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

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

[Out]

-((b*c + 3*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(40*a*x^4) + (((7*b^2*c)/a - 12*b*d
- (3*a*d^2)/c)*Sqrt[a + b*x]*Sqrt[c + d*x])/(240*a*x^3) - ((35*b^3*c^3 - 61*a*b
^2*c^2*d + 9*a^2*b*c*d^2 - 15*a^3*d^3)*Sqrt[a + b*x]*Sqrt[c + d*x])/(960*a^3*c^2
*x^2) + ((105*b^4*c^4 - 190*a*b^3*c^3*d + 36*a^2*b^2*c^2*d^2 + 30*a^3*b*c*d^3 -
45*a^4*d^4)*Sqrt[a + b*x]*Sqrt[c + d*x])/(1920*a^4*c^3*x) - (Sqrt[a + b*x]*(c +
d*x)^(3/2))/(5*x^5) - ((b*c - a*d)^3*(7*b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*ArcTanh
[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(128*a^(9/2)*c^(7/2))

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Rubi [A]  time = 1.05523, 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 \[ -\frac{\left (3 a^2 d^2+6 a b c d+7 b^2 c^2\right ) (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{128 a^{9/2} c^{7/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (-15 a^3 d^3+9 a^2 b c d^2-61 a b^2 c^2 d+35 b^3 c^3\right )}{960 a^3 c^2 x^2}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (-45 a^4 d^4+30 a^3 b c d^3+36 a^2 b^2 c^2 d^2-190 a b^3 c^3 d+105 b^4 c^4\right )}{1920 a^4 c^3 x}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (\frac{7 b^2 c}{a}-\frac{3 a d^2}{c}-12 b d\right )}{240 a x^3}-\frac{\sqrt{a+b x} (c+d x)^{3/2}}{5 x^5}-\frac{\sqrt{a+b x} \sqrt{c+d x} (3 a d+b c)}{40 a x^4} \]

Antiderivative was successfully verified.

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

[Out]

-((b*c + 3*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(40*a*x^4) + (((7*b^2*c)/a - 12*b*d
- (3*a*d^2)/c)*Sqrt[a + b*x]*Sqrt[c + d*x])/(240*a*x^3) - ((35*b^3*c^3 - 61*a*b
^2*c^2*d + 9*a^2*b*c*d^2 - 15*a^3*d^3)*Sqrt[a + b*x]*Sqrt[c + d*x])/(960*a^3*c^2
*x^2) + ((105*b^4*c^4 - 190*a*b^3*c^3*d + 36*a^2*b^2*c^2*d^2 + 30*a^3*b*c*d^3 -
45*a^4*d^4)*Sqrt[a + b*x]*Sqrt[c + d*x])/(1920*a^4*c^3*x) - (Sqrt[a + b*x]*(c +
d*x)^(3/2))/(5*x^5) - ((b*c - a*d)^3*(7*b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*ArcTanh
[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(128*a^(9/2)*c^(7/2))

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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((d*x+c)**(3/2)*(b*x+a)**(1/2)/x**6,x)

[Out]

Timed out

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Mathematica [A]  time = 0.341576, size = 319, normalized size = 0.94 \[ \frac{15 x^5 \log (x) (b c-a d)^3 \left (3 a^2 d^2+6 a b c d+7 b^2 c^2\right )-15 x^5 (b c-a d)^3 \left (3 a^2 d^2+6 a b c d+7 b^2 c^2\right ) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )-2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} \left (3 a^4 \left (128 c^4+176 c^3 d x+8 c^2 d^2 x^2-10 c d^3 x^3+15 d^4 x^4\right )+6 a^3 b c x \left (8 c^3+16 c^2 d x+3 c d^2 x^2-5 d^3 x^3\right )-2 a^2 b^2 c^2 x^2 \left (28 c^2+61 c d x+18 d^2 x^2\right )+10 a b^3 c^3 x^3 (7 c+19 d x)-105 b^4 c^4 x^4\right )}{3840 a^{9/2} c^{7/2} x^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[a + b*x]*(c + d*x)^(3/2))/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 + 19*d*x) - 2*a^2*b^2*c^2*x^2*(28*c^2 + 61*c*d*x + 18*d^2*x^2) + 6*a^3
*b*c*x*(8*c^3 + 16*c^2*d*x + 3*c*d^2*x^2 - 5*d^3*x^3) + 3*a^4*(128*c^4 + 176*c^3
*d*x + 8*c^2*d^2*x^2 - 10*c*d^3*x^3 + 15*d^4*x^4)) + 15*(b*c - a*d)^3*(7*b^2*c^2
+ 6*a*b*c*d + 3*a^2*d^2)*x^5*Log[x] - 15*(b*c - a*d)^3*(7*b^2*c^2 + 6*a*b*c*d +
3*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^(9/2)*c^(7/2)*x^5)

