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

Optimal. Leaf size=116 \[ \frac{\sqrt{d} (3 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{3/2}}-\frac{2 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{\sqrt{a}}+\frac{d \sqrt{a+b x} \sqrt{c+d x}}{b} \]

[Out]

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

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Rubi [A]  time = 0.294748, antiderivative size = 116, 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 \[ \frac{\sqrt{d} (3 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{3/2}}-\frac{2 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{\sqrt{a}}+\frac{d \sqrt{a+b x} \sqrt{c+d x}}{b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 27.2542, size = 107, normalized size = 0.92 \[ \frac{d \sqrt{a + b x} \sqrt{c + d x}}{b} - \frac{\sqrt{d} \left (a d - 3 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{b^{\frac{3}{2}}} - \frac{2 c^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{\sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.213642, size = 159, normalized size = 1.37 \[ \frac{\sqrt{d} (3 b c-a d) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{2 b^{3/2}}-\frac{c^{3/2} \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{\sqrt{a}}+\frac{d \sqrt{a+b x} \sqrt{c+d x}}{b}+\frac{c^{3/2} \log (x)}{\sqrt{a}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.029, size = 219, normalized size = 1.9 \[ -{\frac{1}{2\,b}\sqrt{bx+a}\sqrt{dx+c} \left ( \ln \left ({\frac{1}{2} \left ( 2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ) a{d}^{2}\sqrt{ac}-3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) bcd\sqrt{ac}+2\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ) b{c}^{2}\sqrt{bd}-2\,d\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(d*x+c)^(1/2)*(b*x+a)^(1/2)*(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))*a*d^2*(a*c)^(1/2)-3*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))*b*c*d*(a*c)^(1/2)+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)*b*c^2*(b*d)^(1/2)-2*d*(a*c)^(1
/2)*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2))/((b*x+a)*(d*x+c))^(1/2)/(b*d)^(1/2)/(a*
c)^(1/2)/b

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.874266, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*(2*b*c*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) - (3*b*c - a*d)*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)/b, 1/2*(b*c*s
qrt(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) + (3*b*c - a*d)*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)/b, -1/4*(4*b*c*
sqrt(-c/a)*arctan(1/2*(2*a*c + (b*c + a*d)*x)/(sqrt(b*x + a)*sqrt(d*x + c)*a*sqr
t(-c/a))) + (3*b*c - a*d)*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)/b, -1/2*(2*b*c*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)*sqrt(-d/b)*arctan(1/2*(2*b*d*x + b*c + a*d)/(sqrt(b*x + a)*sq
rt(d*x + c)*b*sqrt(-d/b))) - 2*sqrt(b*x + a)*sqrt(d*x + c)*d)/b]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.268284, size = 259, normalized size = 2.23 \[ -\frac{2 \, \sqrt{b d} c^{2}{\left | b \right |} \arctan \left (-\frac{b^{2} c + a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt{-a b c d} b}\right )}{\sqrt{-a b c d} b} + \frac{\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \sqrt{b x + a} d{\left | b \right |}}{b^{3}} - \frac{{\left (3 \, \sqrt{b d} b c{\left | b \right |} - \sqrt{b d} a d{\left | b \right |}\right )}{\rm ln}\left ({\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{2 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(b*d)*c^2*abs(b)*arctan(-1/2*(b^2*c + a*b*d - (sqrt(b*d)*sqrt(b*x + a) -
sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/(sqrt(-a*b*c*d)*b))/(sqrt(-a*b*c*d)*b) +
sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sqrt(b*x + a)*d*abs(b)/b^3 - 1/2*(3*sqrt(b*
d)*b*c*abs(b) - sqrt(b*d)*a*d*abs(b))*ln((sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c +
(b*x + a)*b*d - a*b*d))^2)/b^3