Optimal. Leaf size=346 \[ -\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (-231 a^2 d^2-2 b d x (5 b c-99 a d)+30 a b c d+5 b^2 c^2\right )}{80 b^4 d^2}+\frac {3 (b c-a d)^2 \left (-231 a^3 d^3+63 a^2 b c d^2+7 a b^2 c^2 d+b^3 c^3\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{13/2} d^{5/2}}+\frac {3 \sqrt {a+b x} \sqrt {c+d x} (b c-a d) \left (-231 a^3 d^3+63 a^2 b c d^2+7 a b^2 c^2 d+b^3 c^3\right )}{128 b^6 d^2}+\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (-231 a^3 d^3+63 a^2 b c d^2+7 a b^2 c^2 d+b^3 c^3\right )}{64 b^5 d^2}+\frac {11 x^2 \sqrt {a+b x} (c+d x)^{5/2}}{5 b^2}-\frac {2 x^3 (c+d x)^{5/2}}{b \sqrt {a+b x}} \]
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Rubi [A] time = 0.30, antiderivative size = 346, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {97, 153, 147, 50, 63, 217, 206} \[ \frac {\sqrt {a+b x} (c+d x)^{3/2} \left (63 a^2 b c d^2-231 a^3 d^3+7 a b^2 c^2 d+b^3 c^3\right )}{64 b^5 d^2}-\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (-231 a^2 d^2-2 b d x (5 b c-99 a d)+30 a b c d+5 b^2 c^2\right )}{80 b^4 d^2}+\frac {3 \sqrt {a+b x} \sqrt {c+d x} (b c-a d) \left (63 a^2 b c d^2-231 a^3 d^3+7 a b^2 c^2 d+b^3 c^3\right )}{128 b^6 d^2}+\frac {3 (b c-a d)^2 \left (63 a^2 b c d^2-231 a^3 d^3+7 a b^2 c^2 d+b^3 c^3\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{13/2} d^{5/2}}+\frac {11 x^2 \sqrt {a+b x} (c+d x)^{5/2}}{5 b^2}-\frac {2 x^3 (c+d x)^{5/2}}{b \sqrt {a+b x}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 97
Rule 147
Rule 153
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \frac {x^3 (c+d x)^{5/2}}{(a+b x)^{3/2}} \, dx &=-\frac {2 x^3 (c+d x)^{5/2}}{b \sqrt {a+b x}}+\frac {2 \int \frac {x^2 (c+d x)^{3/2} \left (3 c+\frac {11 d x}{2}\right )}{\sqrt {a+b x}} \, dx}{b}\\ &=-\frac {2 x^3 (c+d x)^{5/2}}{b \sqrt {a+b x}}+\frac {11 x^2 \sqrt {a+b x} (c+d x)^{5/2}}{5 b^2}+\frac {2 \int \frac {x (c+d x)^{3/2} \left (-11 a c d+\frac {1}{4} d (5 b c-99 a d) x\right )}{\sqrt {a+b x}} \, dx}{5 b^2 d}\\ &=-\frac {2 x^3 (c+d x)^{5/2}}{b \sqrt {a+b x}}+\frac {11 x^2 \sqrt {a+b x} (c+d x)^{5/2}}{5 b^2}-\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (5 b^2 c^2+30 a b c d-231 a^2 d^2-2 b d (5 b c-99 a d) x\right )}{80 b^4 d^2}+\frac {\left (b^3 c^3+7 a b^2 c^2 d+63 a^2 b c d^2-231 a^3 d^3\right ) \int \frac {(c+d x)^{3/2}}{\sqrt {a+b x}} \, dx}{32 b^4 d^2}\\ &=\frac {\left (b^3 c^3+7 a b^2 c^2 d+63 a^2 b c d^2-231 a^3 d^3\right ) \sqrt {a+b x} (c+d x)^{3/2}}{64 b^5 d^2}-\frac {2 x^3 (c+d x)^{5/2}}{b \sqrt {a+b x}}+\frac {11 x^2 \sqrt {a+b x} (c+d x)^{5/2}}{5 b^2}-\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (5 b^2 c^2+30 a b c d-231 a^2 d^2-2 b d (5 b