Optimal. Leaf size=151 \[ -\frac {a \left (a^2-2 b^2 c\right ) x}{b^4 d^2}+\frac {2 \left (a^2-b^2 c\right )^2 \sqrt {c+d x}}{b^5 d^3}+\frac {2 \left (a^2-2 b^2 c\right ) (c+d x)^{3/2}}{3 b^3 d^3}-\frac {a (c+d x)^2}{2 b^2 d^3}+\frac {2 (c+d x)^{5/2}}{5 b d^3}-\frac {2 a \left (a^2-b^2 c\right )^2 \log \left (a+b \sqrt {c+d x}\right )}{b^6 d^3} \]
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Rubi [A]
time = 0.12, 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 = {378, 1412, 786}
\begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 378
Rule 786
Rule 1412
Rubi steps
\begin {align*} \int \frac {x^2}{a+b \sqrt {c+d x}} \, dx &=\frac {\text {Subst}\left (\int \frac {(-c+x)^2}{a+b \sqrt {x}} \, dx,x,c+d x\right )}{d^3}\\ &=\frac {2 \text {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 \text {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 = 0.11, size = 151, normalized size = 1.00 \begin {gather*} \frac {b \left (60 a^4 \sqrt {c+d x}-20 a^2 b^2 (5 c-d x) \sqrt {c+d x}-30 a^3 b (c+d x)+15 a b^3 \left (3 c^2+2 c d x-d^2 x^2\right )+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} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 166, normalized size = 1.10
method | result | size |
derivativedivides | \(\frac {\frac {2 \left (\frac {\left (d x +c \right )^{\frac {5}{2}} b^{4}}{5}-\frac {a \left (d x +c \right )^{2} b^{3}}{4}-\frac {2 b^{4} c \left (d x +c \right )^{\frac {3}{2}}}{3}+\frac {a^{2} b^{2} \left (d x +c \right )^{\frac {3}{2}}}{3}+a \,b^{3} c \left (d x +c \right )+b^{4} c^{2} \sqrt {d x +c}-\frac {a^{3} b \left (d x +c \right )}{2}-2 a^{2} b^{2} c \sqrt {d x +c}+a^{4} \sqrt {d x +c}\right )}{b^{5}}-\frac {2 a \left (b^{4} c^{2}-2 a^{2} b^{2} c +a^{4}\right ) \ln \left (a +b \sqrt {d x +c}\right )}{b^{6}}}{d^{3}}\) | \(166\) |
default | \(\frac {\frac {2 \left (\frac {\left (d x +c \right )^{\frac {5}{2}} b^{4}}{5}-\frac {a \left (d x +c \right )^{2} b^{3}}{4}-\frac {2 b^{4} c \left (d x +c \right )^{\frac {3}{2}}}{3}+\frac {a^{2} b^{2} \left (d x +c \right )^{\frac {3}{2}}}{3}+a \,b^{3} c \left (d x +c \right )+b^{4} c^{2} \sqrt {d x +c}-\frac {a^{3} b \left (d x +c \right )}{2}-2 a^{2} b^{2} c \sqrt {d x +c}+a^{4} \sqrt {d x +c}\right )}{b^{5}}-\frac {2 a \left (b^{4} c^{2}-2 a^{2} b^{2} c +a^{4}\right ) \ln \left (a +b \sqrt {d x +c}\right )}{b^{6}}}{d^{3}}\) | \(166\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 148, normalized size = 0.98 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 138, normalized size = 0.91 \begin {gather*} -\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}} \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}}{a + b \sqrt {c + d x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.90, size = 198, normalized size = 1.31 \begin {gather*} -\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}} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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