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

Optimal. Leaf size=144 \[ -\frac{\sqrt{a+b x} (c+d x)^{3/2}}{x}+2 d \sqrt{a+b x} \sqrt{c+d x}-\frac{\sqrt{c} (3 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{\sqrt{a}}+\frac{\sqrt{d} (a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{\sqrt{b}} \]

[Out]

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

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Rubi [A]  time = 0.41615, antiderivative size = 144, 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 \[ -\frac{\sqrt{a+b x} (c+d x)^{3/2}}{x}+2 d \sqrt{a+b x} \sqrt{c+d x}-\frac{\sqrt{c} (3 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{\sqrt{a}}+\frac{\sqrt{d} (a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 47.7169, size = 133, normalized size = 0.92 \[ 2 d \sqrt{a + b x} \sqrt{c + d x} - \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}}}{x} + \frac{\sqrt{d} \left (a d + 3 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{\sqrt{b}} - \frac{\sqrt{c} \left (3 a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{\sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.357311, size = 180, normalized size = 1.25 \[ \frac{1}{2} \left (2 \sqrt{a+b x} \sqrt{c+d x} \left (d-\frac{c}{x}\right )+\frac{\sqrt{c} \log (x) (3 a d+b c)}{\sqrt{a}}-\frac{\sqrt{c} (3 a d+b c) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{\sqrt{a}}+\frac{\sqrt{d} (a d+3 b c) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{\sqrt{b}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*(d - c/x)*Sqrt[a + b*x]*Sqrt[c + d*x] + (Sqrt[c]*(b*c + 3*a*d)*Log[x])/Sqrt[a
] - (Sqrt[c]*(b*c + 3*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]])/Sqrt[a] + (Sqrt[d]*(3*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]])/Sqrt[b])/2

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Maple [B]  time = 0.021, size = 347, normalized size = 2.4 \[{\frac{1}{2\,x}\sqrt{bx+a}\sqrt{dx+c} \left ( \ln \left ({\frac{1}{2} \left ( 2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ) xa{d}^{2}\sqrt{ac}+3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) xbcd\sqrt{ac}-3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ) xacd\sqrt{bd}-\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac \right ) } \right ) xb{c}^{2}\sqrt{bd}+2\,xd\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}-2\,c\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd} \right ){\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.919185, 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^2,x, algorithm="fricas")

[Out]

[1/4*((3*b*c + a*d)*x*sqrt(d/b)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^
2 + 4*(2*b^2*d*x + b^2*c + a*b*d)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(d/b) + 8*(b^2
*c*d + a*b*d^2)*x) + (b*c + 3*a*d)*x*sqrt(c/a)*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b
*c*d + a^2*d^2)*x^2 - 4*(2*a^2*c + (a*b*c + a^2*d)*x)*sqrt(b*x + a)*sqrt(d*x + c
)*sqrt(c/a) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) + 4*sqrt(b*x + a)*sqrt(d*x + c)*(d*x
- c))/x, 1/4*(2*(3*b*c + a*d)*x*sqrt(-d/b)*arctan(1/2*(2*b*d*x + b*c + a*d)/(sq
rt(b*x + a)*sqrt(d*x + c)*b*sqrt(-d/b))) + (b*c + 3*a*d)*x*sqrt(c/a)*log((8*a^2*
c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 - 4*(2*a^2*c + (a*b*c + a^2*d)*x)*sqrt
(b*x + a)*sqrt(d*x + c)*sqrt(c/a) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) + 4*sqrt(b*x +
a)*sqrt(d*x + c)*(d*x - c))/x, -1/4*(2*(b*c + 3*a*d)*x*sqrt(-c/a)*arctan(1/2*(2
*a*c + (b*c + a*d)*x)/(sqrt(b*x + a)*sqrt(d*x + c)*a*sqrt(-c/a))) - (3*b*c + a*d
)*x*sqrt(d/b)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b^2*d*x +
b^2*c + a*b*d)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(d/b) + 8*(b^2*c*d + a*b*d^2)*x)
- 4*sqrt(b*x + a)*sqrt(d*x + c)*(d*x - c))/x, -1/2*((b*c + 3*a*d)*x*sqrt(-c/a)*
arctan(1/2*(2*a*c + (b*c + a*d)*x)/(sqrt(b*x + a)*sqrt(d*x + c)*a*sqrt(-c/a))) -
(3*b*c + a*d)*x*sqrt(-d/b)*arctan(1/2*(2*b*d*x + b*c + a*d)/(sqrt(b*x + a)*sqrt
(d*x + c)*b*sqrt(-d/b))) - 2*sqrt(b*x + a)*sqrt(d*x + c)*(d*x - c))/x]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.601154, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x