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

Optimal. Leaf size=164 \[ -2 a^{3/2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )-\frac{\left (-3 a^2 d^2-6 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 \sqrt{b} d^{3/2}}+\frac{1}{2} (a+b x)^{3/2} \sqrt{c+d x}+\frac{\sqrt{a+b x} \sqrt{c+d x} (3 a d+b c)}{4 d} \]

[Out]

((b*c + 3*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(4*d) + ((a + b*x)^(3/2)*Sqrt[c + d*
x])/2 - 2*a^(3/2)*Sqrt[c]*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x]
)] - ((b^2*c^2 - 6*a*b*c*d - 3*a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]
*Sqrt[c + d*x])])/(4*Sqrt[b]*d^(3/2))

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Rubi [A]  time = 0.45831, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -2 a^{3/2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )-\frac{\left (-3 a^2 d^2-6 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 \sqrt{b} d^{3/2}}+\frac{1}{2} (a+b x)^{3/2} \sqrt{c+d x}+\frac{\sqrt{a+b x} \sqrt{c+d x} (3 a d+b c)}{4 d} \]

Antiderivative was successfully verified.

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

[Out]

((b*c + 3*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(4*d) + ((a + b*x)^(3/2)*Sqrt[c + d*
x])/2 - 2*a^(3/2)*Sqrt[c]*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x]
)] - ((b^2*c^2 - 6*a*b*c*d - 3*a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]
*Sqrt[c + d*x])])/(4*Sqrt[b]*d^(3/2))

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Rubi in Sympy [A]  time = 40.956, size = 151, normalized size = 0.92 \[ - 2 a^{\frac{3}{2}} \sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )} + \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x}}{2} + \frac{\sqrt{a + b x} \sqrt{c + d x} \left (3 a d + b c\right )}{4 d} + \frac{\left (3 a^{2} d^{2} + 6 a b c d - b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{4 \sqrt{b} d^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**(3/2)*sqrt(c)*atanh(sqrt(c)*sqrt(a + b*x)/(sqrt(a)*sqrt(c + d*x))) + (a +
b*x)**(3/2)*sqrt(c + d*x)/2 + sqrt(a + b*x)*sqrt(c + d*x)*(3*a*d + b*c)/(4*d) +
(3*a**2*d**2 + 6*a*b*c*d - b**2*c**2)*atanh(sqrt(b)*sqrt(c + d*x)/(sqrt(d)*sqrt(
a + b*x)))/(4*sqrt(b)*d**(3/2))

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Mathematica [A]  time = 0.298467, size = 188, normalized size = 1.15 \[ -a^{3/2} \sqrt{c} \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )+a^{3/2} \sqrt{c} \log (x)+\frac{\left (3 a^2 d^2+6 a b c d-b^2 c^2\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{8 \sqrt{b} d^{3/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} (5 a d+b (c+2 d x))}{4 d} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(5*a*d + b*(c + 2*d*x)))/(4*d) + a^(3/2)*Sqrt[c]*Lo
g[x] - a^(3/2)*Sqrt[c]*Log[2*a*c + b*c*x + a*d*x + 2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*
x]*Sqrt[c + d*x]] + ((-(b^2*c^2) + 6*a*b*c*d + 3*a^2*d^2)*Log[b*c + a*d + 2*b*d*
x + 2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(8*Sqrt[b]*d^(3/2))

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Maple [B]  time = 0.017, size = 389, normalized size = 2.4 \[{\frac{1}{8\,d}\sqrt{bx+a}\sqrt{dx+c} \left ( 3\,{d}^{2}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{2}\sqrt{ac}+6\,d\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) acb\sqrt{ac}-{b}^{2}\ln \left ({\frac{1}{2} \left ( 2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ){c}^{2}\sqrt{ac}-8\,{a}^{2}c\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ) d\sqrt{bd}+4\,d\sqrt{d{x}^{2}b+adx+bcx+ac}xb\sqrt{bd}\sqrt{ac}+10\,d\sqrt{d{x}^{2}b+adx+bcx+ac}a\sqrt{bd}\sqrt{ac}+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}cb\sqrt{bd}\sqrt{ac} \right ){\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(3*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*(a*c)^(1/2)+6*d*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*c*b*(a*c)^(
1/2)-b^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))*c^2*(a*c)^(1/2)-8*a^2*c*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)*d*(b*d)^(1/2)+4*d*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x
*b*(b*d)^(1/2)*(a*c)^(1/2)+10*d*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a*(b*d)^(1/2)*(a
*c)^(1/2)+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*c*b*(b*d)^(1/2)*(a*c)^(1/2))/(b*d*x^
2+a*d*x+b*c*x+a*c)^(1/2)/d/(b*d)^(1/2)/(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,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.98067, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(8*sqrt(a*c)*sqrt(b*d)*a*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*(2*b*d*x + b*c + 5*a*d)*sqrt(b*d)*sqrt(b*x + a)*sqrt
(d*x + c) - (b^2*c^2 - 6*a*b*c*d - 3*a^2*d^2)*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)*d), 1/8*(4*sqrt(a*c)*sqrt(-
b*d)*a*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*(2*b*d*x + b*c + 5*a*d)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c) - (b^2*c^2 -
6*a*b*c*d - 3*a^2*d^2)*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)*d), -1/16*(16*sqrt(-a*c)*sqrt(b*d)*a*d*arctan
(1/2*(2*a*c + (b*c + a*d)*x)/(sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c))) - 4*(2*b*
d*x + b*c + 5*a*d)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + (b^2*c^2 - 6*a*b*c*d
- 3*a^2*d^2)*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)*sqr
t(b*d)))/(sqrt(b*d)*d), -1/8*(8*sqrt(-a*c)*sqrt(-b*d)*a*d*arctan(1/2*(2*a*c + (b
*c + a*d)*x)/(sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c))) - 2*(2*b*d*x + b*c + 5*a*
d)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c) + (b^2*c^2 - 6*a*b*c*d - 3*a^2*d^2)*ar
ctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)/(sqrt(b*x + a)*sqrt(d*x + c)*b*d)))/(s
qrt(-b*d)*d)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.261314, size = 336, normalized size = 2.05 \[ -\frac{2 \, \sqrt{b d} a^{2} c{\left | b \right |} \arctan \left (-\frac{b^{2} c + a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt{-a b c d} b}\right )}{\sqrt{-a b c d} b} + \frac{1}{4} \, \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \sqrt{b x + a}{\left (\frac{2 \,{\left (b x + a\right )}{\left | b \right |}}{b^{2}} + \frac{b^{2} c d{\left | b \right |} + 3 \, a b d^{2}{\left | b \right |}}{b^{3} d^{2}}\right )} + \frac{{\left (\sqrt{b d} b^{2} c^{2}{\left | b \right |} - 6 \, \sqrt{b d} a b c d{\left | b \right |} - 3 \, \sqrt{b d} a^{2} d^{2}{\left | b \right |}\right )}{\rm ln}\left ({\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{8 \, b^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(b*d)*a^2*c*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)
+ 1/4*sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sqrt(b*x + a)*(2*(b*x + a)*abs(b)/b^2
+ (b^2*c*d*abs(b) + 3*a*b*d^2*abs(b))/(b^3*d^2)) + 1/8*(sqrt(b*d)*b^2*c^2*abs(b
) - 6*sqrt(b*d)*a*b*c*d*abs(b) - 3*sqrt(b*d)*a^2*d^2*abs(b))*ln((sqrt(b*d)*sqrt(
b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/(b^2*d^2)