Optimal. Leaf size=166 \[ \frac {2 a \left (a^2-b^2 c\right )^2}{b^6 d^3 \left (a+b \sqrt {c+d x}\right )}-\frac {8 a \left (a^2-b^2 c\right ) \sqrt {c+d x}}{b^5 d^3}+\frac {x \left (3 a^2-2 b^2 c\right )}{b^4 d^2}+\frac {2 \left (5 a^4-6 a^2 b^2 c+b^4 c^2\right ) \log \left (a+b \sqrt {c+d x}\right )}{b^6 d^3}-\frac {4 a (c+d x)^{3/2}}{3 b^3 d^3}+\frac {(c+d x)^2}{2 b^2 d^3} \]
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Rubi [A] time = 0.17, antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {371, 1398, 772} \begin {gather*} \frac {2 \left (-6 a^2 b^2 c+5 a^4+b^4 c^2\right ) \log \left (a+b \sqrt {c+d x}\right )}{b^6 d^3}+\frac {2 a \left (a^2-b^2 c\right )^2}{b^6 d^3 \left (a+b \sqrt {c+d x}\right )}-\frac {8 a \left (a^2-b^2 c\right ) \sqrt {c+d x}}{b^5 d^3}+\frac {x \left (3 a^2-2 b^2 c\right )}{b^4 d^2}-\frac {4 a (c+d x)^{3/2}}{3 b^3 d^3}+\frac {(c+d x)^2}{2 b^2 d^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 772
Rule 1398
Rubi steps
\begin {align*} \int \frac {x^2}{\left (a+b \sqrt {c+d x}\right )^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(-c+x)^2}{\left (a+b \sqrt {x}\right )^2} \, dx,x,c+d x\right )}{d^3}\\ &=\frac {2 \operatorname {Subst}\left (\int \frac {x \left (-c+x^2\right )^2}{(a+b x)^2} \, dx,x,\sqrt {c+d x}\right )}{d^3}\\ &=\frac {2 \operatorname {Subst}\left (\int \left (-\frac {4 a \left (a^2-b^2 c\right )}{b^5}-\frac {\left (-3 a^2+2 b^2 c\right ) x}{b^4}-\frac {2 a x^2}{b^3}+\frac {x^3}{b^2}-\frac {a \left (a^2-b^2 c\right )^2}{b^5 (a+b x)^2}+\frac {5 a^4-6 a^2 b^2 c+b^4 c^2}{b^5 (a+b x)}\right ) \, dx,x,\sqrt {c+d x}\right )}{d^3}\\ &=\frac {\left (3 a^2-2 b^2 c\right ) x}{b^4 d^2}-\frac {8 a \left (a^2-b^2 c\right ) \sqrt {c+d x}}{b^5 d^3}-\frac {4 a (c+d x)^{3/2}}{3 b^3 d^3}+\frac {(c+d x)^2}{2 b^2 d^3}+\frac {2 a \left (a^2-b^2 c\right )^2}{b^6 d^3 \left (a+b \sqrt {c+d x}\right )}+\frac {2 \left (5 a^4-6 a^2 b^2 c+b^4 c^2\right ) \log \left (a+b \sqrt {c+d x}\right )}{b^6 d^3}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 185, normalized size = 1.11 \begin {gather*} \frac {16 a^5-44 a^4 b \sqrt {c+d x}-2 a^3 b^2 (38 c+15 d x)+2 a^2 b^3 \sqrt {c+d x} (18 c+5 d x)+12 \left (5 a^4-6 a^2 b^2 c+b^4 c^2\right ) \left (a+b \sqrt {c+d x}\right ) \log \left (a+b \sqrt {c+d x}\right )+a b^4 \left (52 c^2+26 c d x-5 d^2 x^2\right )+3 b^5 d x (d x-2 c) \sqrt {c+d x}}{6 b^6 d^3 \left (a+b \sqrt {c+d x}\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.14, size = 211, normalized size = 1.27 \begin {gather*} \frac {2 \left (5 a^4-6 a^2 b^2 c+b^4 c^2\right ) \log \left (a+b \sqrt {c+d x}\right )}{b^6 d^3}+\frac {12 a^5-48 a^4 b \sqrt {c+d x}-30 a^3 b^2 (c+d x)-24 a^3 b^2 c+10 a^2 b^3 (c+d x)^{3/2}+48 a^2 b^3 c \sqrt {c+d x}+12 a b^4 c^2-5 a b^4 (c+d x)^2+36 a b^4 c (c+d x)+3 b^5 (c+d x)^{5/2}-12 b^5 c (c+d x)^{3/2}}{6 b^6 d^3 \left (a+b \sqrt {c+d x}\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 269, normalized size = 1.62 \begin {gather*} \frac {3 \, b^{6} d^{3} x^{3} - 9 \, b^{6} c^{3} + 15 \, a^{2} b^{4} c^{2} + 6 \, a^{4} b^{2} c - 12 \, a^{6} - 3 \, {\left (b^{6} c - 5 \, a^{2} b^{4}\right )} d^{2} x^{2} - 3 \, {\left (5 \, b^{6} c^{2} - 14 \, a^{2} b^{4} c + 6 \, a^{4} b^{2}\right )} d x + 12 \, {\left (b^{6} c^{3} - 7 \, a^{2} b^{4} c^{2} + 11 \, a^{4} b^{2} c - 5 \, a^{6} + {\left (b^{6} c^{2} - 6 \, a^{2} b^{4} c + 5 \, a^{4} b^{2}\right )} d x\right )} \log \left (\sqrt {d x + c} b + a\right ) - 4 \, {\left (2 \, a b^{5} d^{2} x^{2} - 13 \, a b^{5} c^{2} + 28 \, a^{3} b^{3} c - 15 \, a^{5} b - 2 \, {\left (4 \, a b^{5} c - 5 \, a^{3} b^{3}\right )} d x\right )} \sqrt {d x + c}}{6 \, {\left (b^{8} d^{4} x + {\left (b^{8} c - a^{2} b^{6}\right )} d^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 191, normalized size = 1.15 \begin {gather*} \frac {2 \, {\left (b^{4} c^{2} - 6 \, a^{2} b^{2} c + 5 \, a^{4}\right )} \log \left ({\left | \sqrt {d x + c} b + a \right |}\right )}{b^{6} d^{3}} + \frac {2 \, {\left (a b^{4} c^{2} - 2 \, a^{3} b^{2} c + a^{5}\right )}}{{\left (\sqrt {d x + c} b + a\right )} b^{6} d^{3}} + \frac {3 \, {\left (d x + c\right )}^{2} b^{6} d^{9} - 12 \, {\left (d x + c\right )} b^{6} c d^{9} - 8 \, {\left (d x + c\right )}^{\frac {3}{2}} a b^{5} d^{9} + 48 \, \sqrt {d x + c} a b^{5} c d^{9} + 18 \, {\left (d x + c\right )} a^{2} b^{4} d^{9} - 48 \, \sqrt {d x + c} a^{3} b^{3} d^{9}}{6 \, b^{8} d^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 253, normalized size = 1.52 \begin {gather*} \frac {x^{2}}{2 b^{2} d}+\frac {2 a \,c^{2}}{\left (a +\sqrt {d x +c}\, b \right ) b^{2} d^{3}}+\frac {2 c^{2} \ln \left (a +\sqrt {d x +c}\, b \right )}{b^{2} d^{3}}-\frac {c x}{b^{2} d^{2}}-\frac {4 a^{3} c}{\left (a +\sqrt {d x +c}\, b \right ) b^{4} d^{3}}-\frac {12 a^{2} c \ln \left (a +\sqrt {d x +c}\, b \right )}{b^{4} d^{3}}+\frac {3 a^{2} x}{b^{4} d^{2}}-\frac {3 c^{2}}{2 b^{2} d^{3}}+\frac {2 a^{5}}{\left (a +\sqrt {d x +c}\, b \right ) b^{6} d^{3}}+\frac {10 a^{4} \ln \left (a +\sqrt {d x +c}\, b \right )}{b^{6} d^{3}}+\frac {3 a^{2} c}{b^{4} d^{3}}+\frac {8 \sqrt {d x +c}\, a c}{b^{3} d^{3}}-\frac {8 \sqrt {d x +c}\, a^{3}}{b^{5} d^{3}}-\frac {4 \left (d x +c \right )^{\frac {3}{2}} a}{3 b^{3} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.93, size = 158, normalized size = 0.95 \begin {gather*} \frac {\frac {12 \, {\left (a b^{4} c^{2} - 2 \, a^{3} b^{2} c + a^{5}\right )}}{\sqrt {d x + c} b^{7} + a b^{6}} + \frac {3 \, {\left (d x + c\right )}^{2} b^{3} - 8 \, {\left (d x + c\right )}^{\frac {3}{2}} a b^{2} - 6 \, {\left (2 \, b^{3} c - 3 \, a^{2} b\right )} {\left (d x + c\right )} + 48 \, {\left (a b^{2} c - a^{3}\right )} \sqrt {d x + c}}{b^{5}} + \frac {12 \, {\left (b^{4} c^{2} - 6 \, a^{2} b^{2} c + 5 \, a^{4}\right )} \log \left (\sqrt {d x + c} b + a\right )}{b^{6}}}{6 \, d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.20, size = 197, normalized size = 1.19 \begin {gather*} \left (\frac {4\,a^3}{b^5\,d^3}+\frac {2\,a\,\left (\frac {4\,c}{b^2\,d^3}-\frac {6\,a^2}{b^4\,d^3}\right )}{b}\right )\,\sqrt {c+d\,x}+\frac {2\,\left (a^5-2\,a^3\,b^2\,c+a\,b^4\,c^2\right )}{b\,\left (b^6\,d^3\,\sqrt {c+d\,x}+a\,b^5\,d^3\right )}+\frac {{\left (c+d\,x\right )}^2}{2\,b^2\,d^3}-d\,x\,\left (\frac {2\,c}{b^2\,d^3}-\frac {3\,a^2}{b^4\,d^3}\right )-\frac {4\,a\,{\left (c+d\,x\right )}^{3/2}}{3\,b^3\,d^3}+\frac {\ln \left (a+b\,\sqrt {c+d\,x}\right )\,\left (10\,a^4-12\,a^2\,b^2\,c+2\,b^4\,c^2\right )}{b^6\,d^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\left (a + b \sqrt {c + d x}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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