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

Optimal. Leaf size=213 \[ -2 a^{3/2} c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )-\frac{(a d+b c) \left (a^2 d^2-10 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 b^{3/2} d^{3/2}}+\frac{1}{8} \sqrt{a+b x} \sqrt{c+d x} \left (\frac{a^2 d}{b}+8 a c-\frac{b c^2}{d}\right )+\frac{1}{3} (a+b x)^{3/2} (c+d x)^{3/2}+\frac{\sqrt{a+b x} (c+d x)^{3/2} (a d+b c)}{4 d} \]

[Out]

((8*a*c - (b*c^2)/d + (a^2*d)/b)*Sqrt[a + b*x]*Sqrt[c + d*x])/8 + ((b*c + a*d)*S
qrt[a + b*x]*(c + d*x)^(3/2))/(4*d) + ((a + b*x)^(3/2)*(c + d*x)^(3/2))/3 - 2*a^
(3/2)*c^(3/2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])] - ((b*c +
a*d)*(b^2*c^2 - 10*a*b*c*d + a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*
Sqrt[c + d*x])])/(8*b^(3/2)*d^(3/2))

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Rubi [A]  time = 0.670329, antiderivative size = 213, 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 \[ -2 a^{3/2} c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )-\frac{(a d+b c) \left (a^2 d^2-10 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 b^{3/2} d^{3/2}}+\frac{1}{8} \sqrt{a+b x} \sqrt{c+d x} \left (\frac{a^2 d}{b}+8 a c-\frac{b c^2}{d}\right )+\frac{1}{3} (a+b x)^{3/2} (c+d x)^{3/2}+\frac{\sqrt{a+b x} (c+d x)^{3/2} (a d+b c)}{4 d} \]

Antiderivative was successfully verified.

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

[Out]

((8*a*c - (b*c^2)/d + (a^2*d)/b)*Sqrt[a + b*x]*Sqrt[c + d*x])/8 + ((b*c + a*d)*S
qrt[a + b*x]*(c + d*x)^(3/2))/(4*d) + ((a + b*x)^(3/2)*(c + d*x)^(3/2))/3 - 2*a^
(3/2)*c^(3/2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])] - ((b*c +
a*d)*(b^2*c^2 - 10*a*b*c*d + a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*
Sqrt[c + d*x])])/(8*b^(3/2)*d^(3/2))

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Rubi in Sympy [A]  time = 65.3372, size = 199, normalized size = 0.93 \[ - 2 a^{\frac{3}{2}} c^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )} + \frac{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}}}{3} + \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a d + b c\right )}{4 b} - \frac{\sqrt{a + b x} \sqrt{c + d x} \left (a^{2} d^{2} - 8 a b c d - b^{2} c^{2}\right )}{8 b d} - \frac{\left (a d + b c\right ) \left (a^{2} d^{2} - 10 a b c d + b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{8 b^{\frac{3}{2}} d^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**(3/2)*c**(3/2)*atanh(sqrt(c)*sqrt(a + b*x)/(sqrt(a)*sqrt(c + d*x))) + (a +
b*x)**(3/2)*(c + d*x)**(3/2)/3 + (a + b*x)**(3/2)*sqrt(c + d*x)*(a*d + b*c)/(4*
b) - sqrt(a + b*x)*sqrt(c + d*x)*(a**2*d**2 - 8*a*b*c*d - b**2*c**2)/(8*b*d) - (
a*d + b*c)*(a**2*d**2 - 10*a*b*c*d + b**2*c**2)*atanh(sqrt(d)*sqrt(a + b*x)/(sqr
t(b)*sqrt(c + d*x)))/(8*b**(3/2)*d**(3/2))

