3.1515 \(\int \frac{(a+b x)^{3/2}}{(c+d x)^{5/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0970595, 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 \[ \frac{2 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{d^{5/2}}-\frac{2 b \sqrt{a+b x}}{d^2 \sqrt{c+d x}}-\frac{2 (a+b x)^{3/2}}{3 d (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(a + b*x)^(3/2))/(3*d*(c + d*x)^(3/2)) - (2*b*Sqrt[a + b*x])/(d^2*Sqrt[c + d
*x]) + (2*b^(3/2)*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 = 14.4513, size = 85, normalized size = 0.92 \[ \frac{2 b^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{d^{\frac{5}{2}}} - \frac{2 b \sqrt{a + b x}}{d^{2} \sqrt{c + d x}} - \frac{2 \left (a + b x\right )^{\frac{3}{2}}}{3 d \left (c + d x\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.173791, size = 93, normalized size = 1.01 \[ \frac{b^{3/2} \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{d^{5/2}}-\frac{2 \sqrt{a+b x} (a d+3 b c+4 b d x)}{3 d^2 (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.063, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{{\frac{3}{2}}} \left ( dx+c \right ) ^{-{\frac{5}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.378357, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \sqrt{\frac{b}{d}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \,{\left (2 \, b d^{2} x + b c d + a d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{b}{d}} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \,{\left (4 \, b d x + 3 \, b c + a d\right )} \sqrt{b x + a} \sqrt{d x + c}}{6 \,{\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )}}, \frac{3 \,{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \sqrt{-\frac{b}{d}} \arctan \left (\frac{2 \, b d x + b c + a d}{2 \, \sqrt{b x + a} \sqrt{d x + c} d \sqrt{-\frac{b}{d}}}\right ) - 2 \,{\left (4 \, b d x + 3 \, b c + a d\right )} \sqrt{b x + a} \sqrt{d x + c}}{3 \,{\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/6*(3*(b*d^2*x^2 + 2*b*c*d*x + b*c^2)*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*(4*b*d*x + 3*b*c + a*d)*sqrt(b*x + a)*s
qrt(d*x + c))/(d^4*x^2 + 2*c*d^3*x + c^2*d^2), 1/3*(3*(b*d^2*x^2 + 2*b*c*d*x + b
*c^2)*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*(4*b*d*x + 3*b*c + a*d)*sqrt(b*x + a)*sqrt(d*x + c))/(d^4*x^2
+ 2*c*d^3*x + c^2*d^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((b*x+a)**(3/2)/(d*x+c)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.250972, size = 296, normalized size = 3.22 \[ \frac{\sqrt{b d}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{16 \,{\left (b^{5} c d^{4} - a b^{4} d^{5}\right )}} + \frac{\sqrt{b x + a}{\left (\frac{4 \,{\left (b^{5} c d^{2} - a b^{4} d^{3}\right )}{\left (b x + a\right )}}{b^{8} c^{2} d^{4} - 2 \, a b^{7} c d^{5} + a^{2} b^{6} d^{6}} + \frac{3 \,{\left (b^{6} c^{2} d - 2 \, a b^{5} c d^{2} + a^{2} b^{4} d^{3}\right )}}{b^{8} c^{2} d^{4} - 2 \, a b^{7} c d^{5} + a^{2} b^{6} d^{6}}\right )}}{48 \,{\left (b^{2} c +{\left (b x + a\right )} b d - a b d\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*sqrt(b*d)*ln(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*
b*d)))/(b^5*c*d^4 - a*b^4*d^5) + 1/48*sqrt(b*x + a)*(4*(b^5*c*d^2 - a*b^4*d^3)*(
b*x + a)/(b^8*c^2*d^4 - 2*a*b^7*c*d^5 + a^2*b^6*d^6) + 3*(b^6*c^2*d - 2*a*b^5*c*
d^2 + a^2*b^4*d^3)/(b^8*c^2*d^4 - 2*a*b^7*c*d^5 + a^2*b^6*d^6))/(b^2*c + (b*x +
a)*b*d - a*b*d)^(3/2)