Optimal. Leaf size=174 \[ \frac{\sqrt{a+b x} \left (c \left (3 a^2 d^2-22 a b c d+15 b^2 c^2\right )+d x (5 b c-3 a d) (b c-a d)\right )}{3 b d^3 \sqrt{c+d x} (b c-a d)^2}-\frac{(a d+5 b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{3/2} d^{7/2}}-\frac{2 c x^2 \sqrt{a+b x}}{3 d (c+d x)^{3/2} (b c-a d)} \]
[Out]
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Rubi [A] time = 0.373151, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{\sqrt{a+b x} \left (c \left (3 a^2 d^2-22 a b c d+15 b^2 c^2\right )+d x (5 b c-3 a d) (b c-a d)\right )}{3 b d^3 \sqrt{c+d x} (b c-a d)^2}-\frac{(a d+5 b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{3/2} d^{7/2}}-\frac{2 c x^2 \sqrt{a+b x}}{3 d (c+d x)^{3/2} (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[x^3/(Sqrt[a + b*x]*(c + d*x)^(5/2)),x]
[Out]
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Rubi in Sympy [A] time = 27.2585, size = 165, normalized size = 0.95 \[ \frac{2 c x^{2} \sqrt{a + b x}}{3 d \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )} + \frac{4 \sqrt{a + b x} \left (\frac{c \left (3 a^{2} d^{2} - 22 a b c d + 15 b^{2} c^{2}\right )}{4} + \frac{d x \left (a d - b c\right ) \left (3 a d - 5 b c\right )}{4}\right )}{3 b d^{3} \sqrt{c + d x} \left (a d - b c\right )^{2}} - \frac{\left (a d + 5 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{b^{\frac{3}{2}} d^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3/(d*x+c)**(5/2)/(b*x+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.546933, size = 150, normalized size = 0.86 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (\frac{2 c^3}{(c+d x)^2 (a d-b c)}+\frac{2 c^2 (7 b c-9 a d)}{(c+d x) (b c-a d)^2}+\frac{3}{b}\right )}{3 d^3}-\frac{(a d+5 b c) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{2 b^{3/2} d^{7/2}} \]
Antiderivative was successfully verified.
[In] Integrate[x^3/(Sqrt[a + b*x]*(c + d*x)^(5/2)),x]
[Out]
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Maple [B] time = 0.037, size = 928, normalized size = 5.3 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3/(d*x+c)^(5/2)/(b*x+a)^(1/2),x)
[Out]
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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(x^3/(sqrt(b*x + a)*(d*x + c)^(5/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.519775, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/(sqrt(b*x + a)*(d*x + c)^(5/2)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{\sqrt{a + b x} \left (c + d x\right )^{\frac{5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3/(d*x+c)**(5/2)/(b*x+a)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.25613, size = 504, normalized size = 2.9 \[ \frac{{\left ({\left (b x + a\right )}{\left (\frac{3 \,{\left (b^{6} c^{2} d^{4}{\left | b \right |} - 2 \, a b^{5} c d^{5}{\left | b \right |} + a^{2} b^{4} d^{6}{\left | b \right |}\right )}{\left (b x + a\right )}}{b^{7} c^{2} d^{5} - 2 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}} + \frac{2 \,{\left (10 \, b^{7} c^{3} d^{3}{\left | b \right |} - 18 \, a b^{6} c^{2} d^{4}{\left | b \right |} + 9 \, a^{2} b^{5} c d^{5}{\left | b \right |} - 3 \, a^{3} b^{4} d^{6}{\left | b \right |}\right )}}{b^{7} c^{2} d^{5} - 2 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}}\right )} + \frac{3 \,{\left (5 \, b^{8} c^{4} d^{2}{\left | b \right |} - 14 \, a b^{7} c^{3} d^{3}{\left | b \right |} + 12 \, a^{2} b^{6} c^{2} d^{4}{\left | b \right |} - 4 \, a^{3} b^{5} c d^{5}{\left | b \right |} + a^{4} b^{4} d^{6}{\left | b \right |}\right )}}{b^{7} c^{2} d^{5} - 2 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}}\right )} \sqrt{b x + a}}{3 \,{\left (b^{2} c +{\left (b x + a\right )} b d - a b d\right )}^{\frac{3}{2}}} + \frac{{\left (5 \, b c{\left | b \right |} + a d{\left | b \right |}\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} b^{2} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/(sqrt(b*x + a)*(d*x + c)^(5/2)),x, algorithm="giac")
[Out]