Integrand size = 22, antiderivative size = 377 \[ \int \frac {x^3 (a+b x)^{5/2}}{(c+d x)^{5/2}} \, dx=-\frac {2 x^3 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}}-\frac {2 (11 b c-6 a d) x^2 (a+b x)^{5/2}}{3 d^2 (b c-a d) \sqrt {c+d x}}-\frac {5 \left (231 b^3 c^3-189 a b^2 c^2 d+21 a^2 b c d^2+a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b d^6}+\frac {5 \left (231 b^3 c^3-189 a b^2 c^2 d+21 a^2 b c d^2+a^3 d^3\right ) (a+b x)^{3/2} \sqrt {c+d x}}{96 b d^5 (b c-a d)}-\frac {(a+b x)^{5/2} \sqrt {c+d x} \left (231 b^2 c^2-156 a b c d+5 a^2 d^2-2 b d (99 b c-59 a d) x\right )}{24 b d^4 (b c-a d)}+\frac {5 (b c-a d) \left (231 b^3 c^3-189 a b^2 c^2 d+21 a^2 b c d^2+a^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{3/2} d^{13/2}} \]
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Time = 0.29 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {99, 155, 152, 52, 65, 223, 212} \[ \int \frac {x^3 (a+b x)^{5/2}}{(c+d x)^{5/2}} \, dx=-\frac {(a+b x)^{5/2} \sqrt {c+d x} \left (5 a^2 d^2-2 b d x (99 b c-59 a d)-156 a b c d+231 b^2 c^2\right )}{24 b d^4 (b c-a d)}+\frac {5 (b c-a d) \left (a^3 d^3+21 a^2 b c d^2-189 a b^2 c^2 d+231 b^3 c^3\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{3/2} d^{13/2}}-\frac {5 \sqrt {a+b x} \sqrt {c+d x} \left (a^3 d^3+21 a^2 b c d^2-189 a b^2 c^2 d+231 b^3 c^3\right )}{64 b d^6}+\frac {5 (a+b x)^{3/2} \sqrt {c+d x} \left (a^3 d^3+21 a^2 b c d^2-189 a b^2 c^2 d+231 b^3 c^3\right )}{96 b d^5 (b c-a d)}-\frac {2 x^2 (a+b x)^{5/2} (11 b c-6 a d)}{3 d^2 \sqrt {c+d x} (b c-a d)}-\frac {2 x^3 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}} \]
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Rule 52
Rule 65
Rule 99
Rule 152
Rule 155
Rule 212
Rule 223
Rubi steps \begin{align*} \text {integral}& = -\frac {2 x^3 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}}+\frac {2 \int \frac {x^2 (a+b x)^{3/2} \left (3 a+\frac {11 b x}{2}\right )}{(c+d x)^{3/2}} \, dx}{3 d} \\ & = -\frac {2 x^3 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}}-\frac {2 (11 b c-6 a d) x^2 (a+b x)^{5/2}}{3 d^2 (b c-a d) \sqrt {c+d x}}-\frac {4 \int \frac {x (a+b x)^{3/2} \left (-a (11 b c-6 a d)-\frac {1}{4} b (99 b c-59 a d) x\right )}{\sqrt {c+d x}} \, dx}{3 d^2 (b c-a d)} \\ & = -\frac {2 x^3 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}}-\frac {2 (11 b c-6 a d) x^2 (a+b x)^{5/2}}{3 d^2 (b c-a d) \sqrt {c+d x}}-\frac {(a+b x)^{5/2} \sqrt {c+d x} \left (231 b^2 c^2-156 a b c d+5 a^2 d^2-2 b d (99 b c-59 a d) x\right )}{24 b d^4 (b c-a d)}+\frac {\left (5 \left (231 b^3 c^3-189 a b^2 c^2 d+21 a^2 b c d^2+a^3 d^3\right )\right ) \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx}{48 b d^4 (b c-a d)} \\ & = -\frac {2 x^3 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}}-\frac {2 (11 b c-6 a d) x^2 (a+b x)^{5/2}}{3 d^2 (b c-a d) \sqrt {c+d x}}+\frac {5 \left (231 b^3 c^3-189 a b^2 c^2 d+21 a^2 b c d^2+a^3 d^3\right ) (a+b x)^{3/2} \sqrt {c+d x}}{96 b d^5 (b c-a d)}-\frac {(a+b x)^{5/2} \sqrt {c+d x} \left (231 b^2 c^2-156 a b c d+5 a^2 d^2-2 b d (99 b c-59 a d) x\right )}{24 b d^4 (b c-a d)}-\frac {\left (5 \left (231 b^3 c^3-189 a b^2 c^2 d+21 a^2 b c d^2+a^3 d^3\right )\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{64 b d^5} \\ & = -\frac {2 x^3 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}}-\frac {2 (11 b c-6 a d) x^2 (a+b x)^{5/2}}{3 d^2 (b c-a d) \sqrt {c+d x}}-\frac {5 \left (231 b^3 c^3-189 a b^2 c^2 d+21 a^2 b c d^2+a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b d^6}+\frac {5 \left (231 b^3 c^3-189 a b^2 c^2 d+21 a^2 b c d^2+a^3 d^3\right ) (a+b x)^{3/2} \sqrt {c+d x}}{96 b d^5 (b c-a d)}-\frac {(a+b x)^{5/2} \sqrt {c+d x} \left (231 b^2 c^2-156 a b c d+5 a^2 d^2-2 b d (99 b c-59 a d) x\right )}{24 b d^4 (b c-a d)}+\frac {\left (5 (b c-a d) \left (231 b^3 c^3-189 a b^2 c^2 d+21 a^2 b c d^2+a^3 d^3\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{128 b d^6} \\ & = -\frac {2 x^3 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}}-\frac {2 (11 b c-6 a d) x^2 (a+b x)^{5/2}}{3 d^2 (b c-a d) \sqrt {c+d x}}-\frac {5 \left (231 b^3 c^3-189 a b^2 c^2 d+21 a^2 b c d^2+a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b d^6}+\frac {5 \left (231 b^3 c^3-189 a b^2 c^2 d+21 a^2 b c d^2+a^3 d^3\right ) (a+b x)^{3/2} \sqrt {c+d x}}{96 b d^5 (b c-a d)}-\frac {(a+b x)^{5/2} \sqrt {c+d x} \left (231 b^2 c^2-156 a b c d+5 a^2 d^2-2 b d (99 b c-59 a d) x\right )}{24 b d^4 (b c-a d)}+\frac {\left (5 (b c-a d) \left (231 b^3 c^3-189 a b^2 c^2 d+21 a^2 b c d^2+a^3 d^3\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 )}{64 b^2 d^6} \\ & = -\frac {2 x^3 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}}-\frac {2 (11 b c-6 a d) x^2 (a+b x)^{5/2}}{3 d^2 (b c-a d) \sqrt {c+d x}}-\frac {5 \left (231 b^3 c^3-189 a b^2 c^2 d+21 a^2 b c d^2+a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b d^6}+\frac {5 \left (231 b^3 c^3-189 a b^2 c^2 d+21 a^2 b c d^2+a^3 d^3\right ) (a+b x)^{3/2} \sqrt {c+d x}}{96 b d^5 (b c-a d)}-\frac {(a+b x)^{5/2} \sqrt {c+d x} \left (231 b^2 c^2-156 a b c d+5 a^2 d^2-2 b d (99 b c-59 a d) x\right )}{24 b d^4 (b c-a d)}+\frac {\left (5 (b c-a d) \left (231 b^3 c^3-189 a b^2 c^2 d+21 a^2 b c d^2+a^3 d^3\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 )}{64 b^2 d^6} \\ & = -\frac {2 x^3 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}}-\frac {2 (11 b c-6 a d) x^2 (a+b x)^{5/2}}{3 d^2 (b c-a d) \sqrt {c+d x}}-\frac {5 \left (231 b^3 c^3-189 a b^2 c^2 d+21 a^2 b c d^2+a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b d^6}+\frac {5 \left (231 b^3 c^3-189 a b^2 c^2 d+21 a^2 b c d^2+a^3 d^3\right ) (a+b x)^{3/2} \sqrt {c+d x}}{96 b d^5 (b c-a d)}-\frac {(a+b x)^{5/2} \sqrt {c+d x} \left (231 b^2 c^2-156 a b c d+5 a^2 d^2-2 b d (99 b c-59 a d) x\right )}{24 b d^4 (b c-a d)}+\frac {5 (b c-a d) \left (231 b^3 c^3-189 a b^2 c^2 d+21 a^2 b c d^2+a^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{3/2} d^{13/2}} \\ \end{align*}
Time = 11.73 (sec) , antiderivative size = 331, normalized size of antiderivative = 0.88 \[ \int \frac {x^3 (a+b x)^{5/2}}{(c+d x)^{5/2}} \, dx=\frac {48 x^2 (a+b x)^4+\frac {16 c (a+b x)^4 \left (3 a^2 d^2 (c+2 d x)-14 a b c d (5 c+6 d x)+11 b^2 c^2 (9 c+10 d x)\right )}{d^2 (b c-a d)^2}-\frac {\left (231 b^3 c^3-189 a b^2 c^2 d+21 a^2 b c d^2+a^3 d^3\right ) (c+d x)^2 \left (\sqrt {d} \sqrt {b c-a d} (a+b x) \sqrt {\frac {b (c+d x)}{b c-a d}} \left (33 a^2 d^2+2 a b d (-20 c+13 d x)+b^2 \left (15 c^2-10 c d x+8 d^2 x^2\right )\right )-15 (b c-a d)^3 \sqrt {a+b x} \text {arcsinh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )\right )}{d^{11/2} (b c-a d)^{5/2} \sqrt {\frac {b (c+d x)}{b c-a d}}}}{192 b d \sqrt {a+b x} (c+d x)^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1365\) vs. \(2(333)=666\).
