3.1474 \(\int \frac{(c+d x)^{3/2}}{\sqrt{a+b x}} \, dx\)

Optimal. Leaf size=113 \[ \frac{3 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 b^{5/2} \sqrt{d}}+\frac{3 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)}{4 b^2}+\frac{\sqrt{a+b x} (c+d x)^{3/2}}{2 b} \]

[Out]

(3*(b*c - a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(4*b^2) + (Sqrt[a + b*x]*(c + d*x)^(
3/2))/(2*b) + (3*(b*c - a*d)^2*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c +
d*x])])/(4*b^(5/2)*Sqrt[d])

_______________________________________________________________________________________

Rubi [A]  time = 0.117152, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{3 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 b^{5/2} \sqrt{d}}+\frac{3 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)}{4 b^2}+\frac{\sqrt{a+b x} (c+d x)^{3/2}}{2 b} \]

Antiderivative was successfully verified.

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

[Out]

(3*(b*c - a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(4*b^2) + (Sqrt[a + b*x]*(c + d*x)^(
3/2))/(2*b) + (3*(b*c - a*d)^2*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c +
d*x])])/(4*b^(5/2)*Sqrt[d])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 15.9804, size = 100, normalized size = 0.88 \[ \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}}}{2 b} - \frac{3 \sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )}{4 b^{2}} + \frac{3 \left (a d - b c\right )^{2} \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{4 b^{\frac{5}{2}} \sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(a + b*x)*(c + d*x)**(3/2)/(2*b) - 3*sqrt(a + b*x)*sqrt(c + d*x)*(a*d - b*c)
/(4*b**2) + 3*(a*d - b*c)**2*atanh(sqrt(d)*sqrt(a + b*x)/(sqrt(b)*sqrt(c + d*x))
)/(4*b**(5/2)*sqrt(d))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0688043, size = 107, normalized size = 0.95 \[ \frac{3 (b c-a d)^2 \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{8 b^{5/2} \sqrt{d}}+\frac{\sqrt{a+b x} \sqrt{c+d x} (-3 a d+5 b c+2 b d x)}{4 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.009, size = 308, normalized size = 2.7 \[{\frac{1}{2\,b}\sqrt{bx+a} \left ( dx+c \right ) ^{{\frac{3}{2}}}}-{\frac{3\,ad}{4\,{b}^{2}}\sqrt{bx+a}\sqrt{dx+c}}+{\frac{3\,c}{4\,b}\sqrt{bx+a}\sqrt{dx+c}}+{\frac{3\,{a}^{2}{d}^{2}}{8\,{b}^{2}}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\ln \left ({1 \left ({\frac{ad}{2}}+{\frac{bc}{2}}+bdx \right ){\frac{1}{\sqrt{bd}}}}+\sqrt{d{x}^{2}b+ \left ( ad+bc \right ) x+ac} \right ){\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{dx+c}}}{\frac{1}{\sqrt{bd}}}}-{\frac{3\,adc}{4\,b}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\ln \left ({1 \left ({\frac{ad}{2}}+{\frac{bc}{2}}+bdx \right ){\frac{1}{\sqrt{bd}}}}+\sqrt{d{x}^{2}b+ \left ( ad+bc \right ) x+ac} \right ){\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{dx+c}}}{\frac{1}{\sqrt{bd}}}}+{\frac{3\,{c}^{2}}{8}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\ln \left ({1 \left ({\frac{ad}{2}}+{\frac{bc}{2}}+bdx \right ){\frac{1}{\sqrt{bd}}}}+\sqrt{d{x}^{2}b+ \left ( ad+bc \right ) x+ac} \right ){\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{dx+c}}}{\frac{1}{\sqrt{bd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(d*x+c)^(3/2)*(b*x+a)^(1/2)/b-3/4/b^2*(d*x+c)^(1/2)*(b*x+a)^(1/2)*a*d+3/4/b*
(d*x+c)^(1/2)*(b*x+a)^(1/2)*c+3/8/b^2*((b*x+a)*(d*x+c))^(1/2)/(d*x+c)^(1/2)/(b*x
+a)^(1/2)*ln((1/2*a*d+1/2*b*c+b*d*x)/(b*d)^(1/2)+(d*x^2*b+(a*d+b*c)*x+a*c)^(1/2)
)/(b*d)^(1/2)*a^2*d^2-3/4/b*((b*x+a)*(d*x+c))^(1/2)/(d*x+c)^(1/2)/(b*x+a)^(1/2)*
ln((1/2*a*d+1/2*b*c+b*d*x)/(b*d)^(1/2)+(d*x^2*b+(a*d+b*c)*x+a*c)^(1/2))/(b*d)^(1
/2)*a*d*c+3/8*((b*x+a)*(d*x+c))^(1/2)/(d*x+c)^(1/2)/(b*x+a)^(1/2)*ln((1/2*a*d+1/
2*b*c+b*d*x)/(b*d)^(1/2)+(d*x^2*b+(a*d+b*c)*x+a*c)^(1/2))/(b*d)^(1/2)*c^2

_______________________________________________________________________________________

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, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.249375, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (2 \, b d x + 5 \, b c - 3 \, a d\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} + 3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right )}{16 \, \sqrt{b d} b^{2}}, \frac{2 \,{\left (2 \, b d x + 5 \, b c - 3 \, a d\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c} + 3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right )}{8 \, \sqrt{-b d} b^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.254618, size = 327, normalized size = 2.89 \[ -\frac{\frac{48 \,{\left (\frac{{\left (b^{2} c - a b d\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d}} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \sqrt{b x + a}\right )} c{\left | b \right |}}{b^{2}} - \frac{{\left (\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \sqrt{b x + a}{\left (\frac{2 \,{\left (b x + a\right )}}{b^{4} d^{2}} + \frac{b c d - 5 \, a d^{2}}{b^{4} d^{4}}\right )} + \frac{{\left (b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2}\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} b^{3} d^{3}}\right )} d{\left | b \right |}}{b^{3}}}{48 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/48*(48*((b^2*c - a*b*d)*ln(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x +
a)*b*d - a*b*d)))/sqrt(b*d) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sqrt(b*x + a)
)*c*abs(b)/b^2 - (sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sqrt(b*x + a)*(2*(b*x + a)
/(b^4*d^2) + (b*c*d - 5*a*d^2)/(b^4*d^4)) + (b^2*c^2 + 2*a*b*c*d - 3*a^2*d^2)*ln
(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)
*b^3*d^3))*d*abs(b)/b^3)/b