Optimal. Leaf size=407 \[ \frac {5 \sqrt {\frac {\pi }{3}} d^{5/2} \cos \left (6 a-\frac {6 b c}{d}\right ) C\left (\frac {2 \sqrt {b} \sqrt {\frac {3}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{18432 b^{7/2}}-\frac {45 \sqrt {\pi } d^{5/2} \cos \left (2 a-\frac {2 b c}{d}\right ) C\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{2048 b^{7/2}}-\frac {5 \sqrt {\frac {\pi }{3}} d^{5/2} \sin \left (6 a-\frac {6 b c}{d}\right ) S\left (\frac {2 \sqrt {b} \sqrt {\frac {3}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{18432 b^{7/2}}+\frac {45 \sqrt {\pi } d^{5/2} \sin \left (2 a-\frac {2 b c}{d}\right ) S\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{2048 b^{7/2}}+\frac {45 d^2 \sqrt {c+d x} \cos (2 a+2 b x)}{1024 b^3}-\frac {5 d^2 \sqrt {c+d x} \cos (6 a+6 b x)}{9216 b^3}+\frac {15 d (c+d x)^{3/2} \sin (2 a+2 b x)}{256 b^2}-\frac {5 d (c+d x)^{3/2} \sin (6 a+6 b x)}{2304 b^2}-\frac {3 (c+d x)^{5/2} \cos (2 a+2 b x)}{64 b}+\frac {(c+d x)^{5/2} \cos (6 a+6 b x)}{192 b} \]
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Rubi [A] time = 0.90, antiderivative size = 407, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {4406, 3296, 3306, 3305, 3351, 3304, 3352} \[ \frac {5 \sqrt {\frac {\pi }{3}} d^{5/2} \cos \left (6 a-\frac {6 b c}{d}\right ) \text {FresnelC}\left (\frac {2 \sqrt {\frac {3}{\pi }} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{18432 b^{7/2}}-\frac {45 \sqrt {\pi } d^{5/2} \cos \left (2 a-\frac {2 b c}{d}\right ) \text {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {\pi } \sqrt {d}}\right )}{2048 b^{7/2}}-\frac {5 \sqrt {\frac {\pi }{3}} d^{5/2} \sin \left (6 a-\frac {6 b c}{d}\right ) S\left (\frac {2 \sqrt {b} \sqrt {\frac {3}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{18432 b^{7/2}}+\frac {45 \sqrt {\pi } d^{5/2} \sin \left (2 a-\frac {2 b c}{d}\right ) S\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{2048 b^{7/2}}+\frac {45 d^2 \sqrt {c+d x} \cos (2 a+2 b x)}{1024 b^3}-\frac {5 d^2 \sqrt {c+d x} \cos (6 a+6 b x)}{9216 b^3}+\frac {15 d (c+d x)^{3/2} \sin (2 a+2 b x)}{256 b^2}-\frac {5 d (c+d x)^{3/2} \sin (6 a+6 b x)}{2304 b^2}-\frac {3 (c+d x)^{5/2} \cos (2 a+2 b x)}{64 b}+\frac {(c+d x)^{5/2} \cos (6 a+6 b x)}{192 b} \]
Antiderivative was successfully verified.
