Optimal. Leaf size=172 \[ -\frac{256 b d^3 \sqrt{a+b x}}{15 \sqrt{c+d x} (b c-a d)^5}-\frac{128 d^3 \sqrt{a+b x}}{15 (c+d x)^{3/2} (b c-a d)^4}-\frac{32 d^2}{5 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^3}+\frac{16 d}{15 (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)^2}-\frac{2}{5 (a+b x)^{5/2} (c+d x)^{3/2} (b c-a d)} \]
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Rubi [A] time = 0.159121, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{256 b d^3 \sqrt{a+b x}}{15 \sqrt{c+d x} (b c-a d)^5}-\frac{128 d^3 \sqrt{a+b x}}{15 (c+d x)^{3/2} (b c-a d)^4}-\frac{32 d^2}{5 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^3}+\frac{16 d}{15 (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)^2}-\frac{2}{5 (a+b x)^{5/2} (c+d x)^{3/2} (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b*x)^(7/2)*(c + d*x)^(5/2)),x]
[Out]
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Rubi in Sympy [A] time = 32.7655, size = 155, normalized size = 0.9 \[ \frac{256 b d^{3} \sqrt{a + b x}}{15 \sqrt{c + d x} \left (a d - b c\right )^{5}} - \frac{128 d^{3} \sqrt{a + b x}}{15 \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )^{4}} + \frac{32 d^{2}}{5 \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )^{3}} + \frac{16 d}{15 \left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )^{2}} + \frac{2}{5 \left (a + b x\right )^{\frac{5}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x+a)**(7/2)/(d*x+c)**(5/2),x)
[Out]
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Mathematica [A] time = 0.385134, size = 124, normalized size = 0.72 \[ \frac{2 \sqrt{a+b x} \sqrt{c+d x} \left (\frac{14 b^2 d (b c-a d)}{(a+b x)^2}-\frac{3 b^2 (b c-a d)^2}{(a+b x)^3}-\frac{73 b^2 d^2}{a+b x}+\frac{5 d^3 (a d-b c)}{(c+d x)^2}-\frac{55 b d^3}{c+d x}\right )}{15 (b c-a d)^5} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b*x)^(7/2)*(c + d*x)^(5/2)),x]
[Out]
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Maple [A] time = 0.014, size = 256, normalized size = 1.5 \[ -{\frac{-256\,{b}^{4}{d}^{4}{x}^{4}-640\,a{b}^{3}{d}^{4}{x}^{3}-384\,{b}^{4}c{d}^{3}{x}^{3}-480\,{a}^{2}{b}^{2}{d}^{4}{x}^{2}-960\,a{b}^{3}c{d}^{3}{x}^{2}-96\,{b}^{4}{c}^{2}{d}^{2}{x}^{2}-80\,{a}^{3}b{d}^{4}x-720\,{a}^{2}{b}^{2}c{d}^{3}x-240\,a{b}^{3}{c}^{2}{d}^{2}x+16\,{b}^{4}{c}^{3}dx+10\,{a}^{4}{d}^{4}-120\,{a}^{3}bc{d}^{3}-180\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}+40\,a{b}^{3}{c}^{3}d-6\,{b}^{4}{c}^{4}}{15\,{a}^{5}{d}^{5}-75\,{a}^{4}bc{d}^{4}+150\,{a}^{3}{b}^{2}{c}^{2}{d}^{3}-150\,{a}^{2}{b}^{3}{c}^{3}{d}^{2}+75\,a{b}^{4}{c}^{4}d-15\,{b}^{5}{c}^{5}} \left ( bx+a \right ) ^{-{\frac{5}{2}}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x+a)^(7/2)/(d*x+c)^(5/2),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)^(7/2)*(d*x + c)^(5/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 2.09969, size = 965, normalized size = 5.61 \[ -\frac{2 \,{\left (128 \, b^{4} d^{4} x^{4} + 3 \, b^{4} c^{4} - 20 \, a b^{3} c^{3} d + 90 \, a^{2} b^{2} c^{2} d^{2} + 60 \, a^{3} b c d^{3} - 5 \, a^{4} d^{4} + 64 \,{\left (3 \, b^{4} c d^{3} + 5 \, a b^{3} d^{4}\right )} x^{3} + 48 \,{\left (b^{4} c^{2} d^{2} + 10 \, a b^{3} c d^{3} + 5 \, a^{2} b^{2} d^{4}\right )} x^{2} - 8 \,{\left (b^{4} c^{3} d - 15 \, a b^{3} c^{2} d^{2} - 45 \, a^{2} b^{2} c d^{3} - 5 \, a^{3} b d^{4}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{15 \,{\left (a^{3} b^{5} c^{7} - 5 \, a^{4} b^{4} c^{6} d + 10 \, a^{5} b^{3} c^{5} d^{2} - 10 \, a^{6} b^{2} c^{4} d^{3} + 5 \, a^{7} b c^{3} d^{4} - a^{8} c^{2} d^{5} +{\left (b^{8} c^{5} d^{2} - 5 \, a b^{7} c^{4} d^{3} + 10 \, a^{2} b^{6} c^{3} d^{4} - 10 \, a^{3} b^{5} c^{2} d^{5} + 5 \, a^{4} b^{4} c d^{6} - a^{5} b^{3} d^{7}\right )} x^{5} +{\left (2 \, b^{8} c^{6} d - 7 \, a b^{7} c^{5} d^{2} + 5 \, a^{2} b^{6} c^{4} d^{3} + 10 \, a^{3} b^{5} c^{3} d^{4} - 20 \, a^{4} b^{4} c^{2} d^{5} + 13 \, a^{5} b^{3} c d^{6} - 3 \, a^{6} b^{2} d^{7}\right )} x^{4} +{\left (b^{8} c^{7} + a b^{7} c^{6} d - 17 \, a^{2} b^{6} c^{5} d^{2} + 35 \, a^{3} b^{5} c^{4} d^{3} - 25 \, a^{4} b^{4} c^{3} d^{4} - a^{5} b^{3} c^{2} d^{5} + 9 \, a^{6} b^{2} c d^{6} - 3 \, a^{7} b d^{7}\right )} x^{3} +{\left (3 \, a b^{7} c^{7} - 9 \, a^{2} b^{6} c^{6} d + a^{3} b^{5} c^{5} d^{2} + 25 \, a^{4} b^{4} c^{4} d^{3} - 35 \, a^{5} b^{3} c^{3} d^{4} + 17 \, a^{6} b^{2} c^{2} d^{5} - a^{7} b c d^{6} - a^{8} d^{7}\right )} x^{2} +{\left (3 \, a^{2} b^{6} c^{7} - 13 \, a^{3} b^{5} c^{6} d + 20 \, a^{4} b^{4} c^{5} d^{2} - 10 \, a^{5} b^{3} c^{4} d^{3} - 5 \, a^{6} b^{2} c^{3} d^{4} + 7 \, a^{7} b c^{2} d^{5} - 2 \, a^{8} c d^{6}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^(7/2)*(d*x + c)^(5/2)),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(1/(b*x+a)**(7/2)/(d*x+c)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.826491, size = 1, normalized size = 0.01 \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^(7/2)*(d*x + c)^(5/2)),x, algorithm="giac")
[Out]