Optimal. Leaf size=410 \[ \frac {5 \sqrt {\frac {\pi }{6}} d^{5/2} \sin \left (3 a-\frac {3 b c}{d}\right ) C\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{144 b^{7/2}}+\frac {45 \sqrt {\frac {\pi }{2}} d^{5/2} \sin \left (a-\frac {b c}{d}\right ) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{7/2}}+\frac {45 \sqrt {\frac {\pi }{2}} d^{5/2} \cos \left (a-\frac {b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{7/2}}+\frac {5 \sqrt {\frac {\pi }{6}} d^{5/2} \cos \left (3 a-\frac {3 b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{144 b^{7/2}}-\frac {45 d^2 \sqrt {c+d x} \sin (a+b x)}{16 b^3}-\frac {5 d^2 \sqrt {c+d x} \sin (3 a+3 b x)}{144 b^3}+\frac {5 d (c+d x)^{3/2} \cos ^3(a+b x)}{18 b^2}+\frac {5 d (c+d x)^{3/2} \cos (a+b x)}{3 b^2}+\frac {2 (c+d x)^{5/2} \sin (a+b x)}{3 b}+\frac {(c+d x)^{5/2} \sin (a+b x) \cos ^2(a+b x)}{3 b} \]
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Rubi [A] time = 1.14, antiderivative size = 410, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 8, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {3311, 3296, 3306, 3305, 3351, 3304, 3352, 3312} \[ \frac {5 \sqrt {\frac {\pi }{6}} d^{5/2} \sin \left (3 a-\frac {3 b c}{d}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{144 b^{7/2}}+\frac {45 \sqrt {\frac {\pi }{2}} d^{5/2} \sin \left (a-\frac {b c}{d}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{7/2}}+\frac {45 \sqrt {\frac {\pi }{2}} d^{5/2} \cos \left (a-\frac {b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{7/2}}+\frac {5 \sqrt {\frac {\pi }{6}} d^{5/2} \cos \left (3 a-\frac {3 b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{144 b^{7/2}}-\frac {45 d^2 \sqrt {c+d x} \sin (a+b x)}{16 b^3}-\frac {5 d^2 \sqrt {c+d x} \sin (3 a+3 b x)}{144 b^3}+\frac {5 d (c+d x)^{3/2} \cos ^3(a+b x)}{18 b^2}+\frac {5 d (c+d x)^{3/2} \cos (a+b x)}{3 b^2}+\frac {2 (c+d x)^{5/2} \sin (a+b x)}{3 b}+\frac {(c+d x)^{5/2} \sin (a+b x) \cos ^2(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 3296
Rule 3304
Rule 3305
Rule 3306
Rule 3311
Rule 3312
Rule 3351
Rule 3352
Rubi steps
\begin {align*} \int (c+d x)^{5/2} \cos ^3(a+b x) \, dx &=\frac {5 d (c+d x)^{3/2} \cos ^3(a+b x)}{18 b^2}+\frac {(c+d x)^{5/2} \cos ^2(a+b x) \sin (a+b x)}{3 b}+\frac {2}{3} \int (c+d x)^{5/2} \cos (a+b x) \, dx-\frac {\left (5 d^2\right ) \int \sqrt {c+d x} \cos ^3(a+b x) \, dx}{12 b^2}\\ &=\frac {5 d (c+d x)^{3/2} \cos ^3(a+b x)}{18 b^2}+\frac {2 (c+d x)^{5/2} \sin (a+b x)}{3 b}+\frac {(c+d x)^{5/2} \cos ^2(a+b x) \sin (a+b x)}{3 b}-\frac {(5 d) \int (c+d x)^{3/2} \sin (a+b x) \, dx}{3 b}-\frac {\left (5 d^2\right ) \int \left (\frac {3}{4} \sqrt {c+d x} \cos (a+b x)+\frac {1}{4} \sqrt {c+d x} \cos (3 a+3 b x)\right ) \, dx}{12 b^2}\\ &=\frac {5 d (c+d x)^{3/2} \cos (a+b x)}{3 b^2}+\frac {5 d (c+d x)^{3/2} \cos ^3(a+b x)}{18 b^2}+\frac {2 (c+d x)^{5/2} \sin (a+b x)}{3 b}+\frac {(c+d x)^{5/2} \cos ^2(a+b x) \sin (a+b x)}{3 b}-\frac {\left (5 d^2\right ) \int \sqrt {c+d x} \cos (3 a+3 b x) \, dx}{48 b^2}-\frac {\left (5 d^2\right ) \int \sqrt {c+d x} \cos (a+b x) \, dx}{16 b^2}-\frac {\left (5 d^2\right ) \int \sqrt {c+d x} \cos (a+b x) \, dx}{2 b^2}\\ &=\frac {5 d (c+d x)^{3/2} \cos (a+b x)}{3 b^2}+\frac {5 d (c+d x)^{3/2} \cos ^3(a+b x)}{18 b^2}-\frac {45 d^2 \sqrt {c+d x} \sin (a+b x)}{16 b^3}+\frac {2 (c+d x)^{5/2} \sin (a+b x)}{3 b}+\frac {(c+d x)^{5/2} \cos ^2(a+b x) \sin (a+b x)}{3 b}-\frac {5 d^2 \sqrt {c+d x} \sin (3 a+3 b x)}{144 b^3}+\frac {\left (5 d^3\right ) \int \frac {\sin (3 a+3 b x)}{\sqrt {c+d x}} \, dx}{288 b^3}+\frac {\left (5 d^3\right ) \int \frac {\sin (a+b x)}{\sqrt {c+d x}} \, dx}{32 b^3}+\frac {\left (5 d^3\right ) \int \frac {\sin (a+b x)}{\sqrt {c+d x}} \, dx}{4 b^3}\\ &=\frac {5 d (c+d x)^{3/2} \cos (a+b x)}{3 b^2}+\frac {5 d (c+d x)^{3/2} \cos ^3(a+b x)}{18 b^2}-\frac {45 d^2 \sqrt {c+d x} \sin (a+b x)}{16 b^3}+\frac {2 (c+d x)^{5/2} \sin (a+b x)}{3 b}+\frac {(c+d x)^{5/2} \cos ^2(a+b x) \sin (a+b x)}{3 b}-\frac {5 d^2 \sqrt {c+d x} \sin (3 a+3 b x)}{144 b^3}+\frac {\left (5 d^3 \cos \left (3 a-\frac {3 b c}{d}\right )\right ) \int \frac {\sin \left (\frac {3 b c}{d}+3 b x\right )}{\sqrt {c+d x}} \, dx}{288 b^3}+\frac {\left (5 d^3 \cos \left (a-\frac {b c}{d}\right )\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}} \, dx}{32 b^3}+\frac {\left (5 d^3 \cos \left (a-\frac {b c}{d}\right )\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}} \, dx}{4 b^3}+\frac {\left (5 d^3 \sin \left (3 a-\frac {3 b c}{d}\right )\right ) \int \frac {\cos \left (\frac {3 b c}{d}+3 b x\right )}{\sqrt {c+d x}} \, dx}{288 b^3}+\frac {\left (5 d^3 \sin \left (a-\frac {b c}{d}\right )\right ) \int \frac {\cos \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}} \, dx}{32 b^3}+\frac {\left (5 d^3 \sin \left (a-\frac {b c}{d}\right )\right ) \int \frac {\cos \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}} \, dx}{4 b^3}\\ &=\frac {5 d (c+d x)^{3/2} \cos (a+b x)}{3 b^2}+\frac {5 d (c+d x)^{3/2} \cos ^3(a+b x)}{18 b^2}-\frac {45 d^2 \sqrt {c+d x} \sin (a+b x)}{16 b^3}+\frac {2 (c+d x)^{5/2} \sin (a+b x)}{3 b}+\frac {(c+d x)^{5/2} \cos ^2(a+b x) \sin (a+b x)}{3 b}-\frac {5 d^2 \sqrt {c+d x} \sin (3 a+3 b x)}{144 b^3}+\frac {\left (5 d^2 \cos \left (3 a-\frac {3 b c}{d}\right )\right ) \operatorname {Subst}\left (\int \sin \left (\frac {3 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{144 b^3}+\frac {\left (5 d^2 \cos \left (a-\frac {b c}{d}\right )\right ) \operatorname {Subst}\left (\int \sin \left (\frac {b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{16 b^3}+\frac {\left (5 d^2 \cos \left (a-\frac {b c}{d}\right )\right ) \operatorname {Subst}\left (\int \sin \left (\frac {b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{2 b^3}+\frac {\left (5 d^2 \sin \left (3 a-\frac {3 b c}{d}\right )\right ) \operatorname {Subst}\left (\int \cos \left (\frac {3 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{144 b^3}+\frac {\left (5 d^2 \sin \left (a-\frac {b c}{d}\right )\right ) \operatorname {Subst}\left (\int \cos \left (\frac {b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{16 b^3}+\frac {\left (5 d^2 \sin \left (a-\frac {b c}{d}\right )\right ) \operatorname {Subst}\left (\int \cos \left (\frac {b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{2 b^3}\\ &=\frac {5 d (c+d x)^{3/2} \cos (a+b x)}{3 b^2}+\frac {5 d (c+d x)^{3/2} \cos ^3(a+b x)}{18 b^2}+\frac {45 d^{5/2} \sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{7/2}}+\frac {5 d^{5/2} \sqrt {\frac {\pi }{6}} \cos \left (3 a-\frac {3 b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{144 