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

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

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.118568, 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 \[ \frac{3 \sqrt{d} (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{5/2}}+\frac{3 d \sqrt{a+b x} \sqrt{c+d x}}{b^2}-\frac{2 (c+d x)^{3/2}}{b \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 13.8312, size = 90, normalized size = 0.92 \[ - \frac{2 \left (c + d x\right )^{\frac{3}{2}}}{b \sqrt{a + b x}} + \frac{3 d \sqrt{a + b x} \sqrt{c + d x}}{b^{2}} - \frac{3 \sqrt{d} \left (a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.167674, size = 101, normalized size = 1.03 \[ \frac{3 \sqrt{d} (b c-a d) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{2 b^{5/2}}+\frac{\sqrt{c+d x} (3 a d-2 b c+b d x)}{b^2 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.334096, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (a b c - a^{2} d +{\left (b^{2} c - a b d\right )} x\right )} \sqrt{\frac{d}{b}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} - 4 \,{\left (2 \, b^{2} d x + b^{2} c + a b d\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{d}{b}} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \,{\left (b d x - 2 \, b c + 3 \, a d\right )} \sqrt{b x + a} \sqrt{d x + c}}{4 \,{\left (b^{3} x + a b^{2}\right )}}, \frac{3 \,{\left (a b c - a^{2} d +{\left (b^{2} c - a b d\right )} x\right )} \sqrt{-\frac{d}{b}} \arctan \left (\frac{2 \, b d x + b c + a d}{2 \, \sqrt{b x + a} \sqrt{d x + c} b \sqrt{-\frac{d}{b}}}\right ) + 2 \,{\left (b d x - 2 \, b c + 3 \, a d\right )} \sqrt{b x + a} \sqrt{d x + c}}{2 \,{\left (b^{3} x + a b^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.582928, size = 4, normalized size = 0.04 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x