Optimal. Leaf size=195 \[ \frac {15 d^2 \sqrt {c+d x} \cos (a+b x)}{4 b^3}-\frac {(c+d x)^{5/2} \cos (a+b x)}{b}-\frac {15 d^{5/2} \sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b c}{d}\right ) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 b^{7/2}}+\frac {15 d^{5/2} \sqrt {\frac {\pi }{2}} S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (a-\frac {b c}{d}\right )}{4 b^{7/2}}+\frac {5 d (c+d x)^{3/2} \sin (a+b x)}{2 b^2} \]
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Rubi [A]
time = 0.31, antiderivative size = 195, 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 = {3377, 3387,
3386, 3432, 3385, 3433} \begin {gather*} -\frac {15 \sqrt {\frac {\pi }{2}} d^{5/2} \cos \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} \sin \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} \cos (a+b x)}{4 b^3}+\frac {5 d (c+d x)^{3/2} \sin (a+b x)}{2 b^2}-\frac {(c+d x)^{5/2} \cos (a+b x)}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 3377
Rule 3385
Rule 3386
Rule 3387
Rule 3432
Rule 3433
Rubi steps
\begin {align*} \int (c+d x)^{5/2} \sin (a+b x) \, dx &=-\frac {(c+d x)^{5/2} \cos (a+b x)}{b}+\frac {(5 d) \int (c+d x)^{3/2} \cos (a+b x) \, dx}{2 b}\\ &=-\frac {(c+d x)^{5/2} \cos (a+b x)}{b}+\frac {5 d (c+d x)^{3/2} \sin (a+b x)}{2 b^2}-\frac {\left (15 d^2\right ) \int \sqrt {c+d x} \sin (a+b x) \, dx}{4 b^2}\\ &=\frac {15 d^2 \sqrt {c+d x} \cos (a+b x)}{4 b^3}-\frac {(c+d x)^{5/2} \cos (a+b x)}{b}+\frac {5 d (c+d x)^{3/2} \sin (a+b x)}{2 b^2}-\frac {\left (15 d^3\right ) \int \frac {\cos (a+b x)}{\sqrt {c+d x}} \, dx}{8 b^3}\\ &=\frac {15 d^2 \sqrt {c+d x} \cos (a+b x)}{4 b^3}-\frac {(c+d x)^{5/2} \cos (a+b x)}{b}+\frac {5 d (c+d x)^{3/2} \sin (a+b x)}{2 b^2}-\frac {\left (15 d^3 \cos \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 {\left (15 d^3 \sin \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 {15 d^2 \sqrt {c+d x} \cos (a+b x)}{4 b^3}-\frac {(c+d x)^{5/2} \cos (a+b x)}{b}+\frac {5 d (c+d x)^{3/2} \sin (a+b x)}{2 b^2}-\frac {\left (15 d^2 \cos \left (a-\frac {b c}{d}\right )\right ) \text {Subst}\left (\int \cos \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 ) \text {Subst}\left (\int \sin \left (\frac {b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{4 b^3}\\ &=\frac {15 d^2 \sqrt {c+d x} \cos (a+b x)}{4 b^3}-\frac {(c+d x)^{5/2} \cos (a+b x)}{b}-\frac {15 d^{5/2} \sqrt {\frac {\pi }{2}} \cos \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 d^{5/2} \sqrt {\frac {\pi }{2}} S\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 {5 d (c+d x)^{3/2} \sin (a+b x)}{2 b^2}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.08, size = 124, normalized size = 0.64 \begin {gather*} \frac {d^2 e^{-\frac {i (b c+a d)}{d}} \sqrt {c+d x} \left (\frac {e^{2 i a} \Gamma \left (\frac {7}{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 {7}{2},\frac {i b (c+d x)}{d}\right )}{\sqrt {\frac {i b (c+d x)}{d}}}\right )}{2 b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 233, normalized size = 1.19
method | result | size |
derivativedivides | \(\frac {-\frac {d \left (d x +c \right )^{\frac {5}{2}} \cos \left (\frac {b \left (d x +c \right )}{d}+\frac {d a -c b}{d}\right )}{b}+\frac {5 d \left (\frac {d \left (d x +c \right )^{\frac {3}{2}} \sin \left (\frac {b \left (d x +c \right )}{d}+\frac {d a -c b}{d}\right )}{2 b}-\frac {3 d \left (-\frac {d \sqrt {d x +c}\, \cos \left (\frac {b \left (d x +c \right )}{d}+\frac {d a -c b}{d}\right )}{2 b}+\frac {d \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {d a -c b}{d}\right ) \FresnelC \left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {d a -c b}{d}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{4 b \sqrt {\frac {b}{d}}}\right )}{2 b}\right )}{b}}{d}\) | \(233\) |
