Optimal. Leaf size=151 \[ -\frac {2 a \left (a^2-b^2 c\right )^2 \log \left (a+b \sqrt {c+d x}\right )}{b^6 d^3}+\frac {2 \left (a^2-b^2 c\right )^2 \sqrt {c+d x}}{b^5 d^3}-\frac {a x \left (a^2-2 b^2 c\right )}{b^4 d^2}+\frac {2 \left (a^2-2 b^2 c\right ) (c+d x)^{3/2}}{3 b^3 d^3}-\frac {a (c+d x)^2}{2 b^2 d^3}+\frac {2 (c+d x)^{5/2}}{5 b d^3} \]
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Rubi [A] time = 0.16, antiderivative size = 151, 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} \[ \frac {2 \left (a^2-2 b^2 c\right ) (c+d x)^{3/2}}{3 b^3 d^3}+\frac {2 \left (a^2-b^2 c\right )^2 \sqrt {c+d x}}{b^5 d^3}-\frac {a x \left (a^2-2 b^2 c\right )}{b^4 d^2}-\frac {2 a \left (a^2-b^2 c\right )^2 \log \left (a+b \sqrt {c+d x}\right )}{b^6 d^3}-\frac {a (c+d x)^2}{2 b^2 d^3}+\frac {2 (c+d x)^{5/2}}{5 b d^3} \]
Antiderivative was successfully verified.
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Rule 371
Rule 772
Rule 1398
Rubi steps
\begin {align*} \int \frac {x^2}{a+b \sqrt {c+d x}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(-c+x)^2}{a+b \sqrt {x}} \, 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} \, dx,x,\sqrt {c+d x}\right )}{d^3}\\ &=\frac {2 \operatorname {Subst}\left (\int \left (\frac {\left (-a^2+b^2 c\right )^2}{b^5}-\frac {a \left (a^2-2 b^2 c\right ) x}{b^4}-\frac {\left (-a^2+2 b^2 c\right ) x^2}{b^3}-\frac {a x^3}{b^2}+\frac {x^4}{b}-\frac {a \left (a^2-b^2 c\right )^2}{b^5 (a+b x)}\right ) \, dx,x,\sqrt {c+d x}\right )}{d^3}\\ &=-\frac {a \left (a^2-2 b^2 c\right ) x}{b^4 d^2}+\frac {2 \left (a^2-b^2 c\right )^2 \sqrt {c+d x}}{b^5 d^3}+\frac {2 \left (a^2-2 b^2 c\right ) (c+d x)^{3/2}}{3 b^3 d^3}-\frac {a (c+d x)^2}{2 b^2 d^3}+\frac {2 (c+d x)^{5/2}}{5 b d^3}-\frac {2 a \left (a^2-b^2 c\right )^2 \log \left (a+b \sqrt {c+d x}\right )}{b^6 d^3}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 138, normalized size = 0.91 \[ \frac {b \left (60 a^4 \sqrt {c+d x}-30 a^3 b d x-20 a^2 b^2 (5 c-d x) \sqrt {c+d x}-15 a b^3 d x (d x-2 c)+4 b^4 \sqrt {c+d x} \left (8 c^2-4 c d x+3 d^2 x^2\right )\right )-60 a \left (a^2-b^2 c\right )^2 \log \left (a+b \sqrt {c+d x}\right )}{30 b^6 d^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 138, normalized size = 0.91 \[ -\frac {15 \, a b^{4} d^{2} x^{2} - 30 \, {\left (a b^{4} c - a^{3} b^{2}\right )} d x + 60 \, {\left (a b^{4} c^{2} - 2 \, a^{3} b^{2} c + a^{5}\right )} \log \left (\sqrt {d x + c} b + a\right ) - 4 \, {\left (3 \, b^{5} d^{2} x^{2} + 8 \, b^{5} c^{2} - 25 \, a^{2} b^{3} c + 15 \, a^{4} b - {\left (4 \, b^{5} c - 5 \, a^{2} b^{3}\right )} d x\right )} \sqrt {d x + c}}{30 \, b^{6} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.41, size = 198, normalized size = 1.31 \[ -\frac {2 \, {\left (a b^{4} c^{2} - 2 \, a^{3} b^{2} c + a^{5}\right )} \log \left ({\left | \sqrt {d x + c} b + a \right |}\right )}{b^{6} d^{3}} + \frac {12 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{4} d^{12} - 40 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{4} c d^{12} + 60 \, \sqrt {d x + c} b^{4} c^{2} d^{12} - 15 \, {\left (d x + c\right )}^{2} a b^{3} d^{12} + 60 \, {\left (d x + c\right )} a b^{3} c d^{12} + 20 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{2} b^{2} d^{12} - 120 \, \sqrt {d x + c} a^{2} b^{2} c d^{12} - 30 \, {\left (d x + c\right )} a^{3} b d^{12} + 60 \, \sqrt {d x + c} a^{4} d^{12}}{30 \, b^{5} d^{15}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 235, normalized size = 1.56 \[ -\frac {a \,x^{2}}{2 b^{2} d}-\frac {2 a \,c^{2} \ln \left (a +\sqrt {d x +c}\, b \right )}{b^{2} d^{3}}+\frac {a c x}{b^{2} d^{2}}+\frac {4 a^{3} c \ln \left (a +\sqrt {d x +c}\, b \right )}{b^{4} d^{3}}-\frac {a^{3} x}{b^{4} d^{2}}+\frac {3 a \,c^{2}}{2 b^{2} d^{3}}+\frac {2 \sqrt {d x +c}\, c^{2}}{b \,d^{3}}-\frac {2 a^{5} \ln \left (a +\sqrt {d x +c}\, b \right )}{b^{6} d^{3}}-\frac {a^{3} c}{b^{4} d^{3}}-\frac {4 \sqrt {d x +c}\, a^{2} c}{b^{3} d^{3}}-\frac {4 \left (d x +c \right )^{\frac {3}{2}} c}{3 b \,d^{3}}+\frac {2 \sqrt {d x +c}\, a^{4}}{b^{5} d^{3}}+\frac {2 \left (d x +c \right )^{\frac {3}{2}} a^{2}}{3 b^{3} d^{3}}+\frac {2 \left (d x +c \right )^{\frac {5}{2}}}{5 b \,d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.88, size = 148, normalized size = 0.98 \[ \frac {\frac {12 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{4} - 15 \, {\left (d x + c\right )}^{2} a b^{3} - 20 \, {\left (2 \, b^{4} c - a^{2} b^{2}\right )} {\left (d x + c\right )}^{\frac {3}{2}} + 30 \, {\left (2 \, a b^{3} c - a^{3} b\right )} {\left (d x + c\right )} + 60 \, {\left (b^{4} c^{2} - 2 \, a^{2} b^{2} c + a^{4}\right )} \sqrt {d x + c}}{b^{5}} - \frac {60 \, {\left (a b^{4} c^{2} - 2 \, a^{3} b^{2} c + a^{5}\right )} \log \left (\sqrt {d x + c} b + a\right )}{b^{6}}}{30 \, d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.21, size = 184, normalized size = 1.22 \[ \frac {2\,{\left (c+d\,x\right )}^{5/2}}{5\,b\,d^3}-\left (\frac {a^2\,\left (\frac {4\,c}{b\,d^3}-\frac {2\,a^2}{b^3\,d^3}\right )}{b^2}-\frac {2\,c^2}{b\,d^3}\right )\,\sqrt {c+d\,x}-\left (\frac {4\,c}{3\,b\,d^3}-\frac {2\,a^2}{3\,b^3\,d^3}\right )\,{\left (c+d\,x\right )}^{3/2}-\frac {\ln \left (a+b\,\sqrt {c+d\,x}\right )\,\left (2\,a^5-4\,a^3\,b^2\,c+2\,a\,b^4\,c^2\right )}{b^6\,d^3}-\frac {a\,{\left (c+d\,x\right )}^2}{2\,b^2\,d^3}+\frac {a\,d\,x\,\left (\frac {4\,c}{b\,d^3}-\frac {2\,a^2}{b^3\,d^3}\right )}{2\,b} \]
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 \[ \int \frac {x^{2}}{a + b \sqrt {c + d x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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