Optimal. Leaf size=406 \[ -\frac {5 \sqrt {\frac {\pi }{6}} d^{5/2} \sin \left (3 a-\frac {3 b c}{d}\right ) C\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{144 b^{7/2}}+\frac {15 \sqrt {\frac {\pi }{2}} d^{5/2} \sin \left (a-\frac {b c}{d}\right ) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{7/2}}+\frac {15 \sqrt {\frac {\pi }{2}} d^{5/2} \cos \left (a-\frac {b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{7/2}}-\frac {5 \sqrt {\frac {\pi }{6}} d^{5/2} \cos \left (3 a-\frac {3 b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{144 b^{7/2}}-\frac {15 d^2 \sqrt {c+d x} \sin (a+b x)}{16 b^3}+\frac {5 d^2 \sqrt {c+d x} \sin (3 a+3 b x)}{144 b^3}+\frac {5 d (c+d x)^{3/2} \cos (a+b x)}{8 b^2}-\frac {5 d (c+d x)^{3/2} \cos (3 a+3 b x)}{72 b^2}+\frac {(c+d x)^{5/2} \sin (a+b x)}{4 b}-\frac {(c+d x)^{5/2} \sin (3 a+3 b x)}{12 b} \]
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Rubi [A] time = 0.67, antiderivative size = 406, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {4406, 3296, 3306, 3305, 3351, 3304, 3352} \[ -\frac {5 \sqrt {\frac {\pi }{6}} d^{5/2} \sin \left (3 a-\frac {3 b c}{d}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{144 b^{7/2}}+\frac {15 \sqrt {\frac {\pi }{2}} d^{5/2} \sin \left (a-\frac {b c}{d}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{7/2}}+\frac {15 \sqrt {\frac {\pi }{2}} d^{5/2} \cos \left (a-\frac {b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{7/2}}-\frac {5 \sqrt {\frac {\pi }{6}} d^{5/2} \cos \left (3 a-\frac {3 b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{144 b^{7/2}}-\frac {15 d^2 \sqrt {c+d x} \sin (a+b x)}{16 b^3}+\frac {5 d^2 \sqrt {c+d x} \sin (3 a+3 b x)}{144 b^3}+\frac {5 d (c+d x)^{3/2} \cos (a+b x)}{8 b^2}-\frac {5 d (c+d x)^{3/2} \cos (3 a+3 b x)}{72 b^2}+\frac {(c+d x)^{5/2} \sin (a+b x)}{4 b}-\frac {(c+d x)^{5/2} \sin (3 a+3 b x)}{12 b} \]
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 (a+b x) \sin ^2(a+b x) \, dx &=\int \left (\frac {1}{4} (c+d x)^{5/2} \cos (a+b x)-\frac {1}{4} (c+d x)^{5/2} \cos (3 a+3 b x)\right ) \, dx\\ &=\frac {1}{4} \int (c+d x)^{5/2} \cos (a+b x) \, dx-\frac {1}{4} \int (c+d x)^{5/2} \cos (3 a+3 b x) \, dx\\ &=\frac {(c+d x)^{5/2} \sin (a+b x)}{4 b}-\frac {(c+d x)^{5/2} \sin (3 a+3 b x)}{12 b}+\frac {(5 d) \int (c+d x)^{3/2} \sin (3 a+3 b x) \, dx}{24 b}-\frac {(5 d) \int (c+d x)^{3/2} \sin (a+b x) \, dx}{8 b}\\ &=\frac {5 d (c+d x)^{3/2} \cos (a+b x)}{8 b^2}-\frac {5 d (c+d x)^{3/2} \cos (3 a+3 b x)}{72 b^2}+\frac {(c+d x)^{5/2} \sin (a+b x)}{4 b}-\frac {(c+d x)^{5/2} \sin (3 a+3 b x)}{12 b}+\frac {\left (5 d^2\right ) \int \sqrt {c+d x} \cos (3 a+3 b x) \, dx}{48 b^2}-\frac {\left (15 d^2\right ) \int \sqrt {c+d x} \cos (a+b x) \, dx}{16 b^2}\\ &=\frac {5 d (c+d x)^{3/2} \cos (a+b x)}{8 b^2}-\frac {5 d (c+d x)^{3/2} \cos (3 a+3 b x)}{72 b^2}-\frac {15 d^2 \sqrt {c+d x} \sin (a+b x)}{16 b^3}+\frac {(c+d x)^{5/2} \sin (a+b x)}{4 b}+\frac {5 d^2 \sqrt {c+d x} \sin (3 a+3 b x)}{144 b^3}-\frac {(c+d x)^{5/2} \sin (3 a+3 b x)}{12 b}-\frac {\left (5 d^3\right ) \int \frac {\sin (3 a+3 b x)}{\sqrt {c+d x}} \, dx}{288 b^3}+\frac {\left (15 d^3\right ) \int \frac {\sin (a+b x)}{\sqrt {c+d x}} \, dx}{32 b^3}\\ &=\frac {5 d (c+d x)^{3/2} \cos (a+b x)}{8 b^2}-\frac {5 d (c+d x)^{3/2} \cos (3 a+3 b x)}{72 b^2}-\frac {15 d^2 \sqrt {c+d x} \sin (a+b x)}{16 b^3}+\frac {(c+d x)^{5/2} \sin (a+b x)}{4 b}+\frac {5 d^2 \sqrt {c+d x} \sin (3 a+3 b x)}{144 b^3}-\frac {(c+d x)^{5/2} \sin (3 a+3 b x)}{12 b}-\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 (15 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 \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 (15 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 {5 d (c+d