Optimal. Leaf size=306 \[ \frac{a b^2 d^2 \left (a^2+11 b^2 c\right )}{2 c \left (a^2-b^2 c\right )^3 \left (a+b \sqrt{c+d x}\right )}-\frac{a-b \sqrt{c+d x}}{2 x^2 \left (a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )}-\frac{b d \left (3 a b c-\left (a^2+2 b^2 c\right ) \sqrt{c+d x}\right )}{2 c x \left (a^2-b^2 c\right )^2 \left (a+b \sqrt{c+d x}\right )}+\frac{b^4 d^2 \log (x) \left (5 a^2+b^2 c\right )}{\left (a^2-b^2 c\right )^4}-\frac{2 b^4 d^2 \left (5 a^2+b^2 c\right ) \log \left (a+b \sqrt{c+d x}\right )}{\left (a^2-b^2 c\right )^4}-\frac{a b d^2 \left (a^4-10 a^2 b^2 c-15 b^4 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{2 c^{3/2} \left (a^2-b^2 c\right )^4} \]
[Out]
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Rubi [A] time = 0.837213, antiderivative size = 306, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368 \[ \frac{a b^2 d^2 \left (a^2+11 b^2 c\right )}{2 c \left (a^2-b^2 c\right )^3 \left (a+b \sqrt{c+d x}\right )}-\frac{a-b \sqrt{c+d x}}{2 x^2 \left (a^2-b^2 c\right ) \left (a+b \sqrt{c+d x}\right )}-\frac{b d \left (3 a b c-\left (a^2+2 b^2 c\right ) \sqrt{c+d x}\right )}{2 c x \left (a^2-b^2 c\right )^2 \left (a+b \sqrt{c+d x}\right )}+\frac{b^4 d^2 \log (x) \left (5 a^2+b^2 c\right )}{\left (a^2-b^2 c\right )^4}-\frac{2 b^4 d^2 \left (5 a^2+b^2 c\right ) \log \left (a+b \sqrt{c+d x}\right )}{\left (a^2-b^2 c\right )^4}-\frac{a b d^2 \left (a^4-10 a^2 b^2 c-15 b^4 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{2 c^{3/2} \left (a^2-b^2 c\right )^4} \]
Antiderivative was successfully verified.
[In] Int[1/(x^3*(a + b*Sqrt[c + d*x])^2),x]
[Out]
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Rubi in Sympy [A] time = 71.6826, size = 275, normalized size = 0.9 \[ \frac{a b^{2} d^{2} \left (a^{2} + 11 b^{2} c\right )}{2 c \left (a + b \sqrt{c + d x}\right ) \left (a^{2} - b^{2} c\right )^{3}} - \frac{a b d^{2} \left (a^{4} - 10 a^{2} b^{2} c - 15 b^{4} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{c}} \right )}}{2 c^{\frac{3}{2}} \left (a^{2} - b^{2} c\right )^{4}} + \frac{b^{4} d^{2} \left (5 a^{2} + b^{2} c\right ) \log{\left (- d x \right )}}{\left (a^{2} - b^{2} c\right )^{4}} - \frac{2 b^{4} d^{2} \left (5 a^{2} + b^{2} c\right ) \log{\left (a + b \sqrt{c + d x} \right )}}{\left (a^{2} - b^{2} c\right )^{4}} + \frac{b d \left (- 6 a b c + \left (2 a^{2} + 4 b^{2} c\right ) \sqrt{c + d x}\right )}{4 c x \left (a + b \sqrt{c + d x}\right ) \left (a^{2} - b^{2} c\right )^{2}} - \frac{a - b \sqrt{c + d x}}{2 x^{2} \left (a + b \sqrt{c + d x}\right ) \left (a^{2} - b^{2} c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**3/(a+b*(d*x+c)**(1/2))**2,x)
[Out]
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Mathematica [A] time = 0.616692, size = 390, normalized size = 1.27 \[ \frac{1}{2} \left (-\frac{2 b^2 d \left (3 a^2+b^2 c\right )}{x \left (a^2-b^2 c\right )^3}-\frac{a^2+b^2 c}{x^2 \left (a^2-b^2 c\right )^2}+\frac{4 a^2 b^4 d^2}{\left (a^2-b^2 c\right )^3 \left (a^2-b^2 (c+d x)\right )}+\frac{2 b^4 d^2 \log (x) \left (5 a^2+b^2 c\right )}{\left (a^2-b^2 c\right )^4}-\frac{2 b^4 d^2 \left (5 a^2+b^2 c\right ) \log \left (a^2-b^2 (c+d x)\right )}{\left (a^2-b^2 c\right )^4}-\frac{4 b^4 d^2 \left (5 a^2+b^2 c\right ) \tanh ^{-1}\left (\frac{b \sqrt{c+d x}}{a}\right )}{\left (a^2-b^2 c\right )^4}+\frac{a \sqrt{c+d x} \left (a^4 b (2 c+d x)-a^2 b^3 (d x-2 c)^2+b^5 c \left (2 c^2-5 c d x-11 d^2 x^2\right )\right )}{c x^2 \left (b^2 c-a^2\right )^3 \left (b^2 (c+d x)-a^2\right )}+\frac{d^2 \left (a^5 (-b)+10 a^3 b^3 c+15 a b^5 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{c^{3/2} \left (a^2-b^2 c\right )^4}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^3*(a + b*Sqrt[c + d*x])^2),x]
[Out]
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Maple [B] time = 0.027, size = 612, normalized size = 2. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^3/(a+b*(d*x+c)^(1/2))^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/((sqrt(d*x + c)*b + a)^2*x^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 1.60741, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((sqrt(d*x + c)*b + a)^2*x^3),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{3} \left (a + b \sqrt{c + d x}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**3/(a+b*(d*x+c)**(1/2))**2,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \mathit{undef} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((sqrt(d*x + c)*b + a)^2*x^3),x, algorithm="giac")
[Out]