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Rule 3296
Rule 3304
Rule 3305
Rule 3306
Rule 3351
Rule 3352
Rule 4406
Rubi steps
\begin {align*} \int (c+d x)^{5/2} \cos ^3(a+b x) \sin ^3(a+b x) \, dx &=\int \left (\frac {3}{32} (c+d x)^{5/2} \sin (2 a+2 b x)-\frac {1}{32} (c+d x)^{5/2} \sin (6 a+6 b x)\right ) \, dx\\ &=-\left (\frac {1}{32} \int (c+d x)^{5/2} \sin (6 a+6 b x) \, dx\right )+\frac {3}{32} \int (c+d x)^{5/2} \sin (2 a+2 b x) \, dx\\ &=-\frac {3 (c+d x)^{5/2} \cos (2 a+2 b x)}{64 b}+\frac {(c+d x)^{5/2} \cos (6 a+6 b x)}{192 b}-\frac {(5 d) \int (c+d x)^{3/2} \cos (6 a+6 b x) \, dx}{384 b}+\frac {(15 d) \int (c+d x)^{3/2} \cos (2 a+2 b x) \, dx}{128 b}\\ &=-\frac {3 (c+d x)^{5/2} \cos (2 a+2 b x)}{64 b}+\frac {(c+d x)^{5/2} \cos (6 a+6 b x)}{192 b}+\frac {15 d (c+d x)^{3/2} \sin (2 a+2 b x)}{256 b^2}-\frac {5 d (c+d x)^{3/2} \sin (6 a+6 b x)}{2304 b^2}+\frac {\left (5 d^2\right ) \int \sqrt {c+d x} \sin (6 a+6 b x) \, dx}{1536 b^2}-\frac {\left (45 d^2\right ) \int \sqrt {c+d x} \sin (2 a+2 b x) \, dx}{512 b^2}\\ &=\frac {45 d^2 \sqrt {c+d x} \cos (2 a+2 b x)}{1024 b^3}-\frac {3 (c+d x)^{5/2} \cos (2 a+2 b x)}{64 b}-\frac {5 d^2 \sqrt {c+d x} \cos (6 a+6 b x)}{9216 b^3}+\frac {(c+d x)^{5/2} \cos (6 a+6 b x)}{192 b}+\frac {15 d (c+d x)^{3/2} \sin (2 a+2 b x)}{256 b^2}-\frac {5 d (c+d x)^{3/2} \sin (6 a+6 b x)}{2304 b^2}+\frac {\left (5 d^3\right ) \int \frac {\cos (6 a+6 b x)}{\sqrt {c+d x}} \, dx}{18432 b^3}-\frac {\left (45 d^3\right ) \int \frac {\cos (2 a+2 b x)}{\sqrt {c+d x}} \, dx}{2048 b^3}\\ &=\frac {45 d^2 \sqrt {c+d x} \cos (2 a+2 b x)}{1024 b^3}-\frac {3 (c+d x)^{5/2} \cos (2 a+2 b x)}{64 b}-\frac {5 d^2 \sqrt {c+d x} \cos (6 a+6 b x)}{9216 b^3}+\frac {(c+d x)^{5/2} \cos (6 a+6 b x)}{192 b}+\frac {15 d (c+d x)^{3/2} \sin (2 a+2 b x)}{256 b^2}-\frac {5 d (c+d x)^{3/2} \sin (6 a+6 b x)}{2304 b^2}+\frac {\left (5 d^3 \cos \left (6 a-\frac {6 b c}{d}\right )\right ) \int \frac {\cos \left (\frac {6 b c}{d}+6 b x\right )}{\sqrt {c+d x}} \, dx}{18432 b^3}-\frac {\left (45 d^3 \cos \left (2 a-\frac {2 b c}{d}\right )\right ) \int \frac {\cos \left (\frac {2 b c}{d}+2 b x\right )}{\sqrt {c+d x}} \, dx}{2048 b^3}-\frac {\left (5 d^3 \sin \left (6 a-\frac {6 b c}{d}\right )\right ) \int \frac {\sin \left (\frac {6 b c}{d}+6 b x\right )}{\sqrt {c+d x}} \, dx}{18432 b^3}+\frac {\left (45 d^3 \sin \left (2 a-\frac {2 b c}{d}\right )\right ) \int \frac {\sin \left (\frac {2 b c}{d}+2 b x\right )}{\sqrt {c+d x}} \, dx}{2048 b^3}\\ &=\frac {45 d^2 \sqrt {c+d x} \cos (2 a+2 b x)}{1024 b^3}-\frac {3 (c+d x)^{5/2} \cos (2 a+2 b x)}{64 b}-\frac {5 d^2 \sqrt {c+d x} \cos (6 a+6 b x)}{9216 b^3}+\frac {(c+d x)^{5/2} \cos (6 a+6 b x)}{192 b}+\frac {15 d (c+d x)^{3/2} \sin (2 a+2 b x)}{256 b^2}-\frac {5 d (c+d x)^{3/2} \sin (6 a+6 b x)}{2304 b^2}+\frac {\left (5 d^2 \cos \left (6 a-\frac {6 b c}{d}\right )\right ) \operatorname {Subst}\left (\int \cos \left (\frac {6 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{9216 b^3}-\frac {\left (45 d^2 \cos \left (2 a-\frac {2 b c}{d}\right )\right ) \operatorname {Subst}\left (\int \cos \left (\frac {2 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{1024 b^3}-\frac {\left (5 d^2 \sin \left (6 a-\frac {6 b c}{d}\right )\right ) \operatorname {Subst}\left (\int \sin \left (\frac {6 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{9216 b^3}+\frac {\left (45 d^2 \sin \left (2 a-\frac {2 b c}{d}\right )\right ) \operatorname {Subst}\left (\int \sin \left (\frac {2 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{1024 b^3}\\ &=\frac {45 d^2 \sqrt {c+d x} \cos (2 a+2 b x)}{1024 b^3}-\frac {3 (c+d x)^{5/2} \cos (2 a+2 b