}{c}^{2}{d}^{2}-60\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}a{b}^{3}{c}^{3}d+90\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}{b}^{4}{c}^{4}+140\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}{a}^{4}c{d}^{3}-244\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}{a}^{3}b{c}^{2}{d}^{2}+36\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}{a}^{2}{b}^{2}{c}^{3}d-60\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}a{b}^{3}{c}^{4}-112\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{a}^{4}{c}^{2}{d}^{2}+192\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{a}^{3}b{c}^{3}d+48\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{a}^{2}{b}^{2}{c}^{4}+96\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{4}{c}^{3}d+1056\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{3}b{c}^{4}+768\,\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{4}{c}^{4}\sqrt{ac} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(3/2)*(d*x+c)^(1/2)/x^6,x)

[Out]

-1/3840*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^3/c^4*(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*a^5*d^5-225*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+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^5*a^3*b^2*c^2*d^3+
30*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+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*b^4*c^4*d-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*b^5*c^5-210*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)
*x^4*a^4*d^4+380*(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^3-72*
(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^2-60*(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+90*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c
*x+a*c)^(1/2)*x^4*b^4*c^4+140*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^3*a^
4*c*d^3-244*(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+36*(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-60*(a*c)^(1/2)*(b*d*x
^2+a*d*x+b*c*x+a*c)^(1/2)*x^3*a*b^3*c^4-112*(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+192*(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+48*(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+96*(a*c)^
(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a^4*c^3*d+1056*(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((b*x + a)^(3/2)*sqrt(d*x + c)/x^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.74311, size = 1, normalized size = 0. \[ \left [-\frac{15 \,{\left (3 \, b^{5} c^{5} - 3 \, a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2} - 6 \, a^{3} b^{2} c^{2} d^{3} + 15 \, a^{4} b c d^{4} - 7 \, a^{5} d^{5}\right )} x^{5} \log \left (-\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} -{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right ) + 4 \,{\left (384 \, a^{4} c^{4} +{\left (45 \, b^{4} c^{4} - 30 \, a b^{3} c^{3} d - 36 \, a^{2} b^{2} c^{2} d^{2} + 190 \, a^{3} b c d^{3} - 105 \, a^{4} d^{4}\right )} x^{4} - 2 \,{\left (15 \, a b^{3} c^{4} - 9 \, a^{2} b^{2} c^{3} d + 61 \, a^{3} b c^{2} d^{2} - 35 \, a^{4} c d^{3}\right )} x^{3} + 8 \,{\left (3 \, a^{2} b^{2} c^{4} + 12 \, a^{3} b c^{3} d - 7 \, a^{4} c^{2} d^{2}\right )} x^{2} + 48 \,{\left (11 \, a^{3} b c^{4} + a^{4} c^{3} d\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c}}{7680 \, \sqrt{a c} a^{3} c^{4} x^{5}}, \frac{15 \,{\left (3 \, b^{5} c^{5} - 3 \, a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2} - 6 \, a^{3} b^{2} c^{2} d^{3} + 15 \, a^{4} b c d^{4} - 7 \, a^{5} d^{5}\right )} x^{5} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right ) - 2 \,{\left (384 \, a^{4} c^{4} +{\left (45 \, b^{4} c^{4} - 30 \, a b^{3} c^{3} d - 36 \, a^{2} b^{2} c^{2} d^{2} + 190 \, a^{3} b c d^{3} - 105 \, a^{4} d^{4}\right )} x^{4} - 2 \,{\left (15 \, a b^{3} c^{4} - 9 \, a^{2} b^{2} c^{3} d + 61 \, a^{3} b c^{2} d^{2} - 35 \, a^{4} c d^{3}\right )} x^{3} + 8 \,{\left (3 \, a^{2} b^{2} c^{4} + 12 \, a^{3} b c^{3} d - 7 \, a^{4} c^{2} d^{2}\right )} x^{2} + 48 \,{\left (11 \, a^{3} b c^{4} + a^{4} c^{3} d\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c}}{3840 \, \sqrt{-a c} a^{3} c^{4} x^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(3/2)*sqrt(d*x + c)/x^6,x, algorithm="fricas")

[Out]

[-1/7680*(15*(3*b^5*c^5 - 3*a*b^4*c^4*d - 2*a^2*b^3*c^3*d^2 - 6*a^3*b^2*c^2*d^3
+ 15*a^4*b*c*d^4 - 7*a^5*d^5)*x^5*log(-(4*(2*a^2*c^2 + (a*b*c^2 + a^2*c*d)*x)*sq
rt(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 + (45*b^4*c^4 - 30*a*b^
3*c^3*d - 36*a^2*b^2*c^2*d^2 + 190*a^3*b*c*d^3 - 105*a^4*d^4)*x^4 - 2*(15*a*b^3*
c^4 - 9*a^2*b^2*c^3*d + 61*a^3*b*c^2*d^2 - 35*a^4*c*d^3)*x^3 + 8*(3*a^2*b^2*c^4
+ 12*a^3*b*c^3*d - 7*a^4*c^2*d^2)*x^2 + 48*(11*a^3*b*c^4 + a^4*c^3*d)*x)*sqrt(a*
c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(a*c)*a^3*c^4*x^5), 1/3840*(15*(3*b^5*c^5 -
3*a*b^4*c^4*d - 2*a^2*b^3*c^3*d^2 - 6*a^3*b^2*c^2*d^3 + 15*a^4*b*c*d^4 - 7*a^5*
d^5)*x^5*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*(384*a^4*c^4 + (45*b^4*c^4 - 30*a*b^3*c^3*d - 36*a^2*b^2*c^2*d^2 +
190*a^3*b*c*d^3 - 105*a^4*d^4)*x^4 - 2*(15*a*b^3*c^4 - 9*a^2*b^2*c^3*d + 61*a^3
*b*c^2*d^2 - 35*a^4*c*d^3)*x^3 + 8*(3*a^2*b^2*c^4 + 12*a^3*b*c^3*d - 7*a^4*c^2*d
^2)*x^2 + 48*(11*a^3*b*c^4 + a^4*c^3*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c
))/(sqrt(-a*c)*a^3*c^4*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((b*x+a)**(3/2)*(d*x+c)**(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((b*x + a)^(3/2)*sqrt(d*x + c)/x^6,x, algorithm="giac")

[Out]

Exception raised: TypeError