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Mathematica [A]  time = 0.418157, size = 236, normalized size = 1.11 \[ -a^{3/2} c^{3/2} \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )+a^{3/2} c^{3/2} \log (x)+\sqrt{a+b x} \sqrt{c+d x} \left (\frac{3 a^2 d^2+38 a b c d+3 b^2 c^2}{24 b d}+\frac{7}{12} x (a d+b c)+\frac{1}{3} b d x^2\right )-\frac{\left (a^3 d^3-9 a^2 b c d^2-9 a b^2 c^2 d+b^3 c^3\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{16 b^{3/2} d^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[a + b*x]*Sqrt[c + d*x]*((3*b^2*c^2 + 38*a*b*c*d + 3*a^2*d^2)/(24*b*d) + (7*
(b*c + a*d)*x)/12 + (b*d*x^2)/3) + a^(3/2)*c^(3/2)*Log[x] - a^(3/2)*c^(3/2)*Log[
2*a*c + b*c*x + a*d*x + 2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]] - ((b^3*c
^3 - 9*a*b^2*c^2*d - 9*a^2*b*c*d^2 + a^3*d^3)*Log[b*c + a*d + 2*b*d*x + 2*Sqrt[b
]*Sqrt[d]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(16*b^(3/2)*d^(3/2))

