3.1504 \(\int \frac{(a+b x)^{3/2}}{(c+d x)^{3/2}} \, dx\)

Optimal. Leaf size=98 \[ -\frac{3 \sqrt{b} (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{d^{5/2}}+\frac{3 b \sqrt{a+b x} \sqrt{c+d x}}{d^2}-\frac{2 (a+b x)^{3/2}}{d \sqrt{c+d x}} \]

[Out]

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

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Rubi [A]  time = 0.110028, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ -\frac{3 \sqrt{b} (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{d^{5/2}}+\frac{3 b \sqrt{a+b x} \sqrt{c+d x}}{d^2}-\frac{2 (a+b x)^{3/2}}{d \sqrt{c+d x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 15.3823, size = 90, normalized size = 0.92 \[ \frac{3 \sqrt{b} \left (a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{d^{\frac{5}{2}}} + \frac{3 b \sqrt{a + b x} \sqrt{c + d x}}{d^{2}} - \frac{2 \left (a + b x\right )^{\frac{3}{2}}}{d \sqrt{c + d x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.230562, size = 101, normalized size = 1.03 \[ \frac{3 \sqrt{b} (a d-b c) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{2 d^{5/2}}+\frac{\sqrt{a+b x} (-2 a d+3 b c+b d x)}{d^2 \sqrt{c+d x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.066, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{{\frac{3}{2}}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((b*x+a)^(3/2)/(d*x+c)^(3/2),x)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.304488, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (b c^{2} - a c d +{\left (b c d - a d^{2}\right )} x\right )} \sqrt{\frac{b}{d}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \,{\left (2 \, b d^{2} x + b c d + a d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{b}{d}} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \,{\left (b d x + 3 \, b c - 2 \, a d\right )} \sqrt{b x + a} \sqrt{d x + c}}{4 \,{\left (d^{3} x + c d^{2}\right )}}, -\frac{3 \,{\left (b c^{2} - a c d +{\left (b c d - a d^{2}\right )} x\right )} \sqrt{-\frac{b}{d}} \arctan \left (\frac{2 \, b d x + b c + a d}{2 \, \sqrt{b x + a} \sqrt{d x + c} d \sqrt{-\frac{b}{d}}}\right ) - 2 \,{\left (b d x + 3 \, b c - 2 \, a d\right )} \sqrt{b x + a} \sqrt{d x + c}}{2 \,{\left (d^{3} x + c d^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.245835, size = 207, normalized size = 2.11 \[ \frac{{\left (\frac{{\left (b x + a\right )} b^{2} d^{2}}{b^{6} c d^{4} - a b^{5} d^{5}} + \frac{3 \,{\left (b^{3} c d - a b^{2} d^{2}\right )}}{b^{6} c d^{4} - a b^{5} d^{5}}\right )} \sqrt{b x + a}}{32 \, \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}} + \frac{3 \,{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{32 \, \sqrt{b d} b^{3} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/32*((b*x + a)*b^2*d^2/(b^6*c*d^4 - a*b^5*d^5) + 3*(b^3*c*d - a*b^2*d^2)/(b^6*c
*d^4 - a*b^5*d^5))*sqrt(b*x + a)/sqrt(b^2*c + (b*x + a)*b*d - a*b*d) + 3/32*ln(a
bs(-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)