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

Optimal. Leaf size=170 \[ \frac{\left (-3 a^2 d^2-6 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{3/2} \sqrt{c}}+2 \sqrt{b} d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )-\frac{\sqrt{a+b x} (c+d x)^{3/2}}{2 x^2}-\frac{\sqrt{a+b x} \sqrt{c+d x} (3 a d+b c)}{4 a x} \]

[Out]

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

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Rubi [A]  time = 0.416805, antiderivative size = 170, 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 \[ \frac{\left (-3 a^2 d^2-6 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{3/2} \sqrt{c}}+2 \sqrt{b} d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )-\frac{\sqrt{a+b x} (c+d x)^{3/2}}{2 x^2}-\frac{\sqrt{a+b x} \sqrt{c+d x} (3 a d+b c)}{4 a x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 66.1988, size = 156, normalized size = 0.92 \[ 2 \sqrt{b} d^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )} - \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}}}{2 x^{2}} - \frac{\sqrt{a + b x} \sqrt{c + d x} \left (3 a d + b c\right )}{4 a x} - \frac{\left (3 a^{2} d^{2} + 6 a b c d - b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{4 a^{\frac{3}{2}} \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.295547, size = 222, normalized size = 1.31 \[ \frac{\log (x) \left (3 a^2 d^2+6 a b c d-b^2 c^2\right )}{8 a^{3/2} \sqrt{c}}-\frac{\left (3 a^2 d^2+6 a b c d-b^2 c^2\right ) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{8 a^{3/2} \sqrt{c}}+\sqrt{b} d^{3/2} \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )+\sqrt{a+b x} \sqrt{c+d x} \left (\frac{-5 a d-b c}{4 a x}-\frac{c}{2 x^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.021, size = 401, normalized size = 2.4 \[ -{\frac{1}{8\,a{x}^{2}}\sqrt{bx+a}\sqrt{dx+c} \left ( 3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{2}{a}^{2}{d}^{2}\sqrt{bd}+6\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{2}abcd\sqrt{bd}-\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac \right ) } \right ){x}^{2}{b}^{2}{c}^{2}\sqrt{bd}-8\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}ab{d}^{2}\sqrt{ac}+10\,\sqrt{d{x}^{2}b+adx+bcx+ac}dax\sqrt{ac}\sqrt{bd}+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}bxc\sqrt{ac}\sqrt{bd}+4\,\sqrt{d{x}^{2}b+adx+bcx+ac}ac\sqrt{ac}\sqrt{bd} \right ){\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.943529, 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^3,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.629906, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x