Time = 0.56 (sec) , antiderivative size = 1366, normalized size of antiderivative = 3.62
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Time = 1.08 (sec) , antiderivative size = 1066, normalized size of antiderivative = 2.83 \[ \int \frac {x^3 (a+b x)^{5/2}}{(c+d x)^{5/2}} \, dx=\left [-\frac {15 \, {\left (231 \, b^{4} c^{6} - 420 \, a b^{3} c^{5} d + 210 \, a^{2} b^{2} c^{4} d^{2} - 20 \, a^{3} b c^{3} d^{3} - a^{4} c^{2} d^{4} + {\left (231 \, b^{4} c^{4} d^{2} - 420 \, a b^{3} c^{3} d^{3} + 210 \, a^{2} b^{2} c^{2} d^{4} - 20 \, a^{3} b c d^{5} - a^{4} d^{6}\right )} x^{2} + 2 \, {\left (231 \, b^{4} c^{5} d - 420 \, a b^{3} c^{4} d^{2} + 210 \, a^{2} b^{2} c^{3} d^{3} - 20 \, a^{3} b c^{2} d^{4} - a^{4} c d^{5}\right )} x\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 (48 \, b^{4} d^{6} x^{5} - 3465 \, b^{4} c^{5} d + 5145 \, a b^{3} c^{4} d^{2} - 1743 \, a^{2} b^{2} c^{3} d^{3} + 15 \, a^{3} b c^{2} d^{4} - 8 \, {\left (11 \, b^{4} c d^{5} - 17 \, a b^{3} d^{6}\right )} x^{4} + 2 \, {\left (99 \, b^{4} c^{2} d^{4} - 158 \, a b^{3} c d^{5} + 59 \, a^{2} b^{2} d^{6}\right )} x^{3} - 3 \, {\left (231 \, b^{4} c^{3} d^{3} - 387 \, a b^{3} c^{2} d^{4} + 161 \, a^{2} b^{2} c d^{5} - 5 \, a^{3} b d^{6}\right )} x^{2} - 6 \, {\left (770 \, b^{4} c^{4} d^{2} - 1169 \, a b^{3} c^{3} d^{3} + 412 \, a^{2} b^{2} c^{2} d^{4} - 5 \, a^{3} b c d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{768 \, {\left (b^{2} d^{9} x^{2} + 2 \, b^{2} c d^{8} x + b^{2} c^{2} d^{7}\right )}}, -\frac {15 \, {\left (231 \, b^{4} c^{6} - 420 \, a b^{3} c^{5} d + 210 \, a^{2} b^{2} c^{4} d^{2} - 20 \, a^{3} b c^{3} d^{3} - a^{4} c^{2} d^{4} + {\left (231 \, b^{4} c^{4} d^{2} - 420 \, a b^{3} c^{3} d^{3} + 210 \, a^{2} b^{2} c^{2} d^{4} - 20 \, a^{3} b c d^{5} - a^{4} d^{6}\right )} x^{2} + 2 \, {\left (231 \, b^{4} c^{5} d - 420 \, a b^{3} c^{4} d^{2} + 210 \, a^{2} b^{2} c^{3} d^{3} - 20 \, a^{3} b c^{2} d^{4} - a^{4} c d^{5}\right )} x\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 (48 \, b^{4} d^{6} x^{5} - 3465 \, b^{4} c^{5} d + 5145 \, a b^{3} c^{4} d^{2} - 1743 \, a^{2} b^{2} c^{3} d^{3} + 15 \, a^{3} b c^{2} d^{4} - 8 \, {\left (11 \, b^{4} c d^{5} - 17 \, a b^{3} d^{6}\right )} x^{4} + 2 \, {\left (99 \, b^{4} c^{2} d^{4} - 158 \, a b^{3} c d^{5} + 59 \, a^{2} b^{2} d^{6}\right )} x^{3} - 3 \, {\left (231 \, b^{4} c^{3} d^{3} - 387 \, a b^{3} c^{2} d^{4} + 161 \, a^{2} b^{2} c d^{5} - 5 \, a^{3} b d^{6}\right )} x^{2} - 6 \, {\left (770 \, b^{4} c^{4} d^{2} - 1169 \, a b^{3} c^{3} d^{3} + 412 \, a^{2} b^{2} c^{2} d^{4} - 5 \, a^{3} b c d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{384 \, {\left (b^{2} d^{9} x^{2} + 2 \, b^{2} c d^{8} x + b^{2} c^{2} d^{7}\right )}}\right ] \]
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\[ \int \frac {x^3 (a+b x)^{5/2}}{(c+d x)^{5/2}} \, dx=\int \frac {x^{3} \left (a + b x\right )^{\frac {5}{2}}}{\left (c + d x\right )^{\frac {5}{2}}}\, dx \]
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Exception generated. \[ \int \frac {x^3 (a+b x)^{5/2}}{(c+d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 686 vs. \(2 (333) = 666\).