x)}{64 b}-\frac {5 d^2 \sqrt {c+d x} \cos (6 a+6 b x)}{9216 b^3}+\frac {(c+d x)^{5/2} \cos (6 a+6 b x)}{192 b}+\frac {5 d^{5/2} \sqrt {\frac {\pi }{3}} \cos \left (6 a-\frac {6 b c}{d}\right ) C\left (\frac {2 \sqrt {b} \sqrt {\frac {3}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{18432 b^{7/2}}-\frac {45 d^{5/2} \sqrt {\pi } \cos \left (2 a-\frac {2 b c}{d}\right ) C\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{2048 b^{7/2}}-\frac {5 d^{5/2} \sqrt {\frac {\pi }{3}} S\left (\frac {2 \sqrt {b} \sqrt {\frac {3}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (6 a-\frac {6 b c}{d}\right )}{18432 b^{7/2}}+\frac {45 d^{5/2} \sqrt {\pi } S\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{2048 b^{7/2}}+\frac {15 d (c+d x)^{3/2} \sin (2 a+2 b x)}{256 b^2}-\frac {5 d (c+d x)^{3/2} \sin (6 a+6 b x)}{2304 b^2}\\ \end {align*}
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Mathematica [A] time = 5.07, size = 550, normalized size = 1.35 \[ \frac {-2592 b^3 c^2 \sqrt {c+d x} \cos (2 (a+b x))+288 b^3 c^2 \sqrt {c+d x} \cos (6 (a+b x))-2592 b^3 d^2 x^2 \sqrt {c+d x} \cos (2 (a+b x))+288 b^3 d^2 x^2 \sqrt {c+d x} \cos (6 (a+b x))-5184 b^3 c d x \sqrt {c+d x} \cos (2 (a+b x))+576 b^3 c d x \sqrt {c+d x} \cos (6 (a+b x))+3240 b^2 d^2 x \sqrt {c+d x} \sin (2 (a+b x))-120 b^2 d^2 x \sqrt {c+d x} \sin (6 (a+b x))+3240 b^2 c d \sqrt {c+d x} \sin (2 (a+b x))-120 b^2 c d \sqrt {c+d x} \sin (6 (a+b x))+5 \sqrt {3 \pi } d^3 \sqrt {\frac {b}{d}} \cos \left (6 a-\frac {6 b c}{d}\right ) C\left (2 \sqrt {\frac {b}{d}} \sqrt {\frac {3}{\pi }} \sqrt {c+d x}\right )-1215 \sqrt {\pi } d^3 \sqrt {\frac {b}{d}} \cos \left (2 a-\frac {2 b c}{d}\right ) C\left (\frac {2 \sqrt {\frac {b}{d}} \sqrt {c+d x}}{\sqrt {\pi }}\right )-5 \sqrt {3 \pi } d^3 \sqrt {\frac {b}{d}} \sin \left (6 a-\frac {6 b c}{d}\right ) S\left (2 \sqrt {\frac {b}{d}} \sqrt {\frac {3}{\pi }} \sqrt {c+d x}\right )+1215 \sqrt {\pi } d^3 \sqrt {\frac {b}{d}} \sin \left (2 a-\frac {2 b c}{d}\right ) S\left (\frac {2 \sqrt {\frac {b}{d}} \sqrt {c+d x}}{\sqrt {\pi }}\right )+2430 b d^2 \sqrt {c+d x} \cos (2 (a+b x))-30 b d^2 \sqrt {c+d x} \cos (6 (a+b x))}{55296 b^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 445, normalized size = 1.09 \[ \frac {5 \, \sqrt {3} \pi d^{3} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {C}\left (2 \, \sqrt {3} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 5 \, \sqrt {3} \pi d^{3} \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (2 \, \sqrt {3} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right ) - 1215 \, \pi d^{3} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {C}\left (2 \, \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) + 1215 \, \pi d^{3} \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (2 \, \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + 96 \, {\left (24 \, b^{3} d^{2} x^{2} + 2 \, {\left (48 \, b^{3} d^{2} x^{2} + 96 \, b^{3} c d x + 48 \, b^{3} c^{2} - 5 \, b d^{2}\right )} \cos \left (b x + a\right )^{6} + 48 \, b^{3} c d x + 24 \, b^{3} c^{2} + 45 \, b d^{2} \cos \left (b x + a\right )^{2} - 3 \, {\left (48 \, b^{3} d^{2} x^{2} + 96 \, b^{3} c d x + 48 \, b^{3} c^{2} - 5 \, b d^{2}\right )} \cos \left (b x + a\right )^{4} - 25 \, b d^{2} - 20 \, {\left (2 \, {\left (b^{2} d^{2} x + b^{2} c d\right )} \cos \left (b x + a\right )^{5} - 2 \, {\left (b^{2} d^{2} x + b^{2} c d\right )} \cos \left (b x + a\right )^{3} - 3 \, {\left (b^{2} d^{2} x + b^{2} c d\right )} \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )\right )} \sqrt {d x + c}}{55296 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 13.77, size = 2417, normalized