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

Optimal. Leaf size=191 \[ \frac{3 \left (a^2 d^2+6 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 \sqrt{b} \sqrt{d}}-\frac{(a+b x)^{3/2} (c+d x)^{3/2}}{x}+\frac{3}{2} b \sqrt{a+b x} (c+d x)^{3/2}+\frac{3}{4} \sqrt{a+b x} \sqrt{c+d x} (3 a d+b c)-3 \sqrt{a} \sqrt{c} (a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.622082, antiderivative size = 191, 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 \[ \frac{3 \left (a^2 d^2+6 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 \sqrt{b} \sqrt{d}}-\frac{(a+b x)^{3/2} (c+d x)^{3/2}}{x}+\frac{3}{2} b \sqrt{a+b x} (c+d x)^{3/2}+\frac{3}{4} \sqrt{a+b x} \sqrt{c+d x} (3 a d+b c)-3 \sqrt{a} \sqrt{c} (a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 83.3026, size = 182, normalized size = 0.95 \[ - 3 \sqrt{a} \sqrt{c} \left (a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )} + \frac{3 b \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}}}{2} + \sqrt{a + b x} \sqrt{c + d x} \left (\frac{9 a d}{4} + \frac{3 b c}{4}\right ) - \frac{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}}}{x} + \frac{3 \left (a^{2} d^{2} + 6 a b c d + b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{4 \sqrt{b} \sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*sqrt(a)*sqrt(c)*(a*d + b*c)*atanh(sqrt(c)*sqrt(a + b*x)/(sqrt(a)*sqrt(c + d*x
))) + 3*b*sqrt(a + b*x)*(c + d*x)**(3/2)/2 + sqrt(a + b*x)*sqrt(c + d*x)*(9*a*d/
4 + 3*b*c/4) - (a + b*x)**(3/2)*(c + d*x)**(3/2)/x + 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)*sqrt(
d))

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Mathematica [A]  time = 0.128776, size = 212, normalized size = 1.11 \[ \frac{3 \left (a^2 d^2+6 a b c d+b^2 c^2\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{8 \sqrt{b} \sqrt{d}}+\sqrt{a+b x} \sqrt{c+d x} \left (\frac{5}{4} (a d+b c)-\frac{a c}{x}+\frac{b d x}{2}\right )+\frac{3}{2} \sqrt{a} \sqrt{c} \log (x) (a d+b c)-\frac{3}{2} \sqrt{a} \sqrt{c} (a d+b c) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right ) \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[a + b*x]*Sqrt[c + d*x]*((5*(b*c + a*d))/4 - (a*c)/x + (b*d*x)/2) + (3*Sqrt[
a]*Sqrt[c]*(b*c + a*d)*Log[x])/2 - (3*Sqrt[a]*Sqrt[c]*(b*c + a*d)*Log[2*a*c + b*
c*x + a*d*x + 2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]])/2 + (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]*S
qrt[c + d*x]])/(8*Sqrt[b]*Sqrt[d])

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Maple [B]  time = 0.023, size = 489, normalized size = 2.6 \[{\frac{1}{8\,x}\sqrt{bx+a}\sqrt{dx+c} \left ( 3\,{d}^{2}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{2}\sqrt{ac}x+18\,bd\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) ac\sqrt{ac}x+3\,{b}^{2}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){c}^{2}\sqrt{ac}x-12\,{a}^{2}c\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ) d\sqrt{bd}x-12\,a{c}^{2}\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ) b\sqrt{bd}x+4\,bd{x}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}\sqrt{ac}+10\,\sqrt{d{x}^{2}b+adx+bcx+ac}dax\sqrt{ac}\sqrt{bd}+10\,\sqrt{d{x}^{2}b+adx+bcx+ac}bxc\sqrt{ac}\sqrt{bd}-8\,\sqrt{d{x}^{2}b+adx+bcx+ac}ac\sqrt{ac}\sqrt{bd} \right ){\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(3/2)*(d*x+c)^(3/2)/x^2,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)*x+18*b*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*(a*c
)^(1/2)*x+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)*x-12*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)*x-12*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)*x+4*b*d*x^
2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)+10*(b*d*x^2+a*d*x+b*c*
x+a*c)^(1/2)*d*a*x*(a*c)^(1/2)*(b*d)^(1/2)+10*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*b*
x*c*(a*c)^(1/2)*(b*d)^(1/2)-8*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a*c*(a*c)^(1/2)*(b
*d)^(1/2))/(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)/(b*d)^(1/2)/(a*c)^(1/2)/x

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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^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.65911, 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)*(d*x + c)^(3/2)/x^2,x, algorithm="fricas")

[Out]

[1/16*(12*sqrt(a*c)*(b*c + a*d)*sqrt(b*d)*x*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) + 3*(b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x*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)) + 4*(2*b*d*x
^2 - 4*a*c + 5*(b*c + a*d)*x)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(b*d)*
x), 1/8*(6*sqrt(a*c)*(b*c + a*d)*sqrt(-b*d)*x*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) + 3*(b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x*arcta
n(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)/(sqrt(b*x + a)*sqrt(d*x + c)*b*d)) + 2*(2
*b*d*x^2 - 4*a*c + 5*(b*c + a*d)*x)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c))/(sqr
t(-b*d)*x), -1/16*(24*sqrt(-a*c)*(b*c + a*d)*sqrt(b*d)*x*arctan(1/2*(2*a*c + (b*
c + a*d)*x)/(sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c))) - 3*(b^2*c^2 + 6*a*b*c*d +
a^2*d^2)*x*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)) - 4*(2*b*d*x^2 - 4*a*c + 5*(b*c + a*d)*x)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*
x + c))/(sqrt(b*d)*x), -1/8*(12*sqrt(-a*c)*(b*c + a*d)*sqrt(-b*d)*x*arctan(1/2*(
2*a*c + (b*c + a*d)*x)/(sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c))) - 3*(b^2*c^2 +
6*a*b*c*d + a^2*d^2)*x*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)/(sqrt(b*x + a
)*sqrt(d*x + c)*b*d)) - 2*(2*b*d*x^2 - 4*a*c + 5*(b*c + a*d)*x)*sqrt(-b*d)*sqrt(
b*x + a)*sqrt(d*x + c))/(sqrt(-b*d)*x)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.601075, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x