Integrand size = 22, antiderivative size = 191 \[ \int \frac {x^2 \sqrt {a+b x}}{\sqrt {c+d x}} \, dx=\frac {\left (5 b^2 c^2+2 a b c d+a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 b^2 d^3}-\frac {(5 b c+3 a d) (a+b x)^{3/2} \sqrt {c+d x}}{12 b^2 d^2}+\frac {x (a+b x)^{3/2} \sqrt {c+d x}}{3 b d}-\frac {(b c-a d) \left (5 b^2 c^2+2 a b c d+a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{5/2} d^{7/2}} \]
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Time = 0.11 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {92, 81, 52, 65, 223, 212} \[ \int \frac {x^2 \sqrt {a+b x}}{\sqrt {c+d x}} \, dx=-\frac {(b c-a d) \left (a^2 d^2+2 a b c d+5 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{5/2} d^{7/2}}+\frac {\sqrt {a+b x} \sqrt {c+d x} \left (a^2 d^2+2 a b c d+5 b^2 c^2\right )}{8 b^2 d^3}-\frac {(a+b x)^{3/2} \sqrt {c+d x} (3 a d+5 b c)}{12 b^2 d^2}+\frac {x (a+b x)^{3/2} \sqrt {c+d x}}{3 b d} \]
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Rule 52
Rule 65
Rule 81
Rule 92
Rule 212
Rule 223
Rubi steps \begin{align*} \text {integral}& = \frac {x (a+b x)^{3/2} \sqrt {c+d x}}{3 b d}+\frac {\int \frac {\sqrt {a+b x} \left (-a c-\frac {1}{2} (5 b c+3 a d) x\right )}{\sqrt {c+d x}} \, dx}{3 b d} \\ & = -\frac {(5 b c+3 a d) (a+b x)^{3/2} \sqrt {c+d x}}{12 b^2 d^2}+\frac {x (a+b x)^{3/2} \sqrt {c+d x}}{3 b d}+\frac {\left (5 b^2 c^2+2 a b c d+a^2 d^2\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{8 b^2 d^2} \\ & = \frac {\left (5 b^2 c^2+2 a b c d+a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 b^2 d^3}-\frac {(5 b c+3 a d) (a+b x)^{3/2} \sqrt {c+d x}}{12 b^2 d^2}+\frac {x (a+b x)^{3/2} \sqrt {c+d x}}{3 b d}-\frac {\left ((b c-a d) \left (5 b^2 c^2+2 a b c d+a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{16 b^2 d^3} \\ & = \frac {\left (5 b^2 c^2+2 a b c d+a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 b^2 d^3}-\frac {(5 b c+3 a d) (a+b x)^{3/2} \sqrt {c+d x}}{12 b^2 d^2}+\frac {x (a+b x)^{3/2} \sqrt {c+d x}}{3 b d}-\frac {\left ((b c-a d) \left (5 b^2 c^2+2 a b c d+a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{8 b^3 d^3} \\ & = \frac {\left (5 b^2 c^2+2 a b c d+a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 b^2 d^3}-\frac {(5 b c+3 a d) (a+b x)^{3/2} \sqrt {c+d x}}{12 b^2 d^2}+\frac {x (a+b x)^{3/2} \sqrt {c+d x}}{3 b d}-\frac {\left ((b c-a d) \left (5 b^2 c^2+2 a b c d+a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{8 b^3 d^3} \\ & = \frac {\left (5 b^2 c^2+2 a b c d+a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 b^2 d^3}-\frac {(5 b c+3 a d) (a+b x)^{3/2} \sqrt {c+d x}}{12 b^2 d^2}+\frac {x (a+b x)^{3/2} \sqrt {c+d x}}{3 b d}-\frac {(b c-a d) \left (5 b^2 c^2+2 a b c d+a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{5/2} d^{7/2}} \\ \end{align*}
Time = 10.30 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.91 \[ \int \frac {x^2 \sqrt {a+b x}}{\sqrt {c+d x}} \, dx=\frac {-b \sqrt {d} \sqrt {a+b x} (c+d x) \left (3 a^2 d^2-2 a b d (-2 c+d x)+b^2 \left (-15 c^2+10 c d x-8 d^2 x^2\right )\right )-3 (b c-a d)^{3/2} \left (5 b^2 c^2+2 a b c d+a^2 d^2\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \text {arcsinh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{24 b^3 d^{7/2} \sqrt {c+d x}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(394\) vs. \(2(159)=318\).