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Maple [B]  time = 0.03, size = 967, normalized size = 2.8 \[{\frac{1}{3840\,{a}^{4}{c}^{3}{x}^{5}}\sqrt{bx+a}\sqrt{dx+c} \left ( 45\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}{a}^{5}{d}^{5}-45\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}{a}^{4}bc{d}^{4}-30\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}{a}^{3}{b}^{2}{c}^{2}{d}^{3}-90\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}{a}^{2}{b}^{3}{c}^{3}{d}^{2}+225\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}a{b}^{4}{c}^{4}d-105\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}{b}^{5}{c}^{5}-90\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}{a}^{4}{d}^{4}+60\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}{a}^{3}bc{d}^{3}+72\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}{a}^{2}{b}^{2}{c}^{2}{d}^{2}-380\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}a{b}^{3}{c}^{3}d+210\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}{b}^{4}{c}^{4}+60\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}{a}^{4}c{d}^{3}-36\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}{a}^{3}b{c}^{2}{d}^{2}+244\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}{a}^{2}{b}^{2}{c}^{3}d-140\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}a{b}^{3}{c}^{4}-48\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{a}^{4}{c}^{2}{d}^{2}-192\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{a}^{3}b{c}^{3}d+112\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{a}^{2}{b}^{2}{c}^{4}-1056\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{4}{c}^{3}d-96\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{3}b{c}^{4}-768\,\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{4}{c}^{4}\sqrt{ac} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3840*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^4/c^3*(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^5-45*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^4*b*c*d^4-30*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^3*b^2*c^2*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^5*a^2*
b^3*c^3*d^2+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*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+60*(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^2-380*(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^3-36*(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+244*(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^4-48*(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+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^4-1056*(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)^(3/2)/x^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.7925, size = 1, normalized size = 0. \[ \left [-\frac{15 \,{\left (7 \, b^{5} c^{5} - 15 \, a b^{4} c^{4} d + 6 \, a^{2} b^{3} c^{3} d^{2} + 2 \, a^{3} b^{2} c^{2} d^{3} + 3 \, a^{4} b c d^{4} - 3 \, a^{5} d^{5}\right )} x^{5} \log \left (\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right ) + 4 \,{\left (384 \, a^{4} c^{4} -{\left (105 \, b^{4} c^{4} - 190 \, a b^{3} c^{3} d + 36 \, a^{2} b^{2} c^{2} d^{2} + 30 \, a^{3} b c d^{3} - 45 \, a^{4} d^{4}\right )} x^{4} + 2 \,{\left (35 \, a b^{3} c^{4} - 61 \, a^{2} b^{2} c^{3} d + 9 \, a^{3} b c^{2} d^{2} - 15 \, a^{4} c d^{3}\right )} x^{3} - 8 \,{\left (7 \, a^{2} b^{2} c^{4} - 12 \, a^{3} b c^{3} d - 3 \, a^{4} c^{2} d^{2}\right )} x^{2} + 48 \,{\left (a^{3} b c^{4} + 11 \, a^{4} c^{3} d\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c}}{7680 \, \sqrt{a c} a^{4} c^{3} x^{5}}, -\frac{15 \,{\left (7 \, b^{5} c^{5} - 15 \, a b^{4} c^{4} d + 6 \, a^{2} b^{3} c^{3} d^{2} + 2 \, a^{3} b^{2} c^{2} d^{3} + 3 \, a^{4} b c d^{4} - 3 \, a^{5} d^{5}\right )} x^{5} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right ) + 2 \,{\left (384 \, a^{4} c^{4} -{\left (105 \, b^{4} c^{4} - 190 \, a b^{3} c^{3} d + 36 \, a^{2} b^{2} c^{2} d^{2} + 30 \, a^{3} b c d^{3} - 45 \, a^{4} d^{4}\right )} x^{4} + 2 \,{\left (35 \, a b^{3} c^{4} - 61 \, a^{2} b^{2} c^{3} d + 9 \, a^{3} b c^{2} d^{2} - 15 \, a^{4} c d^{3}\right )} x^{3} - 8 \,{\left (7 \, a^{2} b^{2} c^{4} - 12 \, a^{3} b c^{3} d - 3 \, a^{4} c^{2} d^{2}\right )} x^{2} + 48 \,{\left (a^{3} b c^{4} + 11 \, a^{4} c^{3} d\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c}}{3840 \, \sqrt{-a c} a^{4} c^{3} x^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError