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

Optimal. Leaf size=283 \[ -\frac{(7 a d+3 b c) (b c-a d)^4 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{128 a^{5/2} c^{9/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} (7 a d+3 b c) (b c-a d)^3}{128 a^2 c^4 x}+\frac{\sqrt{a+b x} (c+d x)^{3/2} (7 a d+3 b c) (b c-a d)^2}{64 a c^4 x^2}+\frac{(a+b x)^{3/2} (c+d x)^{3/2} (7 a d+3 b c) (b c-a d)}{48 a c^3 x^3}+\frac{(a+b x)^{5/2} (c+d x)^{3/2} (7 a d+3 b c)}{40 a c^2 x^4}-\frac{(a+b x)^{7/2} (c+d x)^{3/2}}{5 a c x^5} \]

[Out]

((b*c - a*d)^3*(3*b*c + 7*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(128*a^2*c^4*x) + ((
b*c - a*d)^2*(3*b*c + 7*a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(64*a*c^4*x^2) + ((b
*c - a*d)*(3*b*c + 7*a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2))/(48*a*c^3*x^3) + ((3*
b*c + 7*a*d)*(a + b*x)^(5/2)*(c + d*x)^(3/2))/(40*a*c^2*x^4) - ((a + b*x)^(7/2)*
(c + d*x)^(3/2))/(5*a*c*x^5) - ((b*c - a*d)^4*(3*b*c + 7*a*d)*ArcTanh[(Sqrt[c]*S
qrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(128*a^(5/2)*c^(9/2))

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Rubi [A]  time = 0.510758, 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 \[ -\frac{(7 a d+3 b c) (b c-a d)^4 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{128 a^{5/2} c^{9/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} (7 a d+3 b c) (b c-a d)^3}{128 a^2 c^4 x}+\frac{\sqrt{a+b x} (c+d x)^{3/2} (7 a d+3 b c) (b c-a d)^2}{64 a c^4 x^2}+\frac{(a+b x)^{3/2} (c+d x)^{3/2} (7 a d+3 b c) (b c-a d)}{48 a c^3 x^3}+\frac{(a+b x)^{5/2} (c+d x)^{3/2} (7 a d+3 b c)}{40 a c^2 x^4}-\frac{(a+b x)^{7/2} (c+d x)^{3/2}}{5 a c x^5} \]

Antiderivative was successfully verified.

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

[Out]

((b*c - a*d)^3*(3*b*c + 7*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(128*a^2*c^4*x) + ((
b*c - a*d)^2*(3*b*c + 7*a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(64*a*c^4*x^2) + ((b
*c - a*d)*(3*b*c + 7*a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2))/(48*a*c^3*x^3) + ((3*
b*c + 7*a*d)*(a + b*x)^(5/2)*(c + d*x)^(3/2))/(40*a*c^2*x^4) - ((a + b*x)^(7/2)*
(c + d*x)^(3/2))/(5*a*c*x^5) - ((b*c - a*d)^4*(3*b*c + 7*a*d)*ArcTanh[(Sqrt[c]*S
qrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(128*a^(5/2)*c^(9/2))

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Rubi in Sympy [A]  time = 50.4023, size = 260, normalized size = 0.92 \[ - \frac{\left (a + b x\right )^{\frac{7}{2}} \left (c + d x\right )^{\frac{3}{2}}}{5 a c x^{5}} + \frac{\left (a + b x\right )^{\frac{7}{2}} \sqrt{c + d x} \left (7 a d + 3 b c\right )}{40 a^{2} c x^{4}} + \frac{\left (a + b x\right )^{\frac{5}{2}} \sqrt{c + d x} \left (a d - b c\right ) \left (7 a d + 3 b c\right )}{240 a^{2} c^{2} x^{3}} - \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a d - b c\right )^{2} \left (7 a d + 3 b c\right )}{192 a^{2} c^{3} x^{2}} + \frac{\sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )^{3} \left (7 a d + 3 b c\right )}{128 a^{2} c^{4} x} - \frac{\left (a d - b c\right )^{4} \left (7 a d + 3 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{128 a^{\frac{5}{2}} c^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a + b*x)**(7/2)*(c + d*x)**(3/2)/(5*a*c*x**5) + (a + b*x)**(7/2)*sqrt(c + d*x)
*(7*a*d + 3*b*c)/(40*a**2*c*x**4) + (a + b*x)**(5/2)*sqrt(c + d*x)*(a*d - b*c)*(
7*a*d + 3*b*c)/(240*a**2*c**2*x**3) - (a + b*x)**(3/2)*sqrt(c + d*x)*(a*d - b*c)
**2*(7*a*d + 3*b*c)/(192*a**2*c**3*x**2) + sqrt(a + b*x)*sqrt(c + d*x)*(a*d - b*
c)**3*(7*a*d + 3*b*c)/(128*a**2*c**4*x) - (a*d - b*c)**4*(7*a*d + 3*b*c)*atanh(s
qrt(c)*sqrt(a + b*x)/(sqrt(a)*sqrt(c + d*x)))/(128*a**(5/2)*c**(9/2))

