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

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

[Out]

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

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Rubi [A]  time = 0.131343, 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 \[ \frac{3 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 \sqrt{b} d^{5/2}}-\frac{3 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)}{4 d^2}+\frac{(a+b x)^{3/2} \sqrt{c+d x}}{2 d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.8179, size = 100, normalized size = 0.88 \[ \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x}}{2 d} + \frac{3 \sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )}{4 d^{2}} + \frac{3 \left (a d - b c\right )^{2} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{4 \sqrt{b} d^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0748191, size = 107, normalized size = 0.95 \[ \frac{3 (b c-a d)^2 \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{8 \sqrt{b} d^{5/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} (5 a d-3 b c+2 b d x)}{4 d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.007, size = 308, normalized size = 2.7 \[{\frac{1}{2\,d} \left ( bx+a \right ) ^{{\frac{3}{2}}}\sqrt{dx+c}}+{\frac{3\,a}{4\,d}\sqrt{bx+a}\sqrt{dx+c}}-{\frac{3\,bc}{4\,{d}^{2}}\sqrt{bx+a}\sqrt{dx+c}}+{\frac{3\,{a}^{2}}{8}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\ln \left ({1 \left ({\frac{ad}{2}}+{\frac{bc}{2}}+bdx \right ){\frac{1}{\sqrt{bd}}}}+\sqrt{d{x}^{2}b+ \left ( ad+bc \right ) x+ac} \right ){\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{dx+c}}}{\frac{1}{\sqrt{bd}}}}-{\frac{3\,abc}{4\,d}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\ln \left ({1 \left ({\frac{ad}{2}}+{\frac{bc}{2}}+bdx \right ){\frac{1}{\sqrt{bd}}}}+\sqrt{d{x}^{2}b+ \left ( ad+bc \right ) x+ac} \right ){\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{dx+c}}}{\frac{1}{\sqrt{bd}}}}+{\frac{3\,{b}^{2}{c}^{2}}{8\,{d}^{2}}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\ln \left ({1 \left ({\frac{ad}{2}}+{\frac{bc}{2}}+bdx \right ){\frac{1}{\sqrt{bd}}}}+\sqrt{d{x}^{2}b+ \left ( ad+bc \right ) x+ac} \right ){\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{dx+c}}}{\frac{1}{\sqrt{bd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(b*x+a)^(3/2)*(d*x+c)^(1/2)/d+3/4/d*(b*x+a)^(1/2)*(d*x+c)^(1/2)*a-3/4/d^2*(b
*x+a)^(1/2)*(d*x+c)^(1/2)*b*c+3/8*((b*x+a)*(d*x+c))^(1/2)/(b*x+a)^(1/2)/(d*x+c)^
(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^2-3/4/d*((b*x+a)*(d*x+c))^(1/2)/(b*x+a)^(1/2)/(d*x+c)^(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*b*
c+3/8/d^2*((b*x+a)*(d*x+c))^(1/2)/(b*x+a)^(1/2)/(d*x+c)^(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)*b^2*c^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, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.252402, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (2 \, b d x - 3 \, b c + 5 \, a d\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} + 3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right )}{16 \, \sqrt{b d} d^{2}}, \frac{2 \,{\left (2 \, b d x - 3 \, b c + 5 \, a d\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c} + 3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right )}{8 \, \sqrt{-b d} d^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.228277, size = 188, normalized size = 1.66 \[ \frac{{\left (\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \sqrt{b x + a}{\left (\frac{2 \,{\left (b x + a\right )}}{b d} - \frac{3 \,{\left (b c d - a d^{2}\right )}}{b d^{3}}\right )} - \frac{3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} d^{2}}\right )} b}{4 \,{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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