Optimal. Leaf size=209 \[ \frac {2 a^3 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{13/2}}-\frac {2 a^3 \sqrt {c+d x} (b c-a d)^2}{b^6}-\frac {2 a^3 (c+d x)^{3/2} (b c-a d)}{3 b^5}-\frac {2 a^3 (c+d x)^{5/2}}{5 b^4}+\frac {2 (c+d x)^{7/2} \left (a^2 d^2+a b c d+b^2 c^2\right )}{7 b^3 d^3}-\frac {2 (c+d x)^{9/2} (a d+2 b c)}{9 b^2 d^3}+\frac {2 (c+d x)^{11/2}}{11 b d^3} \]
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Rubi [A] time = 0.17, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {88, 50, 63, 208} \[ \frac {2 (c+d x)^{7/2} \left (a^2 d^2+a b c d+b^2 c^2\right )}{7 b^3 d^3}-\frac {2 a^3 (c+d x)^{5/2}}{5 b^4}-\frac {2 a^3 (c+d x)^{3/2} (b c-a d)}{3 b^5}-\frac {2 a^3 \sqrt {c+d x} (b c-a d)^2}{b^6}+\frac {2 a^3 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{13/2}}-\frac {2 (c+d x)^{9/2} (a d+2 b c)}{9 b^2 d^3}+\frac {2 (c+d x)^{11/2}}{11 b d^3} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 88
Rule 208
Rubi steps
\begin {align*} \int \frac {x^3 (c+d x)^{5/2}}{a+b x} \, dx &=\int \left (\frac {\left (b^2 c^2+a b c d+a^2 d^2\right ) (c+d x)^{5/2}}{b^3 d^2}-\frac {a^3 (c+d x)^{5/2}}{b^3 (a+b x)}+\frac {(-2 b c-a d) (c+d x)^{7/2}}{b^2 d^2}+\frac {(c+d x)^{9/2}}{b d^2}\right ) \, dx\\ &=\frac {2 \left (b^2 c^2+a b c d+a^2 d^2\right ) (c+d x)^{7/2}}{7 b^3 d^3}-\frac {2 (2 b c+a d) (c+d x)^{9/2}}{9 b^2 d^3}+\frac {2 (c+d x)^{11/2}}{11 b d^3}-\frac {a^3 \int \frac {(c+d x)^{5/2}}{a+b x} \, dx}{b^3}\\ &=-\frac {2 a^3 (c+d x)^{5/2}}{5 b^4}+\frac {2 \left (b^2 c^2+a b c d+a^2 d^2\right ) (c+d x)^{7/2}}{7 b^3 d^3}-\frac {2 (2 b c+a d) (c+d x)^{9/2}}{9 b^2 d^3}+\frac {2 (c+d x)^{11/2}}{11 b d^3}-\frac {\left (a^3 (b c-a d)\right ) \int \frac {(c+d x)^{3/2}}{a+b x} \, dx}{b^4}\\ &=-\frac {2 a^3 (b c-a d) (c+d x)^{3/2}}{3 b^5}-\frac {2 a^3 (c+d x)^{5/2}}{5 b^4}+\frac {2 \left (b^2 c^2+a b c d+a^2 d^2\right ) (c+d x)^{7/2}}{7 b^3 d^3}-\frac {2 (2 b c+a d) (c+d x)^{9/2}}{9 b^2 d^3}+\frac {2 (c+d x)^{11/2}}{11 b d^3}-\frac {\left (a^3 (b c-a d)^2\right ) \int \frac {\sqrt {c+d x}}{a+b x} \, dx}{b^5}\\ &=-\frac {2 a^3 (b c-a d)^2 \sqrt {c+d x}}{b^6}-\frac {2 a^3 (b c-a d) (c+d x)^{3/2}}{3 b^5}-\frac {2 a^3 (c+d x)^{5/2}}{5 b^4}+\frac {2 \left (b^2 c^2+a b c d+a^2 d^2\right ) (c+d x)^{7/2}}{7 b^3 d^3}-\frac {2 (2 b c+a d) (c+d x)^{9/2}}{9 b^2 d^3}+\frac {2 (c+d x)^{11/2}}{11 b d^3}-\frac {\left (a^3 (b c-a d)^3\right ) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{b^6}\\ &=-\frac {2 a^3 (b c-a d)^2 \sqrt {c+d x}}{b^6}-\frac {2 a^3 (b c-a d) (c+d x)^{3/2}}{3 b^5}-\frac {2 a^3 (c+d x)^{5/2}}{5 b^4}+\frac {2 \left (b^2 c^2+a b c d+a^2 