c-99 a d) x\right )}{80 b^4 d^2}+\frac {\left (3 (b c-a d) \left (b^3 c^3+7 a b^2 c^2 d+63 a^2 b c d^2-231 a^3 d^3\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a+b x}} \, dx}{128 b^5 d^2}\\ &=\frac {3 (b c-a d) \left (b^3 c^3+7 a b^2 c^2 d+63 a^2 b c d^2-231 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 b^6 d^2}+\frac {\left (b^3 c^3+7 a b^2 c^2 d+63 a^2 b c d^2-231 a^3 d^3\right ) \sqrt {a+b x} (c+d x)^{3/2}}{64 b^5 d^2}-\frac {2 x^3 (c+d x)^{5/2}}{b \sqrt {a+b x}}+\frac {11 x^2 \sqrt {a+b x} (c+d x)^{5/2}}{5 b^2}-\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (5 b^2 c^2+30 a b c d-231 a^2 d^2-2 b d (5 b c-99 a d) x\right )}{80 b^4 d^2}+\frac {\left (3 (b c-a d)^2 \left (b^3 c^3+7 a b^2 c^2 d+63 a^2 b c d^2-231 a^3 d^3\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{256 b^6 d^2}\\ &=\frac {3 (b c-a d) \left (b^3 c^3+7 a b^2 c^2 d+63 a^2 b c d^2-231 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 b^6 d^2}+\frac {\left (b^3 c^3+7 a b^2 c^2 d+63 a^2 b c d^2-231 a^3 d^3\right ) \sqrt {a+b x} (c+d x)^{3/2}}{64 b^5 d^2}-\frac {2 x^3 (c+d x)^{5/2}}{b \sqrt {a+b x}}+\frac {11 x^2 \sqrt {a+b x} (c+d x)^{5/2}}{5 b^2}-\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (5 b^2 c^2+30 a b c d-231 a^2 d^2-2 b d (5 b c-99 a d) x\right )}{80 b^4 d^2}+\frac {\left (3 (b c-a d)^2 \left (b^3 c^3+7 a b^2 c^2 d+63 a^2 b c d^2-231 a^3 d^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{128 b^7 d^2}\\ &=\frac {3 (b c-a d) \left (b^3 c^3+7 a b^2 c^2 d+63 a^2 b c d^2-231 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 b^6 d^2}+\frac {\left (b^3 c^3+7 a b^2 c^2 d+63 a^2 b c d^2-231 a^3 d^3\right ) \sqrt {a+b x} (c+d x)^{3/2}}{64 b^5 d^2}-\frac {2 x^3 (c+d x)^{5/2}}{b \sqrt {a+b x}}+\frac {11 x^2 \sqrt {a+b x} (c+d x)^{5/2}}{5 b^2}-\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (5 b^2 c^2+30 a b c d-231 a^2 d^2-2 b d (5 b c-99 a d) x\right )}{80 b^4 d^2}+\frac {\left (3 (b c-a d)^2 \left (b^3 c^3+7 a b^2 c^2 d+63 a^2 b c d^2-231 a^3 d^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{128 b^7 d^2}\\ &=\frac {3 (b c-a d) \left (b^3 c^3+7 a b^2 c^2 d+63 a^2 b c d^2-231 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 b^6 d^2}+\frac {\left (b^3 c^3+7 a b^2 c^2 d+63 a^2 b c d^2-231 a^3 d^3\right ) \sqrt {a+b x} (c+d x)^{3/2}}{64 b^5 d^2}-\frac {2 x^3 (c+d x)^{5/2}}{b \sqrt {a+b x}}+\frac {11 x^2 \sqrt {a+b x} (c+d x)^{5/2}}{5 b^2}-\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (5 b^2 c^2+30 a b c d-231 a^2 d^2-2 b d (5 b c-99 a d) x\right )}{80 b^4 d^2}+\frac {3 (b c-a d)^2 \left (b^3 c^3+7 a b^2 c^2 d+63 a^2 b c d^2-231 a^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{13/2} d^{5/2}}\\ \end {align*}
