Optimal. Leaf size=278 \[ -\frac{7 b \sqrt{c+d x} (15 b c-7 a d) (b c-a d)}{24 a^4 (a+b x)^{3/2}}-\frac{\sqrt{c+d x} (21 b c-11 a d) (b c-a d)}{8 a^3 x (a+b x)^{3/2}}+\frac{3 c \sqrt{c+d x} (b c-a d)}{4 a^2 x^2 (a+b x)^{3/2}}+\frac{5 (b c-a d) \left (a^2 d^2-14 a b c d+21 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{11/2} \sqrt{c}}-\frac{b \sqrt{c+d x} \left (113 a^2 d^2-420 a b c d+315 b^2 c^2\right )}{24 a^5 \sqrt{a+b x}}-\frac{c (c+d x)^{3/2}}{3 a x^3 (a+b x)^{3/2}} \]
[Out]
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Rubi [A] time = 1.14685, antiderivative size = 278, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318 \[ -\frac{7 b \sqrt{c+d x} (15 b c-7 a d) (b c-a d)}{24 a^4 (a+b x)^{3/2}}-\frac{\sqrt{c+d x} (21 b c-11 a d) (b c-a d)}{8 a^3 x (a+b x)^{3/2}}+\frac{3 c \sqrt{c+d x} (b c-a d)}{4 a^2 x^2 (a+b x)^{3/2}}+\frac{5 (b c-a d) \left (a^2 d^2-14 a b c d+21 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{11/2} \sqrt{c}}-\frac{b \sqrt{c+d x} \left (113 a^2 d^2-420 a b c d+315 b^2 c^2\right )}{24 a^5 \sqrt{a+b x}}-\frac{c (c+d x)^{3/2}}{3 a x^3 (a+b x)^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x)^(5/2)/(x^4*(a + b*x)^(5/2)),x]
[Out]
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Rubi in Sympy [A] time = 160.972, size = 267, normalized size = 0.96 \[ - \frac{c \left (c + d x\right )^{\frac{3}{2}}}{3 a x^{3} \left (a + b x\right )^{\frac{3}{2}}} - \frac{3 c \sqrt{c + d x} \left (a d - b c\right )}{4 a^{2} x^{2} \left (a + b x\right )^{\frac{3}{2}}} - \frac{\sqrt{c + d x} \left (a d - b c\right ) \left (11 a d - 21 b c\right )}{8 a^{3} x \left (a + b x\right )^{\frac{3}{2}}} - \frac{7 b \sqrt{c + d x} \left (a d - b c\right ) \left (7 a d - 15 b c\right )}{24 a^{4} \left (a + b x\right )^{\frac{3}{2}}} - \frac{b \sqrt{c + d x} \left (113 a^{2} d^{2} - 420 a b c d + 315 b^{2} c^{2}\right )}{24 a^{5} \sqrt{a + b x}} - \frac{5 \left (a d - b c\right ) \left (a^{2} d^{2} - 14 a b c d + 21 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{8 a^{\frac{11}{2}} \sqrt{c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)**(5/2)/x**4/(b*x+a)**(5/2),x)
[Out]
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Mathematica [A] time = 0.463397, size = 264, normalized size = 0.95 \[ \frac{\frac{15 \log (x) (a d-b c) \left (a^2 d^2-14 a b c d+21 b^2 c^2\right )}{\sqrt{c}}+\frac{15 (b c-a d) \left (a^2 d^2-14 a b c d+21 b^2 c^2\right ) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{\sqrt{c}}-\frac{2 \sqrt{a} \sqrt{c+d x} \left (a^4 \left (8 c^2+26 c d x+33 d^2 x^2\right )+6 a^3 b x \left (-3 c^2-16 c d x+27 d^2 x^2\right )+a^2 b^2 x^2 \left (63 c^2-574 c d x+113 d^2 x^2\right )+420 a b^3 c x^3 (c-d x)+315 b^4 c^2 x^4\right )}{x^3 (a+b x)^{3/2}}}{48 a^{11/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x)^(5/2)/(x^4*(a + b*x)^(5/2)),x]
[Out]
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Maple [B] time = 0.048, size = 1009, normalized size = 3.6 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)^(5/2)/x^4/(b*x+a)^(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((d*x + c)^(5/2)/((b*x + a)^(5/2)*x^4),x, algorithm="maxima")
[Out]
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Fricas [A] time = 1.56182, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^(5/2)/((b*x + a)^(5/2)*x^4),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)**(5/2)/x**4/(b*x+a)**(5/2),x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^(5/2)/((b*x + a)^(5/2)*x^4),x, algorithm="giac")
[Out]