Optimal. Leaf size=135 \[ \frac{5}{16} a^4 c^2 x \sqrt{a+b x} \sqrt{a c-b c x}+\frac{5 a^6 c^{5/2} \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{c (a-b x)}}\right )}{8 b}+\frac{5}{24} a^2 c x (a+b x)^{3/2} (a c-b c x)^{3/2}+\frac{1}{6} x (a+b x)^{5/2} (a c-b c x)^{5/2} \]
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Rubi [A] time = 0.0522339, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {38, 63, 217, 203} \[ \frac{5}{16} a^4 c^2 x \sqrt{a+b x} \sqrt{a c-b c x}+\frac{5 a^6 c^{5/2} \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{c (a-b x)}}\right )}{8 b}+\frac{5}{24} a^2 c x (a+b x)^{3/2} (a c-b c x)^{3/2}+\frac{1}{6} x (a+b x)^{5/2} (a c-b c x)^{5/2} \]
Antiderivative was successfully verified.
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Rule 38
Rule 63
Rule 217
Rule 203
Rubi steps
\begin{align*} \int (a+b x)^{5/2} (a c-b c x)^{5/2} \, dx &=\frac{1}{6} x (a+b x)^{5/2} (a c-b c x)^{5/2}+\frac{1}{6} \left (5 a^2 c\right ) \int (a+b x)^{3/2} (a c-b c x)^{3/2} \, dx\\ &=\frac{5}{24} a^2 c x (a+b x)^{3/2} (a c-b c x)^{3/2}+\frac{1}{6} x (a+b x)^{5/2} (a c-b c x)^{5/2}+\frac{1}{8} \left (5 a^4 c^2\right ) \int \sqrt{a+b x} \sqrt{a c-b c x} \, dx\\ &=\frac{5}{16} a^4 c^2 x \sqrt{a+b x} \sqrt{a c-b c x}+\frac{5}{24} a^2 c x (a+b x)^{3/2} (a c-b c x)^{3/2}+\frac{1}{6} x (a+b x)^{5/2} (a c-b c x)^{5/2}+\frac{1}{16} \left (5 a^6 c^3\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{a c-b c x}} \, dx\\ &=\frac{5}{16} a^4 c^2 x \sqrt{a+b x} \sqrt{a c-b c x}+\frac{5}{24} a^2 c x (a+b x)^{3/2} (a c-b c x)^{3/2}+\frac{1}{6} x (a+b x)^{5/2} (a c-b c x)^{5/2}+\frac{\left (5 a^6 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{2 a c-c x^2}} \, dx,x,\sqrt{a+b x}\right )}{8 b}\\ &=\frac{5}{16} a^4 c^2 x \sqrt{a+b x} \sqrt{a c-b c x}+\frac{5}{24} a^2 c x (a+b x)^{3/2} (a c-b c x)^{3/2}+\frac{1}{6} x (a+b x)^{5/2} (a c-b c x)^{5/2}+\frac{\left (5 a^6 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1+c x^2} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c (a-b x)}}\right )}{8 b}\\ &=\frac{5}{16} a^4 c^2 x \sqrt{a+b x} \sqrt{a c-b c x}+\frac{5}{24} a^2 c x (a+b x)^{3/2} (a c-b c x)^{3/2}+\frac{1}{6} x (a+b x)^{5/2} (a c-b c x)^{5/2}+\frac{5 a^6 c^{5/2} \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{c (a-b x)}}\right )}{8 b}\\ \end{align*}
Mathematica [A] time = 0.147235, size = 120, normalized size = 0.89 \[ \frac{c^3 \left (34 a^2 b^5 x^5-59 a^4 b^3 x^3+33 a^6 b x-30 a^{13/2} \sqrt{a-b x} \sqrt{\frac{b x}{a}+1} \sin ^{-1}\left (\frac{\sqrt{a-b x}}{\sqrt{2} \sqrt{a}}\right )-8 b^7 x^7\right )}{48 b \sqrt{a+b x} \sqrt{c (a-b x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.01, size = 243, normalized size = 1.8 \begin{align*} -{\frac{1}{6\,bc} \left ( bx+a \right ) ^{{\frac{5}{2}}} \left ( -bcx+ac \right ) ^{{\frac{7}{2}}}}-{\frac{a}{6\,bc} \left ( bx+a \right ) ^{{\frac{3}{2}}} \left ( -bcx+ac \right ) ^{{\frac{7}{2}}}}-{\frac{{a}^{2}}{8\,bc}\sqrt{bx+a} \left ( -bcx+ac \right ) ^{{\frac{7}{2}}}}+{\frac{{a}^{3}}{24\,b} \left ( -bcx+ac \right ) ^{{\frac{5}{2}}}\sqrt{bx+a}}+{\frac{5\,{a}^{4}c}{48\,b} \left ( -bcx+ac \right ) ^{{\frac{3}{2}}}\sqrt{bx+a}}+{\frac{5\,{a}^{5}{c}^{2}}{16\,b}\sqrt{bx+a}\sqrt{-bcx+ac}}+{\frac{5\,{a}^{6}{c}^{3}}{16}\sqrt{ \left ( bx+a \right ) \left ( -bcx+ac \right ) }\arctan \left ({x\sqrt{{b}^{2}c}{\frac{1}{\sqrt{-{b}^{2}c{x}^{2}+{a}^{2}c}}}} \right ){\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{-bcx+ac}}}{\frac{1}{\sqrt{{b}^{2}c}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70367, size = 524, normalized size = 3.88 \begin{align*} \left [\frac{15 \, a^{6} \sqrt{-c} c^{2} \log \left (2 \, b^{2} c x^{2} + 2 \, \sqrt{-b c x + a c} \sqrt{b x + a} b \sqrt{-c} x - a^{2} c\right ) + 2 \,{\left (8 \, b^{5} c^{2} x^{5} - 26 \, a^{2} b^{3} c^{2} x^{3} + 33 \, a^{4} b c^{2} x\right )} \sqrt{-b c x + a c} \sqrt{b x + a}}{96 \, b}, -\frac{15 \, a^{6} c^{\frac{5}{2}} \arctan \left (\frac{\sqrt{-b c x + a c} \sqrt{b x + a} b \sqrt{c} x}{b^{2} c x^{2} - a^{2} c}\right ) -{\left (8 \, b^{5} c^{2} x^{5} - 26 \, a^{2} b^{3} c^{2} x^{3} + 33 \, a^{4} b c^{2} x\right )} \sqrt{-b c x + a c} \sqrt{b x + a}}{48 \, b}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (- c \left (- a + b x\right )\right )^{\frac{5}{2}} \left (a + b x\right )^{\frac{5}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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