Optimal. Leaf size=194 \[ \frac {15 \sqrt {\frac {\pi }{2}} d^{5/2} \sin \left (a-\frac {b c}{d}\right ) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 b^{7/2}}+\frac {15 \sqrt {\frac {\pi }{2}} d^{5/2} \cos \left (a-\frac {b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 b^{7/2}}-\frac {15 d^2 \sqrt {c+d x} \sin (a+b x)}{4 b^3}+\frac {5 d (c+d x)^{3/2} \cos (a+b x)}{2 b^2}+\frac {(c+d x)^{5/2} \sin (a+b x)}{b} \]
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Rubi [A] time = 0.42, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3296, 3306, 3305, 3351, 3304, 3352} \[ \frac {15 \sqrt {\frac {\pi }{2}} d^{5/2} \sin \left (a-\frac {b c}{d}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 b^{7/2}}+\frac {15 \sqrt {\frac {\pi }{2}} d^{5/2} \cos \left (a-\frac {b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 b^{7/2}}-\frac {15 d^2 \sqrt {c+d x} \sin (a+b x)}{4 b^3}+\frac {5 d (c+d x)^{3/2} \cos (a+b x)}{2 b^2}+\frac {(c+d x)^{5/2} \sin (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 3296
Rule 3304
Rule 3305
Rule 3306
Rule 3351
Rule 3352
Rubi steps
\begin {align*} \int (c+d x)^{5/2} \cos (a+b x) \, dx &=\frac {(c+d x)^{5/2} \sin (a+b x)}{b}-\frac {(5 d) \int (c+d x)^{3/2} \sin (a+b x) \, dx}{2 b}\\ &=\frac {5 d (c+d x)^{3/2} \cos (a+b x)}{2 b^2}+\frac {(c+d x)^{5/2} \sin (a+b x)}{b}-\frac {\left (15 d^2\right ) \int \sqrt {c+d x} \cos (a+b x) \, dx}{4 b^2}\\ &=\frac {5 d (c+d x)^{3/2} \cos (a+b x)}{2 b^2}-\frac {15 d^2 \sqrt {c+d x} \sin (a+b x)}{4 b^3}+\frac {(c+d x)^{5/2} \sin (a+b x)}{b}+\frac {\left (15 d^3\right ) \int \frac {\sin (a+b x)}{\sqrt {c+d x}} \, dx}{8 b^3}\\ &=\frac {5 d (c+d x)^{3/2} \cos (a+b x)}{2 b^2}-\frac {15 d^2 \sqrt {c+d x} \sin (a+b x)}{4 b^3}+\frac {(c+d x)^{5/2} \sin (a+b x)}{b}+\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}{8 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}{8 b^3}\\ &=\frac {5 d (c+d x)^{3/2} \cos (a+b x)}{2 b^2}-\frac {15 d^2 \sqrt {c+d x} \sin (a+b x)}{4 b^3}+\frac {(c+d x)^{5/2} \sin (a+b x)}{b}+\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 )}{4 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 )}{4 b^3}\\ &=\frac {5 d (c+d x)^{3/2} \cos (a+b x)}{2 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 )}{4 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 )}{4 b^{7/2}}-\frac {15 d^2 \sqrt {c+d x} \sin (a+b x)}{4 b^3}+\frac {(c+d x)^{5/2} \sin (a+b x)}{b}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 124, normalized size = 0.64 \[ -\frac {d^3 e^{-\frac {i (a d+b c)}{d}} \left (e^{2 i a} \sqrt {-\frac {i b (c+d x)}{d}} \Gamma \left (\frac {7}{2},-\frac {i b (c+d x)}{d}\right )+e^{\frac {2 i b c}{d}} \sqrt {\frac {i b (c+d x)}{d}} \Gamma \left (\frac {7}{2},\frac {i b (c+d x)}{d}\right )\right )}{2 b^4 \sqrt {c+d x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.07, size = 190, normalized size = 0.98 \[ \frac {15 \, \sqrt {2} \pi d^{3} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {S}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) + 15 \, \sqrt {2} \pi d^{3} \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {b c - a d}{d}\right ) + 2 \, \sqrt {d x + c} {\left (10 \, {\left (b^{2} d^{2} x + b^{2} c d\right )} \cos \left (b x + a\right ) + {\left (4 \, b^{3} d^{2} x^{2} + 8 \, b^{3} c d x + 4 \, b^{3} c^{2} - 15 \, b d^{2}\right )} \sin \left (b x + a\right )\right )}}{8 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.91, size = 1239, normalized size = 6.39 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 232, normalized size = 1.20 \[ \frac {\frac {d \left (d x +c \right )^{\frac {5}{2}} \sin \left (\frac {\left (d x +c \right ) b}{d}+\frac {d a -c b}{d}\right )}{b}-\frac {5 d \left (-\frac {d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {\left (d x +c \right ) b}{d}+\frac {d a -c b}{d}\right )}{2 b}+\frac {3 d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {\left (d x +c \right ) b}{d}+\frac {d a -c b}{d}\right )}{2 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {d a -c b}{d}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {d a -c b}{d}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{4 b \sqrt {\frac {b}{d}}}\right )}{2 b}\right )}{b}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.94, size = 261, normalized size = 1.35 \[ \frac {\sqrt {2} {\left (40 \, \sqrt {2} {\left (d x + c\right )}^{\frac {3}{2}} b^{2} d \cos \left (\frac {{\left (d x + c\right )} b - b c + a d}{d}\right ) + {\left (\left (15 i + 15\right ) \, \sqrt {\pi } d^{3} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {b c - a d}{d}\right ) - \left (15 i - 15\right ) \, \sqrt {\pi } d^{3} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {b c - a d}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {i \, b}{d}}\right ) + {\left (-\left (15 i - 15\right ) \, \sqrt {\pi } d^{3} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {b c - a d}{d}\right ) + \left (15 i + 15\right ) \, \sqrt {\pi } d^{3} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {b c - a d}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {i \, b}{d}}\right ) + 4 \, {\left (4 \, \sqrt {2} {\left (d x + c\right )}^{\frac {5}{2}} b^{3} - 15 \, \sqrt {2} \sqrt {d x + c} b d^{2}\right )} \sin \left (\frac {{\left (d x + c\right )} b - b c + a d}{d}\right )\right )}}{32 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \cos \left (a+b\,x\right )\,{\left (c+d\,x\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c + d x\right )^{\frac {5}{2}} \cos {\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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