x)^{3/2} \cos (a+b x)}{8 b^2}-\frac {5 d (c+d x)^{3/2} \cos (3 a+3 b x)}{72 b^2}-\frac {15 d^2 \sqrt {c+d x} \sin (a+b x)}{16 b^3}+\frac {(c+d x)^{5/2} \sin (a+b x)}{4 b}+\frac {5 d^2 \sqrt {c+d x} \sin (3 a+3 b x)}{144 b^3}-\frac {(c+d x)^{5/2} \sin (3 a+3 b x)}{12 b}-\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 (15 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 \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 (15 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 {5 d (c+d x)^{3/2} \cos (a+b x)}{8 b^2}-\frac {5 d (c+d x)^{3/2} \cos (3 a+3 b x)}{72 b^2}+\frac {15 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 {15 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 {15 d^2 \sqrt {c+d x} \sin (a+b x)}{16 b^3}+\frac {(c+d x)^{5/2} \sin (a+b x)}{4 b}+\frac {5 d^2 \sqrt {c+d x} \sin (3 a+3 b x)}{144 b^3}-\frac {(c+d x)^{5/2} \sin (3 a+3 b x)}{12 b}\\ \end {align*}
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Mathematica [C] time = 13.63, size = 1171, normalized size = 2.88 \[ -\frac {i e^{-\frac {i (b c+a d)}{d}} \sqrt {c+d x} \left (\frac {e^{2 i a} \Gamma \left (\frac {3}{2},-\frac {i b (c+d x)}{d}\right )}{\sqrt {-\frac {i b (c+d x)}{d}}}-\frac {e^{\frac {2 i b c}{d}} \Gamma \left (\frac {3}{2},\frac {i b (c+d x)}{d}\right )}{\sqrt {\frac {i b (c+d x)}{d}}}\right ) c^2}{8 b}-\frac {\left (-\sqrt {2 \pi } \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 )-\sqrt {2 \pi } C\left (\sqrt {\frac {b}{d}} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}\right ) \sin \left (3 a-\frac {3 b c}{d}\right )+2 \sqrt {3} \sqrt {\frac {b}{d}} \sqrt {c+d x} \sin (3 (a+b x))\right ) c^2}{24 \sqrt {3} b \sqrt {\frac {b}{d}}}+\frac {d \left (\sqrt {\frac {b}{d}} \sqrt {2 \pi } C\left (\sqrt {\frac {b}{d}} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}\right ) \left (2 b c \sin \left (a-\frac {b c}{d}\right )-3 d \cos \left (a-\frac {b c}{d}\right )\right )+\sqrt {\frac {b}{d}} \sqrt {2 \pi } S\left (\sqrt {\frac {b}{d}} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}\right ) \left (2 b c \cos \left (a-\frac {b c}{d}\right )+3 d \sin \left (a-\frac {b c}{d}\right )\right )+2 b \sqrt {c+d x} (3 \cos (a+b x)+2 b x \sin (a+b x))\right ) c}{8 b^3}-\frac {d \left (\sqrt {\frac {b}{d}} \sqrt {2 \pi } C\left (\sqrt {\frac {b}{d}} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}\right ) \left (2 b c \sin \left (3 a-\frac {3 b c}{d}\right )-d \cos \left (3 a-\frac {3 b c}{d}\right )\right )+\sqrt {\frac {b}{d}} \sqrt {2 \pi } S\left (\sqrt {\frac {b}{d}} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}\right ) \left (2 b c \cos \left (3 a-\frac {3 b c}{d}\right )+d \sin \left (3 a-\frac {3 b c}{d}\right )\right )+2 \sqrt {3} b \sqrt {c+d x} (\cos (3 (a+b x))+2 b x \sin (3 (a+b x)))\right ) c}{24 \sqrt {3} b^3}+\frac {\left (\frac {b}{d}\right )^{3/2} d^2 \left (-\sqrt {2 \pi } S\left (\sqrt {\frac {b}{d}} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}\right ) \left (\left (4 b^2 c^2-15 d^2\right ) \cos \left (a-\frac {b c}{d}\right )+12 b c d \sin \left (a-\frac {b c}{d}\right )\right )-\sqrt {2 \pi } C\left (\sqrt {\frac {b}{d}} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}\right ) \left (\left (4 b^2 c^2-15 d^2\right ) \sin \left (a-\frac {b c}{d}\right )-12 b c d \cos \left (a-\frac {b c}{d}\right )\right )+2 \sqrt {\frac {b}{d}} d \sqrt {c+d x} \left (d \left (4 b^2 x^2-15\right ) \sin (a+b x)-2 b (c-5 d x) \cos (a+b x)\right )\right )}{32 b^5}-\frac {\left (\frac {b}{d}\right )^{3/2} d^2 \left (-\sqrt {2 \pi } S\left (\sqrt {\frac {b}{d}} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}\right ) \left (\left (12 b^2 c^2-5 d^2\right ) \cos \left (3 a-\frac {3 b c}{d}\right )+12 b c d \sin \left (3 a-\frac {3 b c}{d}\right )\right )-\sqrt {2 \pi } C\left (\sqrt {\frac {b}{d}} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}\right ) \left (\left (12 b^2 c^2-5 d^2\right ) \sin \left (3 a-\frac {3 b c}{d}\right )-12 b c d \cos \left (3 a-\frac {3 b c}{d}\right )\right )+2 \sqrt {3} \sqrt {\frac {b}{d}} d \sqrt {c+d x} \left (d \left (12 b^2 x^2-5\right ) \sin (3 (a+b x))-2 b (c-5 d x) \cos (3 (a+b x))\right )\right )}{288 \sqrt {3} b^5} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.69, size = 370, normalized size = 0.91 \[ -\frac {5 \, \sqrt {6} \pi d^{3} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {S}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 405 \, \sqrt {2} \pi d^{3} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {S}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 405 \, \sqrt {2} \pi d^{3} \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {b c - a d}{d}\right ) + 5 \, \sqrt {6} \pi d^{3} \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + 24 \, {\left (10 \, {\left (b^{2} d^{2} x + b^{2} c d\right )} \cos \left (b x + a\right )^{3} - 30 \, {\left (b^{2} d^{2} x + b^{2} c d\right )} \cos \left (b x + a\right ) - {\left (12 \, b^{3} d^{2} x^{2} + 24 \, b^{3} c d x + 12 \, b^{3} c^{2} - 35 \, b d^{2} - {\left (12 \, b^{3} d^{2} x^{2} + 24 \, b^{3} c d x + 12 \, b^{3} c^{2} - 5 \, b d^{2}\right )} \cos \left (b x + a\right )^{2}\right )} \sin \left (b x + a\right )\right )} \sqrt {d x + c}}{864 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 2.46, size = 2453, normalized size = 6.04 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 474, normalized size = 1.17 \[ \frac {\frac {d \left (d x +c \right )^{\frac {5}{2}} \sin \left (\frac {\left (d x +c \right ) b}{d}+\frac {d a -c b}{d}\right )}{4 b}-\frac {5 d \left (-\frac {d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {\left (d x +c \right ) b}{d}+\frac {d a -c b}{d}\right )}{2 b}+\frac {3 d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {\left (d x +c \right ) b}{d}+\frac {d a -c b}{d}\right )}{2 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {d a -c b}{d}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {d a -c b}{d}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{4 b \sqrt {\frac {b}{d}}}\right )}{2 b}\right )}{4 b}-\frac {d \left (d x +c \right )^{\frac {5}{2}} \sin \left (\frac {3 \left (d x +c \right ) b}{d}+\frac {3 d a -3 c b}{d}\right )}{12 b}+\frac {5 d \left (-\frac {d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {3 \left (d x +c \right ) b}{d}+\frac {3 d a -3 c b}{d}\right )}{6 b}+\frac {d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {3 \left (d x +c \right ) b}{d}+\frac {3 d a -3 c b}{d}\right )}{6 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \left (\cos \left (\frac {3 d a -3 c b}{d}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {3 d a -3 c b}{d}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{36 b \sqrt {\frac {b}{d}}}\right )}{2 b}\right )}{12 b}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.60, size = 543, normalized size = 1.34 \[ -\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 ) - 2160 \, {\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 (405 i + 405\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 (405 i - 405\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 (405 i - 405\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 (405 i + 405\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 ) - 216 \, {\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 \[ \int \cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^{5/2} \,d x \]
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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