d^2\right ) (c+d x)^{7/2}}{7 b^3 d^3}-\frac {2 (2 b c+a d) (c+d x)^{9/2}}{9 b^2 d^3}+\frac {2 (c+d x)^{11/2}}{11 b d^3}-\frac {\left (2 a^3 (b c-a d)^3\right ) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{b^6 d}\\ &=-\frac {2 a^3 (b c-a d)^2 \sqrt {c+d x}}{b^6}-\frac {2 a^3 (b c-a d) (c+d x)^{3/2}}{3 b^5}-\frac {2 a^3 (c+d x)^{5/2}}{5 b^4}+\frac {2 \left (b^2 c^2+a b c d+a^2 d^2\right ) (c+d x)^{7/2}}{7 b^3 d^3}-\frac {2 (2 b c+a d) (c+d x)^{9/2}}{9 b^2 d^3}+\frac {2 (c+d x)^{11/2}}{11 b d^3}+\frac {2 a^3 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{13/2}}\\ \end {align*}
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Mathematica [A] time = 0.45, size = 197, normalized size = 0.94 \[ -\frac {2 a^3 (a d-b c) \left (3 (b c-a d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )-\sqrt {b} \sqrt {c+d x} (-3 a d+4 b c+b d x)\right )}{3 b^{13/2}}-\frac {2 a^3 (c+d x)^{5/2}}{5 b^4}+\frac {2 (c+d x)^{7/2} \left (a^2 d^2+a b c d+b^2 c^2\right )}{7 b^3 d^3}-\frac {2 (c+d x)^{9/2} (a d+2 b c)}{9 b^2 d^3}+\frac {2 (c+d x)^{11/2}}{11 b d^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 713, normalized size = 3.41 \[ \left [\frac {3465 \, {\left (a^{3} b^{2} c^{2} d^{3} - 2 \, a^{4} b c d^{4} + a^{5} d^{5}\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x + 2 \, b c - a d + 2 \, \sqrt {d x + c} b \sqrt {\frac {b c - a d}{b}}}{b x + a}\right ) + 2 \, {\left (315 \, b^{5} d^{5} x^{5} + 40 \, b^{5} c^{5} + 110 \, a b^{4} c^{4} d + 495 \, a^{2} b^{3} c^{3} d^{2} - 5313 \, a^{3} b^{2} c^{2} d^{3} + 8085 \, a^{4} b c d^{4} - 3465 \, a^{5} d^{5} + 35 \, {\left (23 \, b^{5} c d^{4} - 11 \, a b^{4} d^{5}\right )} x^{4} + 5 \, {\left (113 \, b^{5} c^{2} d^{3} - 209 \, a b^{4} c d^{4} + 99 \, a^{2} b^{3} d^{5}\right )} x^{3} + 3 \, {\left (5 \, b^{5} c^{3} d^{2} - 275 \, a b^{4} c^{2} d^{3} + 495 \, a^{2} b^{3} c d^{4} - 231 \, a^{3} b^{2} d^{5}\right )} x^{2} - {\left (20 \, b^{5} c^{4} d + 55 \, a b^{4} c^{3} d^{2} - 1485 \, a^{2} b^{3} c^{2} d^{3} + 2541 \, a^{3} b^{2} c d^{4} - 1155 \, a^{4} b d^{5}\right )} x\right )} \sqrt {d x + c}}{3465 \, b^{6} d^{3}}, \frac {2 \, {\left (3465 \, {\left (a^{3} b^{2} c^{2} d^{3} - 2 \, a^{4} b c d^{4} + a^{5} d^{5}\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right ) + {\left (315 \, b^{5} d^{5} x^{5} + 40 \, b^{5} c^{5} + 110 \, a b^{4} c^{4} d + 495 \, a^{2} b^{3} c^{3} d^{2} - 5313 \, a^{3} b^{2} c^{2} d^{3} + 8085 \, a^{4} b c d^{4} - 3465 \, a^{5} d^{5} + 35 \, {\left (23 \, b^{5} c d^{4} - 11 \, a b^{4} d^{5}\right )} x^{4} + 5 \, {\left (113 \, b^{5} c^{2} d^{3} - 209 \, a b^{4} c d^{4} + 99 \, a^{2} b^{3} d^{5}\right )} x^{3} + 3 \, {\left (5 \, b^{5} c^{3} d^{2} - 275 \, a b^{4} c^{2} d^{3} + 495 \, a^{2} b^{3} c d^{4} - 231 \, a^{3} b^{2} d^{5}\right )} x^{2} - {\left (20 \, b^{5} c^{4} d + 55 \, a b^{4} c^{3} d^{2} - 1485 \, a^{2} b^{3} c^{2} d^{3} + 2541 \, a^{3} b^{2} c d^{4} - 1155 \, a^{4} b d^{5}\right )} x\right )} \sqrt {d x + c}\right )}}{3465 \, b^{6} d^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.34, size = 305, normalized size = 1.46 \[ -\frac {2 \, {\left (a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3}\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} b^{6}} + \frac {2 \, {\left (315 \, {\left (d x + c\right )}^{\frac {11}{2}} b^{10} d^{30} - 770 \, {\left (d x + c\right )}^{\frac {9}{2}} b^{10} c d^{30} + 495 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{10} c^{2} d^{30} - 385 \, {\left (d x + c\right )}^{\frac {9}{2}} a b^{9} d^{31} + 495 \, {\left (d x + c\right )}^{\frac {7}{2}} a b^{9} c d^{31} + 495 \, {\left (d x + c\right )}^{\frac {7}{2}} a^{2} b^{8} d^{32} - 693 \, {\left (d x + c\right )}^{\frac {5}{2}} a^{3} b^{7} d^{33} - 1155 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{3} b^{7} c d^{33} - 3465 \, \sqrt {d x + c} a^{3} b^{7} c^{2} d^{33} + 1155 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{4} b^{6} d^{34} + 6930 \, \sqrt {d x + c} a^{4} b^{6} c d^{34} - 3465 \, \sqrt {d x + c} a^{5} b^{5} d^{35}\right )}}{3465 \, b^{11} d^{33}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 384, normalized size = 1.84 \[ \frac {2 a^{6} d^{3} \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}\, b^{6}}-\frac {6 a^{5} c \,d^{2} \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}\, b^{5}}+\frac {6 a^{4} c^{2} d \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}\, b^{4}}-\frac {2 a^{3} c^{3} \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}\, b^{3}}-\frac {2 \sqrt {d x +c}\, a^{5} d^{2}}{b^{6}}+\frac {4 \sqrt {d x +c}\, a^{4} c d}{b^{5}}-\frac {2 \sqrt {d x +c}\, a^{3} c^{2}}{b^{4}}+\frac {2 \left (d x +c \right )^{\frac {3}{2}} a^{4} d}{3 b^{5}}-\frac {2 \left (d x +c \right )^{\frac {3}{2}} a^{3} c}{3 b^{4}}-\frac {2 \left (d x +c \right )^{\frac {5}{2}} a^{3}}{5 b^{4}}+\frac {2 \left (d x +c \right )^{\frac {7}{2}} a^{2}}{7 b^{3} d}+\frac {2 \left (d x +c \right )^{\frac {7}{2}} a c}{7 b^{2} d^{2}}+\frac {2 \left (d x +c \right )^{\frac {7}{2}} c^{2}}{7 b \,d^{3}}-\frac {2 \left (d x +c \right )^{\frac {9}{2}} a}{9 b^{2} d^{2}}-\frac {4 \left (d x +c \right )^{\frac {9}{2}} c}{9 b \,d^{3}}+\frac {2 \left (d x +c \right )^{\frac {11}{2}}}{11 b \,d^{3}} \]
