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

Optimal. Leaf size=121 \[ \frac{2 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{\sqrt{d}}-\frac{\sqrt{a} (3 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{c^{3/2}}-\frac{a \sqrt{a+b x} \sqrt{c+d x}}{c x} \]

[Out]

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

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Rubi [A]  time = 0.285871, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{2 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{\sqrt{d}}-\frac{\sqrt{a} (3 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{c^{3/2}}-\frac{a \sqrt{a+b x} \sqrt{c+d x}}{c x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 27.7394, size = 109, normalized size = 0.9 \[ \frac{\sqrt{a} \left (a d - 3 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{c^{\frac{3}{2}}} - \frac{a \sqrt{a + b x} \sqrt{c + d x}}{c x} + \frac{2 b^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{\sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.228782, size = 172, normalized size = 1.42 \[ \frac{b^{3/2} \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{\sqrt{d}}-\frac{\sqrt{a} \log (x) (a d-3 b c)}{2 c^{3/2}}+\frac{\sqrt{a} (a d-3 b c) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{2 c^{3/2}}-\frac{a \sqrt{a+b x} \sqrt{c+d x}}{c x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.03, size = 223, normalized size = 1.8 \[{\frac{1}{2\,cx}\sqrt{bx+a}\sqrt{dx+c} \left ( 2\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) x{b}^{2}c\sqrt{ac}+\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ) x{a}^{2}d\sqrt{bd}-3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ) xabc\sqrt{bd}-2\,a\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.724796, 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)/(sqrt(d*x + c)*x^2),x, algorithm="fricas")

[Out]

[1/4*(2*b*c*x*sqrt(b/d)*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^2*x + b*c*d + a*d^2)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(b/d) + 8*(b^2*c*d + a
*b*d^2)*x) - (3*b*c - a*d)*x*sqrt(a/c)*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^2 + (b*c^2 + a*c*d)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(a
/c) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) - 4*sqrt(b*x + a)*sqrt(d*x + c)*a)/(c*x), 1/
4*(4*b*c*x*sqrt(-b/d)*arctan(1/2*(2*b*d*x + b*c + a*d)/(sqrt(b*x + a)*sqrt(d*x +
c)*d*sqrt(-b/d))) - (3*b*c - a*d)*x*sqrt(a/c)*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^2 + (b*c^2 + a*c*d)*x)*sqrt(b*x + a)*sqrt(d*x + c
)*sqrt(a/c) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) - 4*sqrt(b*x + a)*sqrt(d*x + c)*a)/(
c*x), 1/2*(b*c*x*sqrt(b/d)*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^2*x + b*c*d + a*d^2)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(b/d) + 8*(b^2*c*d
+ a*b*d^2)*x) - (3*b*c - a*d)*x*sqrt(-a/c)*arctan(1/2*(2*a*c + (b*c + a*d)*x)/(s
qrt(b*x + a)*sqrt(d*x + c)*c*sqrt(-a/c))) - 2*sqrt(b*x + a)*sqrt(d*x + c)*a)/(c*
x), 1/2*(2*b*c*x*sqrt(-b/d)*arctan(1/2*(2*b*d*x + b*c + a*d)/(sqrt(b*x + a)*sqrt
(d*x + c)*d*sqrt(-b/d))) - (3*b*c - a*d)*x*sqrt(-a/c)*arctan(1/2*(2*a*c + (b*c +
a*d)*x)/(sqrt(b*x + a)*sqrt(d*x + c)*c*sqrt(-a/c))) - 2*sqrt(b*x + a)*sqrt(d*x
+ c)*a)/(c*x)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.548584, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x