3.1476 \(\int \frac{(c+d x)^{3/2}}{(a+b x)^{5/2}} \, dx\)

Optimal. Leaf size=92 \[ \frac{2 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{5/2}}-\frac{2 d \sqrt{c+d x}}{b^2 \sqrt{a+b x}}-\frac{2 (c+d x)^{3/2}}{3 b (a+b x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0978239, antiderivative size = 92, 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 \[ \frac{2 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{5/2}}-\frac{2 d \sqrt{c+d x}}{b^2 \sqrt{a+b x}}-\frac{2 (c+d x)^{3/2}}{3 b (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^(3/2)/(a + b*x)^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 14.6893, size = 85, normalized size = 0.92 \[ - \frac{2 \left (c + d x\right )^{\frac{3}{2}}}{3 b \left (a + b x\right )^{\frac{3}{2}}} - \frac{2 d \sqrt{c + d x}}{b^{2} \sqrt{a + b x}} + \frac{2 d^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(3/2)/(b*x+a)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.168471, size = 93, normalized size = 1.01 \[ \frac{d^{3/2} \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{b^{5/2}}-\frac{2 \sqrt{c+d x} (3 a d+b (c+4 d x))}{3 b^2 (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^(3/2)/(a + b*x)^(5/2),x]

[Out]

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

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Maple [F]  time = 0.053, size = 0, normalized size = 0. \[ \int{1 \left ( dx+c \right ) ^{{\frac{3}{2}}} \left ( bx+a \right ) ^{-{\frac{5}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(3/2)/(b*x+a)^(5/2),x)

[Out]

int((d*x+c)^(3/2)/(b*x+a)^(5/2),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((d*x + c)^(3/2)/(b*x + a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.373974, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (b^{2} d x^{2} + 2 \, a b d x + a^{2} d\right )} \sqrt{\frac{d}{b}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \,{\left (2 \, b^{2} d x + b^{2} c + a b d\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{d}{b}} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \,{\left (4 \, b d x + b c + 3 \, a d\right )} \sqrt{b x + a} \sqrt{d x + c}}{6 \,{\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}}, \frac{3 \,{\left (b^{2} d x^{2} + 2 \, a b d x + a^{2} d\right )} \sqrt{-\frac{d}{b}} \arctan \left (\frac{2 \, b d x + b c + a d}{2 \, \sqrt{b x + a} \sqrt{d x + c} b \sqrt{-\frac{d}{b}}}\right ) - 2 \,{\left (4 \, b d x + b c + 3 \, a d\right )} \sqrt{b x + a} \sqrt{d x + c}}{3 \,{\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(3/2)/(b*x + a)^(5/2),x, algorithm="fricas")

[Out]

[1/6*(3*(b^2*d*x^2 + 2*a*b*d*x + a^2*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*(4*b*d*x + b*c + 3*a*d)*sqrt(b*x + a)*s
qrt(d*x + c))/(b^4*x^2 + 2*a*b^3*x + a^2*b^2), 1/3*(3*(b^2*d*x^2 + 2*a*b*d*x + a
^2*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*(4*b*d*x + b*c + 3*a*d)*sqrt(b*x + a)*sqrt(d*x + c))/(b^4*x^2
+ 2*a*b^3*x + a^2*b^2)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x+c)**(3/2)/(b*x+a)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.578947, 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)^(5/2),x, algorithm="giac")

[Out]

sage0*x