default | \(\frac {-\frac {d \left (d x +c \right )^{\frac {5}{2}} \cos \left (\frac {b \left (d x +c \right )}{d}+\frac {d a -c b}{d}\right )}{b}+\frac {5 d \left (\frac {d \left (d x +c \right )^{\frac {3}{2}} \sin \left (\frac {b \left (d x +c \right )}{d}+\frac {d a -c b}{d}\right )}{2 b}-\frac {3 d \left (-\frac {d \sqrt {d x +c}\, \cos \left (\frac {b \left (d x +c \right )}{d}+\frac {d a -c b}{d}\right )}{2 b}+\frac {d \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {d a -c b}{d}\right ) \FresnelC \left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {d a -c b}{d}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{4 b \sqrt {\frac {b}{d}}}\right )}{2 b}\right )}{b}}{d}\) | \(233\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.31, size = 263, normalized size = 1.35 \begin {gather*} \frac {\sqrt {2} {\left (40 \, \sqrt {2} {\left (d x + c\right )}^{\frac {3}{2}} b^{2} d \sin \left (\frac {{\left (d x + c\right )} b - b c + a d}{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 )} \cos \left (\frac {{\left (d x + c\right )} b - b c + a d}{d}\right ) - 15 \, {\left (-\left (i - 1\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 (i + 1\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 ) - 15 \, {\left (\left (i + 1\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 (i - 1\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 )\right )}}{32 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 190, normalized size = 0.97 \begin {gather*} -\frac {15 \, \sqrt {2} \pi d^{3} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {C}\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 {S}\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 ({\left (4 \, b^{3} d^{2} x^{2} + 8 \, b^{3} c d x + 4 \, b^{3} c^{2} - 15 \, b d^{2}\right )} \cos \left (b x + a\right ) - 10 \, {\left (b^{2} d^{2} x + b^{2} c d\right )} \sin \left (b x + a\right )\right )}}{8 \, b^{4}} \end {gather*}
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 \begin {gather*} \int \left (c + d x\right )^{\frac {5}{2}} \sin {\left (a + b x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 3.14, size = 1246, normalized size = 6.39 \begin {gather*} -\frac {8 \, {\left (\frac {i \, \sqrt {2} \sqrt {\pi } d \operatorname {erf}\left (-\frac {\sqrt {2} \sqrt {b d} \sqrt {d x + c} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{2 \, d}\right ) e^{\left (\frac {i \, b c - i \, a d}{d}\right )}}{\sqrt {b d} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}} - \frac {i \, \sqrt {2} \sqrt {\pi } d \operatorname {erf}\left (-\frac {\sqrt {2} \sqrt {b d} \sqrt {d x + c} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{2 \, d}\right ) e^{\left (\frac {-i \, b c + i \, a d}{d}\right )}}{\sqrt {b d} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}\right )} c^{3} + 6 \, c d^{2} {\left (\frac {\frac {i \, \sqrt {2} \sqrt {\pi } {\left (4 \, b^{2} c^{2} + 4 i \, b c d - 3 \, d^{2}\right )} d \operatorname {erf}\left (-\frac {\sqrt {2} \sqrt {b d} \sqrt {d x + c} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{2 \, d}\right ) e^{\left (\frac {i \, b c - i \, a d}{d}\right )}}{\sqrt {b d} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )} b^{2}} - \frac {2 i \, {\left (2 i \, {\left (d x + c\right )}^{\frac {3}{2}} b d - 4 i \, \sqrt {d x + c} b c d + 3 \, \sqrt {d x + c} d^{2}\right )} e^{\left (\frac {-i \, {\left (d x + c\right )} b + i \, b c - i \, a d}{d}\right )}}{b^{2}}}{d^{2}} + \frac {-\frac {i \, \sqrt {2} \sqrt {\pi } {\left (4 \, b^{2} c^{2} - 4 i \, b c d - 3 \, d^{2}\right )} d \operatorname {erf}\left (-\frac {\sqrt {2} \sqrt {b d} \sqrt {d x + c} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{2 \, d}\right ) e^{\left (\frac {-i \, b c + i \, a d}{d}\right )}}{\sqrt {b d} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )} b^{2}} - \frac {2 i \, {\left (2 i \, {\left (d x + c\right )}^{\frac {3}{2}} b d - 4 i \, \sqrt {d x + c} b c d - 3 \, \sqrt {d x + c} d^{2}\right )} e^{\left (\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{d}\right )}}{b^{2}}}{d^{2}}\right )} + d^{3} {\left (\frac {-\frac {i \, \sqrt {2} \sqrt {\pi } {\left (8 \, b^{3} c^{3} + 12 i \, b^{2} c^{2} d - 18 \, b c d^{2} - 15 i \, d^{3}\right )} d \operatorname {erf}\left (-\frac {\sqrt {2} \sqrt {b d} \sqrt {d x + c} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{2 \, d}\right ) e^{\left (\frac {i \, b c - i \, a d}{d}\right )}}{\sqrt {b d} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )} b^{3}} - \frac {2 i \, {\left (4 i \, {\left (d x + c\right )}^{\frac {5}{2}} b^{2} d - 12 i \, {\left (d x + c\right )}^{\frac {3}{2}} b^{2} c d + 12 i \, \sqrt {d x + c} b^{2} c^{2} d + 10 \, {\left (d x + c\right )}^{\frac {3}{2}} b d^{2} - 18 \, \sqrt {d x + c} b c d^{2} - 15 i \, \sqrt {d x + c} d^{3}\right )} e^{\left (\frac {-i \, {\left (d x + c\right )} b + i \, b c - i \, a d}{d}\right )}}{b^{3}}}{d^{3}} + \frac {\frac {i \, \sqrt {2} \sqrt {\pi } {\left (8 \, b^{3} c^{3} - 12 i \, b^{2} c^{2} d - 18 \, b c d^{2} + 15 i \, d^{3}\right )} d \operatorname {erf}\left (-\frac {\sqrt {2} \sqrt {b d} \sqrt {d x + c} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{2 \, d}\right ) e^{\left (\frac {-i \, b c + i \, a d}{d}\right )}}{\sqrt {b d} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )} b^{3}} - \frac {2 i \, {\left (4 i \, {\left (d x + c\right )}^{\frac {5}{2}} b^{2} d - 12 i \, {\left (d x + c\right )}^{\frac {3}{2}} b^{2} c d + 12 i \, \sqrt {d x + c} b^{2} c^{2} d - 10 \, {\left (d x + c\right )}^{\frac {3}{2}} b d^{2} + 18 \, \sqrt {d x + c} b c d^{2} - 15 i \, \sqrt {d x + c} d^{3}\right )} e^{\left (\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{d}\right )}}{b^{3}}}{d^{3}}\right )} + 12 \, {\left (-\frac {i \, \sqrt {2} \sqrt {\pi } {\left (2 \, b c + i \, d\right )} d \operatorname {erf}\left (-\frac {\sqrt {2} \sqrt {b d} \sqrt {d x + c} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{2 \, d}\right ) e^{\left (\frac {i \, b c - i \, a d}{d}\right )}}{\sqrt {b d} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )} b} + \frac {i \, \sqrt {2} \sqrt {\pi } {\left (2 \, b c - i \, d\right )} d \operatorname {erf}\left (-\frac {\sqrt {2} \sqrt {b d} \sqrt {d x + c} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{2 \, d}\right ) e^{\left (\frac {-i \, b c + i \, a d}{d}\right )}}{\sqrt {b d} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )} b} + \frac {2 \, \sqrt {d x + c} d e^{\left (\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{d}\right )}}{b} + \frac {2 \, \sqrt {d x + c} d e^{\left (\frac {-i \, {\left (d x + c\right )} b + i \, b c - i \, a d}{d}\right )}}{b}\right )} c^{2}}{16 \, d} \end {gather*}
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 \begin {gather*} \int \sin \left (a+b\,x\right )\,{\left (c+d\,x\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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