Optimal. Leaf size=66 \[ \frac{4 d \sqrt{c+d x}}{3 \sqrt{a+b x} (b c-a d)^2}-\frac{2 \sqrt{c+d x}}{3 (a+b x)^{3/2} (b c-a d)} \]
[Out]
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Rubi [A] time = 0.0497228, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{4 d \sqrt{c+d x}}{3 \sqrt{a+b x} (b c-a d)^2}-\frac{2 \sqrt{c+d x}}{3 (a+b x)^{3/2} (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b*x)^(5/2)*Sqrt[c + d*x]),x]
[Out]
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Rubi in Sympy [A] time = 7.91964, size = 56, normalized size = 0.85 \[ \frac{4 d \sqrt{c + d x}}{3 \sqrt{a + b x} \left (a d - b c\right )^{2}} + \frac{2 \sqrt{c + d x}}{3 \left (a + b x\right )^{\frac{3}{2}} \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x+a)**(5/2)/(d*x+c)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0600362, size = 46, normalized size = 0.7 \[ \frac{2 \sqrt{c+d x} (3 a d-b c+2 b d x)}{3 (a+b x)^{3/2} (b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b*x)^(5/2)*Sqrt[c + d*x]),x]
[Out]
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Maple [A] time = 0.009, size = 54, normalized size = 0.8 \[{\frac{4\,bdx+6\,ad-2\,bc}{3\,{a}^{2}{d}^{2}-6\,abcd+3\,{b}^{2}{c}^{2}}\sqrt{dx+c} \left ( bx+a \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x+a)^(5/2)/(d*x+c)^(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(1/((b*x + a)^(5/2)*sqrt(d*x + c)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.250438, size = 159, normalized size = 2.41 \[ \frac{2 \,{\left (2 \, b d x - b c + 3 \, a d\right )} \sqrt{b x + a} \sqrt{d x + c}}{3 \,{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} +{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2} + 2 \,{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^(5/2)*sqrt(d*x + c)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x\right )^{\frac{5}{2}} \sqrt{c + d x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x+a)**(5/2)/(d*x+c)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.232389, size = 163, normalized size = 2.47 \[ \frac{8 \,{\left (b^{2} c - a b d - 3 \,{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )} \sqrt{b d} b^{2} d}{3 \,{\left (b^{2} c - a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}^{3}{\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^(5/2)*sqrt(d*x + c)),x, algorithm="giac")
[Out]