Optimal. Leaf size=32 \[ -\frac{2 (c+d x)^{5/2}}{5 (a+b x)^{5/2} (b c-a d)} \]
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Rubi [A] time = 0.0215797, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ -\frac{2 (c+d x)^{5/2}}{5 (a+b x)^{5/2} (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x)^(3/2)/(a + b*x)^(7/2),x]
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Rubi in Sympy [A] time = 3.88762, size = 26, normalized size = 0.81 \[ \frac{2 \left (c + d x\right )^{\frac{5}{2}}}{5 \left (a + b x\right )^{\frac{5}{2}} \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)**(3/2)/(b*x+a)**(7/2),x)
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Mathematica [A] time = 0.0708334, size = 32, normalized size = 1. \[ -\frac{2 (c+d x)^{5/2}}{5 (a+b x)^{5/2} (b c-a d)} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x)^(3/2)/(a + b*x)^(7/2),x]
[Out]
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Maple [A] time = 0.007, size = 27, normalized size = 0.8 \[{\frac{2}{5\,ad-5\,bc} \left ( dx+c \right ) ^{{\frac{5}{2}}} \left ( bx+a \right ) ^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)^(3/2)/(b*x+a)^(7/2),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((d*x + c)^(3/2)/(b*x + a)^(7/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.338004, size = 140, normalized size = 4.38 \[ -\frac{2 \,{\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \sqrt{b x + a} \sqrt{d x + c}}{5 \,{\left (a^{3} b c - a^{4} d +{\left (b^{4} c - a b^{3} d\right )} x^{3} + 3 \,{\left (a b^{3} c - a^{2} b^{2} d\right )} x^{2} + 3 \,{\left (a^{2} b^{2} c - a^{3} b d\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^(3/2)/(b*x + a)^(7/2),x, algorithm="fricas")
[Out]
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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)**(7/2),x)
[Out]
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GIAC/XCAS [A] time = 0.309561, size = 505, normalized size = 15.78 \[ -\frac{4 \,{\left (\sqrt{b d} b^{8} c^{4} d^{2}{\left | b \right |} - 4 \, \sqrt{b d} a b^{7} c^{3} d^{3}{\left | b \right |} + 6 \, \sqrt{b d} a^{2} b^{6} c^{2} d^{4}{\left | b \right |} - 4 \, \sqrt{b d} a^{3} b^{5} c d^{5}{\left | b \right |} + \sqrt{b d} a^{4} b^{4} d^{6}{\left | b \right |} + 10 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{4} b^{4} c^{2} d^{2}{\left | b \right |} - 20 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{4} a b^{3} c d^{3}{\left | b \right |} + 10 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{4} a^{2} b^{2} d^{4}{\left | b \right |} + 5 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{8} d^{2}{\left | b \right |}\right )}}{5 \,{\left (b^{2} c - a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}^{5} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^(3/2)/(b*x + a)^(7/2),x, algorithm="giac")
[Out]