Time = 0.43 (sec) , antiderivative size = 686, normalized size of antiderivative = 1.82 \[ \int \frac {x^3 (a+b x)^{5/2}}{(c+d x)^{5/2}} \, dx=\frac {{\left ({\left ({\left (2 \, {\left (4 \, {\left (b x + a\right )} {\left (\frac {6 \, {\left (b^{5} c d^{10} {\left | b \right |} - a b^{4} d^{11} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{6} c d^{11} - a b^{5} d^{12}} - \frac {11 \, b^{6} c^{2} d^{9} {\left | b \right |} + 2 \, a b^{5} c d^{10} {\left | b \right |} - 13 \, a^{2} b^{4} d^{11} {\left | b \right |}}{b^{6} c d^{11} - a b^{5} d^{12}}\right )} + \frac {9 \, {\left (11 \, b^{7} c^{3} d^{8} {\left | b \right |} - 9 \, a b^{6} c^{2} d^{9} {\left | b \right |} + a^{2} b^{5} c d^{10} {\left | b \right |} - 3 \, a^{3} b^{4} d^{11} {\left | b \right |}\right )}}{b^{6} c d^{11} - a b^{5} d^{12}}\right )} {\left (b x + a\right )} - \frac {3 \, {\left (231 \, b^{8} c^{4} d^{7} {\left | b \right |} - 420 \, a b^{7} c^{3} d^{8} {\left | b \right |} + 210 \, a^{2} b^{6} c^{2} d^{9} {\left | b \right |} - 20 \, a^{3} b^{5} c d^{10} {\left | b \right |} - a^{4} b^{4} d^{11} {\left | b \right |}\right )}}{b^{6} c d^{11} - a b^{5} d^{12}}\right )} {\left (b x + a\right )} - \frac {20 \, {\left (231 \, b^{9} c^{5} d^{6} {\left | b \right |} - 651 \, a b^{8} c^{4} d^{7} {\left | b \right |} + 630 \, a^{2} b^{7} c^{3} d^{8} {\left | b \right |} - 230 \, a^{3} b^{6} c^{2} d^{9} {\left | b \right |} + 19 \, a^{4} b^{5} c d^{10} {\left | b \right |} + a^{5} b^{4} d^{11} {\left | b \right |}\right )}}{b^{6} c d^{11} - a b^{5} d^{12}}\right )} {\left (b x + a\right )} - \frac {15 \, {\left (231 \, b^{10} c^{6} d^{5} {\left | b \right |} - 882 \, a b^{9} c^{5} d^{6} {\left | b \right |} + 1281 \, a^{2} b^{8} c^{4} d^{7} {\left | b \right |} - 860 \, a^{3} b^{7} c^{3} d^{8} {\left | b \right |} + 249 \, a^{4} b^{6} c^{2} d^{9} {\left | b \right |} - 18 \, a^{5} b^{5} c d^{10} {\left | b \right |} - a^{6} b^{4} d^{11} {\left | b \right |}\right )}}{b^{6} c d^{11} - a b^{5} d^{12}}\right )} \sqrt {b x + a}}{192 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} - \frac {5 \, {\left (231 \, b^{4} c^{4} {\left | b \right |} - 420 \, a b^{3} c^{3} d {\left | b \right |} + 210 \, a^{2} b^{2} c^{2} d^{2} {\left | b \right |} - 20 \, a^{3} b c d^{3} {\left | b \right |} - a^{4} d^{4} {\left | b \right |}\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 )}{64 \, \sqrt {b d} b^{2} d^{6}} \]
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Timed out. \[ \int \frac {x^3 (a+b x)^{5/2}}{(c+d x)^{5/2}} \, dx=\int \frac {x^3\,{\left (a+b\,x\right )}^{5/2}}{{\left (c+d\,x\right )}^{5/2}} \,d x \]
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