size = 5.94 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 477, normalized size = 1.17 \[ \frac {-\frac {3 d \left (d x +c \right )^{\frac {5}{2}} \cos \left (\frac {2 \left (d x +c \right ) b}{d}+\frac {2 d a -2 c b}{d}\right )}{64 b}+\frac {15 d \left (\frac {d \left (d x +c \right )^{\frac {3}{2}} \sin \left (\frac {2 \left (d x +c \right ) b}{d}+\frac {2 d a -2 c b}{d}\right )}{4 b}-\frac {3 d \left (-\frac {d \sqrt {d x +c}\, \cos \left (\frac {2 \left (d x +c \right ) b}{d}+\frac {2 d a -2 c b}{d}\right )}{4 b}+\frac {d \sqrt {\pi }\, \left (\cos \left (\frac {2 d a -2 c b}{d}\right ) \FresnelC \left (\frac {2 \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {2 d a -2 c b}{d}\right ) \mathrm {S}\left (\frac {2 \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{8 b \sqrt {\frac {b}{d}}}\right )}{4 b}\right )}{64 b}+\frac {d \left (d x +c \right )^{\frac {5}{2}} \cos \left (\frac {6 \left (d x +c \right ) b}{d}+\frac {6 d a -6 c b}{d}\right )}{192 b}-\frac {5 d \left (\frac {d \left (d x +c \right )^{\frac {3}{2}} \sin \left (\frac {6 \left (d x +c \right ) b}{d}+\frac {6 d a -6 c b}{d}\right )}{12 b}-\frac {d \left (-\frac {d \sqrt {d x +c}\, \cos \left (\frac {6 \left (d x +c \right ) b}{d}+\frac {6 d a -6 c b}{d}\right )}{12 b}+\frac {d \sqrt {2}\, \sqrt {\pi }\, \sqrt {6}\, \left (\cos \left (\frac {6 d a -6 c b}{d}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {6}\, \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {6 d a -6 c b}{d}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {6}\, \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{144 b \sqrt {\frac {b}{d}}}\right )}{4 b}\right )}{192 b}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.53, size = 557, normalized size = 1.37 \[ -\frac {{\left (1920 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} \sin \left (\frac {6 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right ) - 51840 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} \sin \left (\frac {2 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right ) - 96 \, {\left (\frac {48 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{4}}{d} - 5 \, \sqrt {d x + c} b^{2} d\right )} \cos \left (\frac {6 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right ) + 2592 \, {\left (\frac {16 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{4}}{d} - 15 \, \sqrt {d x + c} b^{2} d\right )} \cos \left (\frac {2 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right ) + {\left (\left (10 i - 10\right ) \cdot 36^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right ) + \left (10 i + 10\right ) \cdot 36^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {6 i \, b}{d}}\right ) + {\left (-\left (2430 i - 2430\right ) \cdot 4^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - \left (2430 i + 2430\right ) \cdot 4^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {2 i \, b}{d}}\right ) + {\left (\left (2430 i + 2430\right ) \cdot 4^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + \left (2430 i - 2430\right ) \cdot 4^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {2 i \, b}{d}}\right ) + {\left (-\left (10 i + 10\right ) \cdot 36^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right ) - \left (10 i - 10\right ) \cdot 36^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {6 i \, b}{d}}\right )\right )} d}{884736 \, 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 \[ \int {\cos \left (a+b\,x\right )}^3\,{\sin \left (a+b\,x\right )}^3\,{\left (c+d\,x\right )}^{5/2} \,d x \]
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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