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Mathematica [A] time = 1.45, size = 324, normalized size = 0.94 \[ \frac {\sqrt {c+d x} \left (\frac {15 \left (-231 a^3 d^3+63 a^2 b c d^2+7 a b^2 c^2 d+b^3 c^3\right ) (b c-a d)^{3/2} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{\sqrt {\frac {b (c+d x)}{b c-a d}}}+\frac {\sqrt {d} \left (3465 a^5 d^4+105 a^4 b d^3 (11 d x-64 c)-42 a^3 b^2 d^2 \left (-79 c^2+57 c d x+11 d^2 x^2\right )+2 a^2 b^3 d \left (-40 c^3+662 c^2 d x+459 c d^2 x^2+132 d^3 x^3\right )-a b^4 \left (15 c^4+70 c^3 d x+466 c^2 d^2 x^2+512 c d^3 x^3+176 d^4 x^4\right )+b^5 x \left (-15 c^4+10 c^3 d x+248 c^2 d^2 x^2+336 c d^3 x^3+128 d^4 x^4\right )\right )}{\sqrt {a+b x}}\right )}{640 b^6 d^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.90, size = 1008, normalized size = 2.91 \[ \left [-\frac {15 \, {\left (a b^{5} c^{5} + 5 \, a^{2} b^{4} c^{4} d + 50 \, a^{3} b^{3} c^{3} d^{2} - 350 \, a^{4} b^{2} c^{2} d^{3} + 525 \, a^{5} b c d^{4} - 231 \, a^{6} d^{5} + {\left (b^{6} c^{5} + 5 \, a b^{5} c^{4} d + 50 \, a^{2} b^{4} c^{3} d^{2} - 350 \, a^{3} b^{3} c^{2} d^{3} + 525 \, a^{4} b^{2} c d^{4} - 231 \, a^{5} b 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 (128 \, b^{6} d^{5} x^{5} - 15 \, a b^{5} c^{4} d - 80 \, a^{2} b^{4} c^{3} d^{2} + 3318 \, a^{3} b^{3} c^{2} d^{3} - 6720 \, a^{4} b^{2} c d^{4} + 3465 \, a^{5} b d^{5} + 16 \, {\left (21 \, b^{6} c d^{4} - 11 \, a b^{5} d^{5}\right )} x^{4} + 8 \, {\left (31 \, b^{6} c^{2} d^{3} - 64 \, a b^{5} c d^{4} + 33 \, a^{2} b^{4} d^{5}\right )} x^{3} + 2 \, {\left (5 \, b^{6} c^{3} d^{2} - 233 \, a b^{5} c^{2} d^{3} + 459 \, a^{2} b^{4} c d^{4} - 231 \, a^{3} b^{3} d^{5}\right )} x^{2} - {\left (15 \, b^{6} c^{4} d + 70 \, a b^{5} c^{3} d^{2} - 1324 \, a^{2} b^{4} c^{2} d^{3} + 2394 \, a^{3} b^{3} c d^{4} - 1155 \, a^{4} b^{2} d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{2560 \, {\left (b^{8} d^{3} x + a b^{7} d^{3}\right )}}, -\frac {15 \, {\left (a b^{5} c^{5} + 5 \, a^{2} b^{4} c^{4} d + 50 \, a^{3} b^{3} c^{3} d^{2} - 350 \, a^{4} b^{2} c^{2} d^{3} + 525 \, a^{5} b c d^{4} - 231 \, a^{6} d^{5} + {\left (b^{6} c^{5} + 5 \, a b^{5} c^{4} d + 50 \, a^{2} b^{4} c^{3} d^{2} - 350 \, a^{3} b^{3} c^{2} d^{3} + 525 \, a^{4} b^{2} c d^{4} - 231 \, a^{5} b 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 (128 \, b^{6} d^{5} x^{5} - 15 \, a b^{5} c^{4} d - 80 \, a^{2} b^{4} c^{3} d^{2} + 3318 \, a^{3} b^{3} c^{2} d^{3} - 6720 \, a^{4} b^{2} c d^{4} + 3465 \, a^{5} b d^{5} + 16 \, {\left (21 \, b^{6} c d^{4} - 11 \, a b^{5} d^{5}\right )} x^{4} + 8 \, {\left (31 \, b^{6} c^{2} d^{3} - 64 \, a b^{5} c d^{4} + 33 \, a^{2} b^{4} d^{5}\right )} x^{3} + 2 \, {\left (5 \, b^{6} c^{3} d^{2} - 233 \, a b^{5} c^{2} d^{3} + 459 \, a^{2} b^{4} c d^{4} - 231 \, a^{3} b^{3} d^{5}\right )} x^{2} - {\left (15 \, b^{6} c^{4} d + 70 \, a b^{5} c^{3} d^{2} - 1324 \, a^{2} b^{4} c^{2} d^{3} + 2394 \, a^{3} b^{3} c d^{4} - 1155 \, a^{4} b^{2} d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{1280 \, {\left (b^{8} d^{3} x + a b^{7} d^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 3.14, size = 570, normalized size = 1.65 \[ \frac {1}{640} \, \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (2 \, {\left (4 \, {\left (b x + a\right )} {\left (2 \, {\left (b x + a\right )} {\left (\frac {8 \, {\left (b x + a\right )} d^{2} {\left | b \right |}}{b^{8}} + \frac {3 \, {\left (7 \, b^{40} c d^{9} {\left | b \right |} - 17 \, a b^{39} d^{10} {\left | b \right |}\right )}}{b^{47} d^{8}}\right )} + \frac {31 \, b^{41} c^{2} d^{8} {\left | b \right |} - 232 \, a b^{40} c d^{9} {\left | b \right |} + 281 \, a^{2} b^{39} d^{10} {\left | b \right |}}{b^{47} d^{8}}\right )} + \frac {5 \, {\left (b^{42} c^{3} d^{7} {\left | b \right |} - 121 \, a b^{41} c^{2} d^{8} {\left | b \right |} + 447 \, a^{2} b^{40} c d^{9} {\left | b \right |} - 359 \, a^{3} b^{39} d^{10} {\left | b \right |}\right )}}{b^{47} d^{8}}\right )} {\left (b x + a\right )} - \frac {15 \, {\left (b^{43} c^{4} d^{6} {\left | b \right |} + 6 \, a b^{42} c^{3} d^{7} {\left | b \right |} - 200 \, a^{2} b^{41} c^{2} d^{8} {\left | b \right |} + 474 \, a^{3} b^{40} c d^{9} {\left | b \right |} - 281 \, a^{4} b^{39} d^{10} {\left | b \right |}\right )}}{b^{47} d^{8}}\right )} \sqrt {b x + a} + \frac {4 \, {\left (\sqrt {b d} a^{3} b^{3} c^{3} {\left | b \right |} - 3 \, \sqrt {b d} a^{4} b^{2} c^{2} d {\left | b \right |} + 3 \, \sqrt {b d} a^{5} b c d^{2} {\left | b \right |} - \sqrt {b d} a^{6} d^{3} {\left | b \right |}\right )}}{{\left (b^{2} c - a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )} b^{7}} - \frac {3 \, {\left (\sqrt {b d} b^{5} c^{5} {\left | b \right |} + 5 \, \sqrt {b d} a b^{4} c^{4} d {\left | b \right |} + 50 \, \sqrt {b d} a^{2} b^{3} c^{3} d^{2} {\left | b \right |} - 350 \, \sqrt {b d} a^{3} b^{2} c^{2} d^{3} {\left | b \right |} + 525 \, \sqrt {b d} a^{4} b c d^{4} {\left | b \right |} - 231 \, \sqrt {b d} a^{5} d^{5} {\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 )}^{2}\right )}{256 \, b^{8} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 1265, normalized size = 3.66 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^3\,{\left (c+d\,x\right )}^{5/2}}{{\left (a+b\,x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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