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 [B] time = 0.10, size = 567, normalized size = 2.71 \[ \left (\frac {6\,c^2}{7\,b\,d^3}+\frac {\left (\frac {6\,c}{b\,d^3}+\frac {2\,\left (a\,d^4-b\,c\,d^3\right )}{b^2\,d^6}\right )\,\left (a\,d^4-b\,c\,d^3\right )}{7\,b\,d^3}\right )\,{\left (c+d\,x\right )}^{7/2}-\left (\frac {2\,c^3}{5\,b\,d^3}+\frac {\left (\frac {6\,c^2}{b\,d^3}+\frac {\left (\frac {6\,c}{b\,d^3}+\frac {2\,\left (a\,d^4-b\,c\,d^3\right )}{b^2\,d^6}\right )\,\left (a\,d^4-b\,c\,d^3\right )}{b\,d^3}\right )\,\left (a\,d^4-b\,c\,d^3\right )}{5\,b\,d^3}\right )\,{\left (c+d\,x\right )}^{5/2}-\left (\frac {2\,c}{3\,b\,d^3}+\frac {2\,\left (a\,d^4-b\,c\,d^3\right )}{9\,b^2\,d^6}\right )\,{\left (c+d\,x\right )}^{9/2}+\frac {2\,{\left (c+d\,x\right )}^{11/2}}{11\,b\,d^3}+\frac {2\,a^3\,\mathrm {atan}\left (\frac {a^3\,\sqrt {b}\,{\left (a\,d-b\,c\right )}^{5/2}\,\sqrt {c+d\,x}}{a^6\,d^3-3\,a^5\,b\,c\,d^2+3\,a^4\,b^2\,c^2\,d-a^3\,b^3\,c^3}\right )\,{\left (a\,d-b\,c\right )}^{5/2}}{b^{13/2}}-\frac {\left (\frac {2\,c^3}{b\,d^3}+\frac {\left (\frac {6\,c^2}{b\,d^3}+\frac {\left (\frac {6\,c}{b\,d^3}+\frac {2\,\left (a\,d^4-b\,c\,d^3\right )}{b^2\,d^6}\right )\,\left (a\,d^4-b\,c\,d^3\right )}{b\,d^3}\right )\,\left (a\,d^4-b\,c\,d^3\right )}{b\,d^3}\right )\,{\left (a\,d^4-b\,c\,d^3\right )}^2\,\sqrt {c+d\,x}}{b^2\,d^6}+\frac {\left (\frac {2\,c^3}{b\,d^3}+\frac {\left (\frac {6\,c^2}{b\,d^3}+\frac {\left (\frac {6\,c}{b\,d^3}+\frac {2\,\left (a\,d^4-b\,c\,d^3\right )}{b^2\,d^6}\right )\,\left (a\,d^4-b\,c\,d^3\right )}{b\,d^3}\right )\,\left (a\,d^4-b\,c\,d^3\right )}{b\,d^3}\right )\,\left (a\,d^4-b\,c\,d^3\right )\,{\left (c+d\,x\right )}^{3/2}}{3\,b\,d^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 74.58, size = 228, normalized size = 1.09 \[ - \frac {2 a^{3} \left (c + d x\right )^{\frac {5}{2}}}{5 b^{4}} + \frac {2 a^{3} \left (a d - b c\right )^{3} \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {\frac {a d - b c}{b}}} \right )}}{b^{7} \sqrt {\frac {a d - b c}{b}}} + \frac {2 \left (c + d x\right )^{\frac {11}{2}}}{11 b d^{3}} + \frac {\left (c + d x\right )^{\frac {9}{2}} \left (- 2 a d - 4 b c\right )}{9 b^{2} d^{3}} + \frac {\left (c + d x\right )^{\frac {7}{2}} \left (2 a^{2} d^{2} + 2 a b c d + 2 b^{2} c^{2}\right )}{7 b^{3} d^{3}} + \frac {\left (c + d x\right )^{\frac {3}{2}} \left (2 a^{4} d - 2 a^{3} b c\right )}{3 b^{5}} + \frac {\sqrt {c + d x} \left (- 2 a^{5} d^{2} + 4 a^{4} b c d - 2 a^{3} b^{2} c^{2}\right )}{b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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