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Maple [B]  time = 0.021, size = 587, normalized size = 2.8 \[ -{\frac{1}{48\,bd}\sqrt{bx+a}\sqrt{dx+c} \left ( -16\,{x}^{2}{b}^{2}{d}^{2}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+3\,{d}^{3}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{3}\sqrt{ac}-27\,{d}^{2}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{2}c\sqrt{ac}b-27\,{c}^{2}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) ad\sqrt{ac}{b}^{2}+3\,{c}^{3}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){b}^{3}\sqrt{ac}+48\,{a}^{2}{c}^{2}\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ) b\sqrt{bd}d-28\,{d}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}xa\sqrt{bd}\sqrt{ac}b-28\,d\sqrt{d{x}^{2}b+adx+bcx+ac}xc\sqrt{bd}\sqrt{ac}{b}^{2}-6\,{d}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{2}\sqrt{bd}\sqrt{ac}-76\,d\sqrt{d{x}^{2}b+adx+bcx+ac}ac\sqrt{bd}\sqrt{ac}b-6\,{c}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}\sqrt{ac}{b}^{2} \right ){\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/48*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(-16*x^2*b^2*d^2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*
c*x+a*c)^(1/2)*(b*d)^(1/2)+3*d^3*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/
2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^3*(a*c)^(1/2)-27*d^2*ln(1/2*(2*b*d*x+2*(b
*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^2*c*(a*c)^(1/2
)*b-27*c^2*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c
)/(b*d)^(1/2))*a*d*(a*c)^(1/2)*b^2+3*c^3*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+
a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*b^3*(a*c)^(1/2)+48*a^2*c^2*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)*b*(b*d)^(1/2)*d-
28*d^2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x*a*(b*d)^(1/2)*(a*c)^(1/2)*b-28*d*(b*d*x
^2+a*d*x+b*c*x+a*c)^(1/2)*x*c*(b*d)^(1/2)*(a*c)^(1/2)*b^2-6*d^2*(b*d*x^2+a*d*x+b
*c*x+a*c)^(1/2)*a^2*(b*d)^(1/2)*(a*c)^(1/2)-76*d*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)
*a*c*(b*d)^(1/2)*(a*c)^(1/2)*b-6*c^2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)
*(a*c)^(1/2)*b^2)/(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)/b/(b*d)^(1/2)/d/(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)*(d*x + c)^(3/2)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 6.46196, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/96*(48*sqrt(a*c)*sqrt(b*d)*a*b*c*d*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^
2*d^2)*x^2 - 4*(2*a*c + (b*c + a*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c) + 8
*(a*b*c^2 + a^2*c*d)*x)/x^2) + 4*(8*b^2*d^2*x^2 + 3*b^2*c^2 + 38*a*b*c*d + 3*a^2
*d^2 + 14*(b^2*c*d + a*b*d^2)*x)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 3*(b^3*
c^3 - 9*a*b^2*c^2*d - 9*a^2*b*c*d^2 + a^3*d^3)*log(-4*(2*b^2*d^2*x + b^2*c*d + a
*b*d^2)*sqrt(b*x + a)*sqrt(d*x + c) + (8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2
*d^2 + 8*(b^2*c*d + a*b*d^2)*x)*sqrt(b*d)))/(sqrt(b*d)*b*d), 1/48*(24*sqrt(a*c)*
sqrt(-b*d)*a*b*c*d*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 - 4*(2*a
*c + (b*c + a*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(a*b*c^2 + a^2*c*d
)*x)/x^2) + 2*(8*b^2*d^2*x^2 + 3*b^2*c^2 + 38*a*b*c*d + 3*a^2*d^2 + 14*(b^2*c*d
+ a*b*d^2)*x)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c) - 3*(b^3*c^3 - 9*a*b^2*c^2*
d - 9*a^2*b*c*d^2 + a^3*d^3)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)/(sqrt(b
*x + a)*sqrt(d*x + c)*b*d)))/(sqrt(-b*d)*b*d), -1/96*(96*sqrt(-a*c)*sqrt(b*d)*a*
b*c*d*arctan(1/2*(2*a*c + (b*c + a*d)*x)/(sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c)
)) - 4*(8*b^2*d^2*x^2 + 3*b^2*c^2 + 38*a*b*c*d + 3*a^2*d^2 + 14*(b^2*c*d + a*b*d
^2)*x)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) - 3*(b^3*c^3 - 9*a*b^2*c^2*d - 9*a^
2*b*c*d^2 + a^3*d^3)*log(-4*(2*b^2*d^2*x + b^2*c*d + a*b*d^2)*sqrt(b*x + a)*sqrt
(d*x + c) + (8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 8*(b^2*c*d + a*b*d^
2)*x)*sqrt(b*d)))/(sqrt(b*d)*b*d), -1/48*(48*sqrt(-a*c)*sqrt(-b*d)*a*b*c*d*arcta
n(1/2*(2*a*c + (b*c + a*d)*x)/(sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c))) - 2*(8*b
^2*d^2*x^2 + 3*b^2*c^2 + 38*a*b*c*d + 3*a^2*d^2 + 14*(b^2*c*d + a*b*d^2)*x)*sqrt
(-b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 3*(b^3*c^3 - 9*a*b^2*c^2*d - 9*a^2*b*c*d^2
+ a^3*d^3)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)/(sqrt(b*x + a)*sqrt(d*x +
c)*b*d)))/(sqrt(-b*d)*b*d)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.305846, size = 447, normalized size = 2.1 \[ -\frac{2 \, \sqrt{b d} a^{2} c^{2}{\left | b \right |} \arctan \left (-\frac{b^{2} c + a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt{-a b c d} b}\right )}{\sqrt{-a b c d} b} + \frac{1}{24} \, \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \sqrt{b x + a}{\left (2 \,{\left (b x + a\right )}{\left (\frac{4 \,{\left (b x + a\right )} d{\left | b \right |}}{b^{3}} + \frac{7 \, b^{5} c d^{4}{\left | b \right |} - a b^{4} d^{5}{\left | b \right |}}{b^{7} d^{4}}\right )} + \frac{3 \,{\left (b^{6} c^{2} d^{3}{\left | b \right |} + 8 \, a b^{5} c d^{4}{\left | b \right |} - a^{2} b^{4} d^{5}{\left | b \right |}\right )}}{b^{7} d^{4}}\right )} + \frac{{\left (\sqrt{b d} b^{3} c^{3}{\left | b \right |} - 9 \, \sqrt{b d} a b^{2} c^{2} d{\left | b \right |} - 9 \, \sqrt{b d} a^{2} b c d^{2}{\left | b \right |} + \sqrt{b d} a^{3} d^{3}{\left | b \right |}\right )}{\rm ln}\left ({\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{16 \, b^{3} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(b*d)*a^2*c^2*abs(b)*arctan(-1/2*(b^2*c + a*b*d - (sqrt(b*d)*sqrt(b*x + a
) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/(sqrt(-a*b*c*d)*b))/(sqrt(-a*b*c*d)*
b) + 1/24*sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b*x
+ a)*d*abs(b)/b^3 + (7*b^5*c*d^4*abs(b) - a*b^4*d^5*abs(b))/(b^7*d^4)) + 3*(b^6
*c^2*d^3*abs(b) + 8*a*b^5*c*d^4*abs(b) - a^2*b^4*d^5*abs(b))/(b^7*d^4)) + 1/16*(
sqrt(b*d)*b^3*c^3*abs(b) - 9*sqrt(b*d)*a*b^2*c^2*d*abs(b) - 9*sqrt(b*d)*a^2*b*c*
d^2*abs(b) + sqrt(b*d)*a^3*d^3*abs(b))*ln((sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c
+ (b*x + a)*b*d - a*b*d))^2)/(b^3*d^2)