3.601 \(\int \frac{(a+b x)^{3/2} (c+d x)^{3/2}}{x^4} \, dx\)

Optimal. Leaf size=222 \[ \frac{(a d+b c) \left (a^2 d^2-10 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{3/2} c^{3/2}}+2 b^{3/2} d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (\frac{b^2 c}{a}-\frac{a d^2}{c}+8 b d\right )}{8 x}-\frac{(a+b x)^{3/2} (c+d x)^{3/2}}{3 x^3}-\frac{\sqrt{a+b x} (c+d x)^{3/2} (a d+b c)}{4 c x^2} \]

[Out]

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

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Rubi [A]  time = 0.654958, antiderivative size = 222, 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 \[ \frac{(a d+b c) \left (a^2 d^2-10 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{3/2} c^{3/2}}+2 b^{3/2} d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (\frac{b^2 c}{a}-\frac{a d^2}{c}+8 b d\right )}{8 x}-\frac{(a+b x)^{3/2} (c+d x)^{3/2}}{3 x^3}-\frac{\sqrt{a+b x} (c+d x)^{3/2} (a d+b c)}{4 c x^2} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^(3/2)*(c + d*x)^(3/2))/x^4,x]

[Out]

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

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Rubi in Sympy [A]  time = 94.41, size = 207, normalized size = 0.93 \[ 2 b^{\frac{3}{2}} d^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )} - \frac{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}}}{3 x^{3}} - \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (a d + b c\right )}{4 c x^{2}} + \frac{\sqrt{a + b x} \sqrt{c + d x} \left (a^{2} d^{2} - 8 a b c d - b^{2} c^{2}\right )}{8 a c x} + \frac{\left (a d + b c\right ) \left (a^{2} d^{2} - 10 a b c d + b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{8 a^{\frac{3}{2}} c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(3/2)*(d*x+c)**(3/2)/x**4,x)

[Out]

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

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Mathematica [A]  time = 0.550424, size = 278, normalized size = 1.25 \[ \sqrt{a+b x} \sqrt{c+d x} \left (\frac{-3 a^2 d^2-38 a b c d-3 b^2 c^2}{24 a c x}-\frac{7 (a d+b c)}{12 x^2}-\frac{a c}{3 x^3}\right )-\frac{\log (x) \left (a^3 d^3-9 a^2 b c d^2-9 a b^2 c^2 d+b^3 c^3\right )}{16 a^{3/2} c^{3/2}}+\frac{\left (a^3 d^3-9 a^2 b c d^2-9 a b^2 c^2 d+b^3 c^3\right ) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{16 a^{3/2} c^{3/2}}+b^{3/2} d^{3/2} \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^(3/2)*(c + d*x)^(3/2))/x^4,x]

[Out]

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

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Maple [B]  time = 0.023, size = 605, normalized size = 2.7 \[{\frac{1}{48\,ac{x}^{3}}\sqrt{bx+a}\sqrt{dx+c} \left ( 3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{3}{a}^{3}{d}^{3}\sqrt{bd}-27\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{3}{a}^{2}bc{d}^{2}\sqrt{bd}-27\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{3}a{b}^{2}{c}^{2}d\sqrt{bd}+3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{3}{b}^{3}{c}^{3}\sqrt{bd}+48\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{3}a{b}^{2}c{d}^{2}\sqrt{ac}-6\,\sqrt{d{x}^{2}b+adx+bcx+ac}{d}^{2}\sqrt{bd}{a}^{2}\sqrt{ac}{x}^{2}-76\,\sqrt{d{x}^{2}b+adx+bcx+ac}db\sqrt{bd}a\sqrt{ac}{x}^{2}c-6\,{c}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}{b}^{2}\sqrt{bd}\sqrt{ac}{x}^{2}-28\,\sqrt{d{x}^{2}b+adx+bcx+ac}d\sqrt{bd}{a}^{2}\sqrt{ac}xc-28\,{c}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}b\sqrt{bd}a\sqrt{ac}x-16\,{c}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}{a}^{2}\sqrt{ac} \right ){\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(3/2)*(d*x+c)^(3/2)/x^4,x)

[Out]

1/48*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a/c*(3*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^3*a^3*d^3*(b*d)^(1/2)-27*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^3*a^2*b*c*d^2*(b*d)^(1/2)-2
7*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^3*a*
b^2*c^2*d*(b*d)^(1/2)+3*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^3*b^3*c^3*(b*d)^(1/2)+48*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))*x^3*a*b^2*c*d^2*(a*c)^(1/2)-6*(b*
d*x^2+a*d*x+b*c*x+a*c)^(1/2)*d^2*(b*d)^(1/2)*a^2*(a*c)^(1/2)*x^2-76*(b*d*x^2+a*d
*x+b*c*x+a*c)^(1/2)*d*b*(b*d)^(1/2)*a*(a*c)^(1/2)*x^2*c-6*c^2*(b*d*x^2+a*d*x+b*c
*x+a*c)^(1/2)*b^2*(b*d)^(1/2)*(a*c)^(1/2)*x^2-28*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)
*d*(b*d)^(1/2)*a^2*(a*c)^(1/2)*x*c-28*c^2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*b*(b*d
)^(1/2)*a*(a*c)^(1/2)*x-16*c^2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*a^2*(
a*c)^(1/2))/(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)/x^3/(a*c)^(1/2)/(b*d)^(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^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.44244, 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^4,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}}}{x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(3/2)*(d*x+c)**(3/2)/x**4,x)

[Out]

Integral((a + b*x)**(3/2)*(c + d*x)**(3/2)/x**4, x)

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GIAC/XCAS [A]  time = 0.655668, 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^4,x, algorithm="giac")

[Out]

sage0*x