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Mathematica [A]  time = 0.340339, size = 288, normalized size = 1.02 \[ \frac{-2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} \left (a^4 \left (384 c^4+48 c^3 d x-56 c^2 d^2 x^2+70 c d^3 x^3-105 d^4 x^4\right )+2 a^3 b c x \left (504 c^3+88 c^2 d x-111 c d^2 x^2+170 d^3 x^3\right )+2 a^2 b^2 c^2 x^2 \left (372 c^2+109 c d x-173 d^2 x^2\right )+30 a b^3 c^3 x^3 (c+2 d x)-45 b^4 c^4 x^4\right )+15 x^5 \log (x) (b c-a d)^4 (7 a d+3 b c)-15 x^5 (b c-a d)^4 (7 a d+3 b c) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{3840 a^{5/2} c^{9/2} x^5} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^(5/2)*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 + 2*d*x) + 2*a^2*b^2*c^2*x^2*(372*c^2 + 109*c*d*x - 173*d^2*x^2) + 2*a^3*
b*c*x*(504*c^3 + 88*c^2*d*x - 111*c*d^2*x^2 + 170*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)^4*(3*b*c
+ 7*a*d)*x^5*Log[x] - 15*(b*c - a*d)^4*(3*b*c + 7*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^(5/2)*c^(9/2)*x^
5)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3840*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^2/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^5-375*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^4*b*c*d^4+450*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
-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^2*b^3*c^3*d^2-75*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*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+680*(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^3-6
92*(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+120*(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^3-444*(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+43
6*(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+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+1488*(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+2016*(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)^(5/2)*sqrt(d*x + c)/x^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.75735, size = 1, normalized size = 0. \[ \left [\frac{15 \,{\left (3 \, b^{5} c^{5} - 5 \, a b^{4} c^{4} d - 10 \, a^{2} b^{3} c^{3} d^{2} + 30 \, a^{3} b^{2} c^{2} d^{3} - 25 \, a^{4} b c d^{4} + 7 \, a^{5} d^{5}\right )} x^{5} \log \left (-\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} -{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right ) - 4 \,{\left (384 \, a^{4} c^{4} -{\left (45 \, b^{4} c^{4} - 60 \, a b^{3} c^{3} d + 346 \, a^{2} b^{2} c^{2} d^{2} - 340 \, a^{3} b c d^{3} + 105 \, a^{4} d^{4}\right )} x^{4} + 2 \,{\left (15 \, a b^{3} c^{4} + 109 \, a^{2} b^{2} c^{3} d - 111 \, a^{3} b c^{2} d^{2} + 35 \, a^{4} c d^{3}\right )} x^{3} + 8 \,{\left (93 \, a^{2} b^{2} c^{4} + 22 \, a^{3} b c^{3} d - 7 \, a^{4} c^{2} d^{2}\right )} x^{2} + 48 \,{\left (21 \, a^{3} b c^{4} + a^{4} c^{3} d\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c}}{7680 \, \sqrt{a c} a^{2} c^{4} x^{5}}, -\frac{15 \,{\left (3 \, b^{5} c^{5} - 5 \, a b^{4} c^{4} d - 10 \, a^{2} b^{3} c^{3} d^{2} + 30 \, a^{3} b^{2} c^{2} d^{3} - 25 \, a^{4} b c d^{4} + 7 \, a^{5} d^{5}\right )} x^{5} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right ) + 2 \,{\left (384 \, a^{4} c^{4} -{\left (45 \, b^{4} c^{4} - 60 \, a b^{3} c^{3} d + 346 \, a^{2} b^{2} c^{2} d^{2} - 340 \, a^{3} b c d^{3} + 105 \, a^{4} d^{4}\right )} x^{4} + 2 \,{\left (15 \, a b^{3} c^{4} + 109 \, a^{2} b^{2} c^{3} d - 111 \, a^{3} b c^{2} d^{2} + 35 \, a^{4} c d^{3}\right )} x^{3} + 8 \,{\left (93 \, a^{2} b^{2} c^{4} + 22 \, a^{3} b c^{3} d - 7 \, a^{4} c^{2} d^{2}\right )} x^{2} + 48 \,{\left (21 \, a^{3} b c^{4} + a^{4} c^{3} d\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c}}{3840 \, \sqrt{-a c} a^{2} c^{4} x^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/7680*(15*(3*b^5*c^5 - 5*a*b^4*c^4*d - 10*a^2*b^3*c^3*d^2 + 30*a^3*b^2*c^2*d^3
- 25*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)*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 - (45*b^4*c^4 - 60*a*b
^3*c^3*d + 346*a^2*b^2*c^2*d^2 - 340*a^3*b*c*d^3 + 105*a^4*d^4)*x^4 + 2*(15*a*b^
3*c^4 + 109*a^2*b^2*c^3*d - 111*a^3*b*c^2*d^2 + 35*a^4*c*d^3)*x^3 + 8*(93*a^2*b^
2*c^4 + 22*a^3*b*c^3*d - 7*a^4*c^2*d^2)*x^2 + 48*(21*a^3*b*c^4 + a^4*c^3*d)*x)*s
qrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(a*c)*a^2*c^4*x^5), -1/3840*(15*(3*b^
5*c^5 - 5*a*b^4*c^4*d - 10*a^2*b^3*c^3*d^2 + 30*a^3*b^2*c^2*d^3 - 25*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)*s
qrt(d*x + c)*a*c)) + 2*(384*a^4*c^4 - (45*b^4*c^4 - 60*a*b^3*c^3*d + 346*a^2*b^2
*c^2*d^2 - 340*a^3*b*c*d^3 + 105*a^4*d^4)*x^4 + 2*(15*a*b^3*c^4 + 109*a^2*b^2*c^
3*d - 111*a^3*b*c^2*d^2 + 35*a^4*c*d^3)*x^3 + 8*(93*a^2*b^2*c^4 + 22*a^3*b*c^3*d
- 7*a^4*c^2*d^2)*x^2 + 48*(21*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^2*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)**(5/2)*(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)^(5/2)*sqrt(d*x + c)/x^6,x, algorithm="giac")

[Out]

Exception raised: TypeError