Time = 0.54 (sec) , antiderivative size = 395, normalized size of antiderivative = 2.07
method | result | size |
default | \(\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (16 b^{2} d^{2} x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{3} d^{3}+3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} b c \,d^{2}+9 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,b^{2} c^{2} d -15 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{3} c^{3}+4 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b \,d^{2} x -20 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c d x -6 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} d^{2}-8 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b c d +30 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c^{2}\right )}{48 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, d^{3} b^{2} \sqrt {b d}}\) | \(395\) |
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Time = 0.26 (sec) , antiderivative size = 414, normalized size of antiderivative = 2.17 \[ \int \frac {x^2 \sqrt {a+b x}}{\sqrt {c+d x}} \, dx=\left [-\frac {3 \, {\left (5 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (8 \, b^{3} d^{3} x^{2} + 15 \, b^{3} c^{2} d - 4 \, a b^{2} c d^{2} - 3 \, a^{2} b d^{3} - 2 \, {\left (5 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{96 \, b^{3} d^{4}}, \frac {3 \, {\left (5 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) + 2 \, {\left (8 \, b^{3} d^{3} x^{2} + 15 \, b^{3} c^{2} d - 4 \, a b^{2} c d^{2} - 3 \, a^{2} b d^{3} - 2 \, {\left (5 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, b^{3} d^{4}}\right ] \]
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\[ \int \frac {x^2 \sqrt {a+b x}}{\sqrt {c+d x}} \, dx=\int \frac {x^{2} \sqrt {a + b x}}{\sqrt {c + d x}}\, dx \]
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Exception generated. \[ \int \frac {x^2 \sqrt {a+b x}}{\sqrt {c+d x}} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.32 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.12 \[ \int \frac {x^2 \sqrt {a+b x}}{\sqrt {c+d x}} \, dx=\frac {{\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a} {\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (b x + a\right )}}{b^{3} d} - \frac {5 \, b^{7} c d^{3} + 7 \, a b^{6} d^{4}}{b^{9} d^{5}}\right )} + \frac {3 \, {\left (5 \, b^{8} c^{2} d^{2} + 2 \, a b^{7} c d^{3} + a^{2} b^{6} d^{4}\right )}}{b^{9} d^{5}}\right )} + \frac {3 \, {\left (5 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} b^{2} d^{3}}\right )} b}{24 \, {\left | b \right |}} \]
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Time = 51.84 (sec) , antiderivative size = 929, normalized size of antiderivative = 4.86 \[ \int \frac {x^2 \sqrt {a+b x}}{\sqrt {c+d x}} \, dx=\frac {\mathrm {atanh}\left (\frac {\sqrt {d}\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\sqrt {b}\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}\right )\,\left (a\,d-b\,c\right )\,\left (a^2\,d^2+2\,a\,b\,c\,d+5\,b^2\,c^2\right )}{4\,b^{5/2}\,d^{7/2}}-\frac {\frac {\left (\sqrt {a+b\,x}-\sqrt {a}\right )\,\left (\frac {a^3\,b^3\,d^3}{4}+\frac {a^2\,b^4\,c\,d^2}{4}+\frac {3\,a\,b^5\,c^2\,d}{4}-\frac {5\,b^6\,c^3}{4}\right )}{d^9\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}-\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3\,\left (\frac {17\,a^3\,b^2\,d^3}{12}+\frac {91\,a^2\,b^3\,c\,d^2}{4}+\frac {17\,a\,b^4\,c^2\,d}{4}-\frac {85\,b^5\,c^3}{12}\right )}{d^8\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^3}-\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^7\,\left (\frac {19\,a^3\,d^3}{2}+\frac {275\,a^2\,b\,c\,d^2}{2}+\frac {313\,a\,b^2\,c^2\,d}{2}+\frac {33\,b^3\,c^3}{2}\right )}{d^6\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^7}-\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^5\,\left (\frac {19\,a^3\,b\,d^3}{2}+\frac {275\,a^2\,b^2\,c\,d^2}{2}+\frac {313\,a\,b^3\,c^2\,d}{2}+\frac {33\,b^4\,c^3}{2}\right )}{d^7\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^5}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^{11}\,\left (\frac {a^3\,d^3}{4}+\frac {a^2\,b\,c\,d^2}{4}+\frac {3\,a\,b^2\,c^2\,d}{4}-\frac {5\,b^3\,c^3}{4}\right )}{b^2\,d^4\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{11}}-\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^9\,\left (\frac {17\,a^3\,d^3}{12}+\frac {91\,a^2\,b\,c\,d^2}{4}+\frac {17\,a\,b^2\,c^2\,d}{4}-\frac {85\,b^3\,c^3}{12}\right )}{b\,d^5\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^9}+\frac {\sqrt {a}\,\sqrt {c}\,\left (32\,d\,a^2+96\,b\,c\,a\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^8}{d^4\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^8}+\frac {\sqrt {a}\,\sqrt {c}\,\left (32\,d\,a^2\,b^2+96\,c\,a\,b^3\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}{d^6\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^4}+\frac {\sqrt {a}\,\sqrt {c}\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^6\,\left (64\,a^2\,b\,d^2+\frac {704\,a\,b^2\,c\,d}{3}+128\,b^3\,c^2\right )}{d^6\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^6}}{\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^{12}}{{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{12}}+\frac {b^6}{d^6}-\frac {6\,b^5\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{d^5\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}+\frac {15\,b^4\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}{d^4\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^4}-\frac {20\,b^3\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^6}{d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^6}+\frac {15\,b^2\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^8}{d^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^8}-\frac {6\,b\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^{10}}{d\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{10}}} \]
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