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

Optimal. Leaf size=209 \[ -\frac{3 \left (a^2 d^2+6 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 \sqrt{a} \sqrt{c}}-\frac{(a+b x)^{3/2} (c+d x)^{3/2}}{2 x^2}-\frac{3 \sqrt{a+b x} (c+d x)^{3/2} (a d+b c)}{4 c x}+\frac{3 d \sqrt{a+b x} \sqrt{c+d x} (a d+3 b c)}{4 c}+3 \sqrt{b} \sqrt{d} (a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.628484, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318 \[ -\frac{3 \left (a^2 d^2+6 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 \sqrt{a} \sqrt{c}}-\frac{(a+b x)^{3/2} (c+d x)^{3/2}}{2 x^2}-\frac{3 \sqrt{a+b x} (c+d x)^{3/2} (a d+b c)}{4 c x}+\frac{3 d \sqrt{a+b x} \sqrt{c+d x} (a d+3 b c)}{4 c}+3 \sqrt{b} \sqrt{d} (a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 93.8123, size = 196, normalized size = 0.94 \[ 3 \sqrt{b} \sqrt{d} \left (a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )} - \frac{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}}}{2 x^{2}} + \frac{3 d \sqrt{a + b x} \sqrt{c + d x} \left (a d + 3 b c\right )}{4 c} - \frac{3 \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (a d + b c\right )}{4 c x} - \frac{3 \left (a^{2} d^{2} + 6 a b c d + b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{4 \sqrt{a} \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.595011, size = 224, normalized size = 1.07 \[ \frac{1}{8} \left (\frac{3 \log (x) \left (a^2 d^2+6 a b c d+b^2 c^2\right )}{\sqrt{a} \sqrt{c}}-\frac{3 \left (a^2 d^2+6 a b c d+b^2 c^2\right ) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{\sqrt{a} \sqrt{c}}-\frac{2 \sqrt{a+b x} \sqrt{c+d x} (a (2 c+5 d x)+b x (5 c-4 d x))}{x^2}+12 \sqrt{b} \sqrt{d} (a d+b c) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-2*Sqrt[a + b*x]*Sqrt[c + d*x]*(b*x*(5*c - 4*d*x) + a*(2*c + 5*d*x)))/x^2 + (3
*(b^2*c^2 + 6*a*b*c*d + a^2*d^2)*Log[x])/(Sqrt[a]*Sqrt[c]) - (3*(b^2*c^2 + 6*a*b
*c*d + a^2*d^2)*Log[2*a*c + b*c*x + a*d*x + 2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt
[c + d*x]])/(Sqrt[a]*Sqrt[c]) + 12*Sqrt[b]*Sqrt[d]*(b*c + a*d)*Log[b*c + a*d + 2
*b*d*x + 2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*Sqrt[c + d*x]])/8

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Maple [B]  time = 0.023, size = 497, normalized size = 2.4 \[{\frac{1}{8\,{x}^{2}}\sqrt{bx+a}\sqrt{dx+c} \left ( 12\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}ab{d}^{2}\sqrt{ac}+12\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}{b}^{2}cd\sqrt{ac}-3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{2}{a}^{2}{d}^{2}\sqrt{bd}-18\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{2}abcd\sqrt{bd}-3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{2}{b}^{2}{c}^{2}\sqrt{bd}+8\,bd{x}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}\sqrt{ac}-10\,\sqrt{d{x}^{2}b+adx+bcx+ac}dax\sqrt{ac}\sqrt{bd}-10\,\sqrt{d{x}^{2}b+adx+bcx+ac}bxc\sqrt{ac}\sqrt{bd}-4\,\sqrt{d{x}^{2}b+adx+bcx+ac}ac\sqrt{ac}\sqrt{bd} \right ){\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(12*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^2*a*b*d^2*(a*c)^(1/2)+12*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^2*b^2*c
*d*(a*c)^(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^2*a^2*d^2*(b*d)^(1/2)-18*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^2*a*b*c*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^2*b^2*c^2*(b*d)^(1/2)+8*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)-4*(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)/x^2/(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)*(d*x + c)^(3/2)/x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.28303, 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^3,x, algorithm="fricas")

[Out]

[1/16*(12*sqrt(a*c)*(b*c + a*d)*sqrt(b*d)*x^2*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^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2*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*(4*
b*d*x^2 - 2*a*c - 5*(b*c + a*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(
a*c)*x^2), 1/16*(24*sqrt(a*c)*(b*c + a*d)*sqrt(-b*d)*x^2*arctan(1/2*(2*b*d*x + 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)*x^2*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*(4*b*d*x^2 - 2*a*c - 5*(b*c + a*d)*x)*sqrt(a*c)*sqrt(b*x +
a)*sqrt(d*x + c))/(sqrt(a*c)*x^2), 1/8*(6*sqrt(-a*c)*(b*c + a*d)*sqrt(b*d)*x^2*
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^2*c^2 + 6*a*
b*c*d + a^2*d^2)*x^2*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*(4*b*d*x^2 - 2*a*c - 5*(b*c + a*d)*x)*sqrt(-a*c)*sqrt(
b*x + a)*sqrt(d*x + c))/(sqrt(-a*c)*x^2), 1/8*(12*sqrt(-a*c)*(b*c + a*d)*sqrt(-b
*d)*x^2*arctan(1/2*(2*b*d*x + 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)*x^2*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sq
rt(-a*c)/(sqrt(b*x + a)*sqrt(d*x + c)*a*c)) + 2*(4*b*d*x^2 - 2*a*c - 5*(b*c + a*
d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(-a*c)*x^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.62525, 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^3,x, algorithm="giac")

[Out]

sage0*x