b^{7/2}}+\frac {5 d^{5/2} \sqrt {\frac {\pi }{6}} C\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (3 a-\frac {3 b c}{d}\right )}{144 b^{7/2}}+\frac {45 d^{5/2} \sqrt {\frac {\pi }{2}} C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (a-\frac {b c}{d}\right )}{16 b^{7/2}}-\frac {45 d^2 \sqrt {c+d x} \sin (a+b x)}{16 b^3}+\frac {2 (c+d x)^{5/2} \sin (a+b x)}{3 b}+\frac {(c+d x)^{5/2} \cos ^2(a+b x) \sin (a+b x)}{3 b}-\frac {5 d^2 \sqrt {c+d x} \sin (3 a+3 b x)}{144 b^3}\\ \end {align*}
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Mathematica [A] time = 3.10, size = 542, normalized size = 1.32 \[ \frac {648 b^3 c^2 \sqrt {c+d x} \sin (a+b x)+72 b^3 c^2 \sqrt {c+d x} \sin (3 (a+b x))+648 b^3 d^2 x^2 \sqrt {c+d x} \sin (a+b x)+72 b^3 d^2 x^2 \sqrt {c+d x} \sin (3 (a+b x))+1296 b^3 c d x \sqrt {c+d x} \sin (a+b x)+144 b^3 c d x \sqrt {c+d x} \sin (3 (a+b x))+1620 b^2 d^2 x \sqrt {c+d x} \cos (a+b x)+60 b^2 d^2 x \sqrt {c+d x} \cos (3 (a+b x))+1620 b^2 c d \sqrt {c+d x} \cos (a+b x)+60 b^2 c d \sqrt {c+d x} \cos (3 (a+b x))+5 \sqrt {6 \pi } d^3 \sqrt {\frac {b}{d}} \sin \left (3 a-\frac {3 b c}{d}\right ) C\left (\sqrt {\frac {b}{d}} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}\right )+1215 \sqrt {2 \pi } d^3 \sqrt {\frac {b}{d}} \sin \left (a-\frac {b c}{d}\right ) C\left (\sqrt {\frac {b}{d}} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}\right )+1215 \sqrt {2 \pi } d^3 \sqrt {\frac {b}{d}} \cos \left (a-\frac {b c}{d}\right ) S\left (\sqrt {\frac {b}{d}} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}\right )+5 \sqrt {6 \pi } d^3 \sqrt {\frac {b}{d}} \cos \left (3 a-\frac {3 b c}{d}\right ) S\left (\sqrt {\frac {b}{d}} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}\right )-2430 b d^2 \sqrt {c+d x} \sin (a+b x)-30 b d^2 \sqrt {c+d x} \sin (3 (a+b x))}{864 b^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 368, normalized size = 0.90 \[ \frac {5 \, \sqrt {6} \pi d^{3} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {S}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) + 1215 \, \sqrt {2} \pi d^{3} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {S}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) + 1215 \, \sqrt {2} \pi d^{3} \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {b c - a d}{d}\right ) + 5 \, \sqrt {6} \pi d^{3} \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + 24 \, {\left (10 \, {\left (b^{2} d^{2} x + b^{2} c d\right )} \cos \left (b x + a\right )^{3} + 60 \, {\left (b^{2} d^{2} x + b^{2} c d\right )} \cos \left (b x + a\right ) + {\left (24 \, b^{3} d^{2} x^{2} + 48 \, b^{3} c d x + 24 \, b^{3} c^{2} - 100 \, b d^{2} + {\left (12 \, b^{3} d^{2} x^{2} + 24 \, b^{3} c d x + 12 \, b^{3} c^{2} - 5 \, b d^{2}\right )} \cos \left (b x + a\right )^{2}\right )} \sin \left (b x + a\right )\right )} \sqrt {d x + c}}{864 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 1.92, size = 2457, normalized size = 5.99 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 474, normalized size = 1.16 \[ \frac {\frac {3 d \left (d x +c \right )^{\frac {5}{2}} \sin \left (\frac {\left (d x +c \right ) b}{d}+\frac {d a -c b}{d}\right )}{4 b}-\frac {15 d \left (-\frac {d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {\left (d x +c \right ) b}{d}+\frac {d a -c b}{d}\right )}{2 b}+\frac {3 d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {\left (d x +c \right ) b}{d}+\frac {d a -c b}{d}\right )}{2 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {d a -c b}{d}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {d a -c b}{d}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{4 b \sqrt {\frac {b}{d}}}\right )}{2 b}\right )}{4 b}+\frac {d \left (d x +c \right )^{\frac {5}{2}} \sin \left (\frac {3 \left (d x +c \right ) b}{d}+\frac {3 d a -3 c b}{d}\right )}{12 b}-\frac {5 d \left (-\frac {d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {3 \left (d x +c \right ) b}{d}+\frac {3 d a -3 c b}{d}\right )}{6 b}+\frac {d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {3 \left (d x +c \right ) b}{d}+\frac {3 d a -3 c b}{d}\right )}{6 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \left (\cos \left (\frac {3 d a -3 c b}{d}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {3 d a -3 c b}{d}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{36 b \sqrt {\frac {b}{d}}}\right )}{2 b}\right )}{12 b}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.41, size = 543, normalized size = 1.32 \[ \frac {{\left (240 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} \cos \left (\frac {3 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right ) + 6480 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} \cos \left (\frac {{\left (d x + c\right )} b - b c + a d}{d}\right ) + {\left (\left (5 i + 5\right ) \cdot 9^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) - \left (5 i - 5\right ) \cdot 9^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {3 i \, b}{d}}\right ) + {\left (\left (1215 i + 1215\right ) \, \sqrt {2} \sqrt {\pi } b d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {b c - a d}{d}\right ) - \left (1215 i - 1215\right ) \, \sqrt {2} \sqrt {\pi } b d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {b c - a d}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {i \, b}{d}}\right ) + {\left (-\left (1215 i - 1215\right ) \, \sqrt {2} \sqrt {\pi } b d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {b c - a d}{d}\right ) + \left (1215 i + 1215\right ) \, \sqrt {2} \sqrt {\pi } b d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {b c - a d}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {i \, b}{d}}\right ) + {\left (-\left (5 i - 5\right ) \cdot 9^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + \left (5 i + 5\right ) \cdot 9^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {3 i \, b}{d}}\right ) + 24 \, {\left (\frac {12 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{4}}{d} - 5 \, \sqrt {d x + c} b^{2} d\right )} \sin \left (\frac {3 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right ) + 648 \, {\left (\frac {4 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{4}}{d} - 15 \, \sqrt {d x + c} b^{2} d\right )} \sin \left (\frac {{\left (d x + c\right )} b - b c + a d}{d}\right )\right )} d}{3456 \, b^{5}} \]
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 \[ \int {\cos \left (a+b\,x\right )}^3\,{\left (c+d\,x\right )}^{5/2} \,d x \]
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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