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

Optimal. Leaf size=165 \[ \frac{\left (-a^2 d^2+6 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 b^{3/2} \sqrt{d}}-2 \sqrt{a} c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )+\frac{1}{2} \sqrt{a+b x} (c+d x)^{3/2}+\frac{\sqrt{a+b x} \sqrt{c+d x} (a d+3 b c)}{4 b} \]

[Out]

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

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Rubi [A]  time = 0.461458, antiderivative size = 165, 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 \[ \frac{\left (-a^2 d^2+6 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 b^{3/2} \sqrt{d}}-2 \sqrt{a} c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )+\frac{1}{2} \sqrt{a+b x} (c+d x)^{3/2}+\frac{\sqrt{a+b x} \sqrt{c+d x} (a d+3 b c)}{4 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 40.1357, size = 151, normalized size = 0.92 \[ - 2 \sqrt{a} c^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )} + \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}}}{2} + \frac{\sqrt{a + b x} \sqrt{c + d x} \left (a d + 3 b c\right )}{4 b} - \frac{\left (a^{2} d^{2} - 6 a b c d - 3 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{4 b^{\frac{3}{2}} \sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.312552, size = 188, normalized size = 1.14 \[ \frac{\left (-a^2 d^2+6 a b c d+3 b^2 c^2\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{8 b^{3/2} \sqrt{d}}-\sqrt{a} c^{3/2} \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )+\frac{\sqrt{a+b x} \sqrt{c+d x} (a d+5 b c+2 b d x)}{4 b}+\sqrt{a} c^{3/2} \log (x) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.02, size = 388, normalized size = 2.4 \[ -{\frac{1}{8\,b}\sqrt{bx+a}\sqrt{dx+c} \left ({d}^{2}\ln \left ({\frac{1}{2} \left ( 2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ){a}^{2}\sqrt{ac}-6\,d\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) acb\sqrt{ac}-3\,{b}^{2}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){c}^{2}\sqrt{ac}+8\,a{c}^{2}\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ) b\sqrt{bd}-4\,d\sqrt{d{x}^{2}b+adx+bcx+ac}xb\sqrt{bd}\sqrt{ac}-2\,d\sqrt{d{x}^{2}b+adx+bcx+ac}a\sqrt{bd}\sqrt{ac}-10\,\sqrt{d{x}^{2}b+adx+bcx+ac}cb\sqrt{bd}\sqrt{ac} \right ){\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.0583, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(8*sqrt(a*c)*sqrt(b*d)*b*c*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 + 5*b*c + a*d)*sqrt(b*d)*sqrt(b*x + a)*sqrt
(d*x + c) - (3*b^2*c^2 + 6*a*b*c*d - 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)*b), 1/8*(4*sqrt(a*c)*sqrt(
-b*d)*b*c*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 + 5*b*c + a*d)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c) + (3*b^2*c^2
+ 6*a*b*c*d - 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)*b), -1/16*(16*sqrt(-a*c)*sqrt(b*d)*b*c*arcta
n(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 + 5*b*c + a*d)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + (3*b^2*c^2 + 6*a*b*c
*d - 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)*s
qrt(b*d)))/(sqrt(b*d)*b), -1/8*(8*sqrt(-a*c)*sqrt(-b*d)*b*c*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 + 5*b*c +
a*d)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c) - (3*b^2*c^2 + 6*a*b*c*d - 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)*b)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.263581, size = 346, normalized size = 2.1 \[ -\frac{2 \, \sqrt{b d} a c^{2}{\left | b \right |} \arctan \left (-\frac{b^{2} c + a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt{-a b c d} b}\right )}{\sqrt{-a b c d} b} + \frac{1}{4} \, \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \sqrt{b x + a}{\left (\frac{2 \,{\left (b x + a\right )} d{\left | b \right |}}{b^{3}} + \frac{5 \, b^{4} c d^{2}{\left | b \right |} - a b^{3} d^{3}{\left | b \right |}}{b^{6} d^{2}}\right )} - \frac{{\left (3 \, \sqrt{b d} b^{2} c^{2}{\left | b \right |} + 6 \, \sqrt{b d} a b c d{\left | b \right |} - \sqrt{b d} a^{2} d^{2}{\left | b \right |}\right )}{\rm ln}\left ({\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{8 \, b^{3} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(b*d)*a*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)
+ 1/4*sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sqrt(b*x + a)*(2*(b*x + a)*d*abs(b)/b
^3 + (5*b^4*c*d^2*abs(b) - a*b^3*d^3*abs(b))/(b^6*d^2)) - 1/8*(3*sqrt(b*d)*b^2*c
^2*abs(b) + 6*sqrt(b*d)*a*b*c*d*abs(b) - 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^3*d)